Slim Tayachi

Fred B Weissler

NEW LIFE-SPAN RESULTS FOR THE NONLINEAR HEAT EQUATION

Keywords: 2010 Mathematics Subject Classification. Primary 35K55; Secondary 35B30, 35B44 Nonlinear heat equation, Hardy-Hénon parabolic equations, Local well-posedness, Blowup, Life-span

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ∂tu -∆u = |u| α u, α > 0 with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of life-span are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for life-span using dilation methods and we give new life-span estimates for Hardy-Hénon parabolic equations.

Introduction and statement of the results

In this paper, we consider the nonlinear heat equation

∂ t u = ∆u + |u| α u, (1.1)
where u = u(t, x) ∈ R, t > 0, x ∈ Ω, a domain of R N not necessarily bounded, N ≥ 1 and α > 0.

In the case where the boundary ∂Ω ̸ = ∅, we suppose ∂Ω sufficiently smooth and we impose Dirichlet conditions on the boundary: u(t, x) = 0, t > 0, x ∈ ∂Ω.

If Ω is not bounded, we impose Dirichlet conditions at infinity: lim |x|→∞, x∈Ω u(t, x) = 0, t > 0, or perhaps other convenient formulation (see for example [START_REF] Quittner | Superlinear parabolic problems. Blow-up, global existence and steady states[END_REF]Definition 15.1,p. 75]). We usually consider the equation (1.1) with the initial value u(0, •) = u 0 .

(1.

2)

The Cauchy problem (1.1)-(1.2) is locally well-posed in various Banach spaces. In other words, each element or initial value u 0 in that space gives rise to a trajectory u(t) = u(t, •) which is a solution in some appropriate sense to the given equation, here equation (1.1), and such that u(0) = u 0 . In many cases, this trajectory cannot exist for all time t, and we denote by T max (u 0 ) the maximal possible existence time of such a trajectory. The term life-span refers to the study of the maximal existence time of solutions with initial data of the form u 0 = λφ for some fixed element φ in the considered Banach space and all λ > 0. Our aim is to establish lower and upper bounds of the life-span for a large class of initial data φ in terms of λ and study the asymptotic behavior of T max (λφ), either as λ → ∞ or as λ → 0.

It is well known that if u 0 ∈ C b (R N ), the Banach space of continuous bounded functions on R N , there exists T max (u 0 ) > 0 such that (1.1)-(1.2) has a unique classical solution u ∈ C 1,2 ((0, T max (u 0 ))× R N )∩C([0, T max (u 0 ))×R N ) which is bounded in [0, T ]×R N for all T < T max (u 0 ), and ∥u(t)∥ L ∞ (R N ) → ∞ as t → T max (u 0 ), if T max (u 0 ) < ∞. It is proved in [15] that if α < 2/N and φ ∈ C b (R N ) with φ ≥ 0, φ ̸ ≡ 0, then T max (λφ) < ∞ for any λ > 0. For all α > 0, if φ ∈ C b (R N ), φ ≥ 0 and lim inf |x|→∞ |x| γ φ(x) > 0 with γ < 2/α, then T max (λφ) < ∞, for all λ > 0, as shown in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]. This last result has been improved in many papers, see [50] for instance and some references therein. If we do not impose the positivity of the initial data, it has been proven in [START_REF] Levine | Some nonexistence and instability theorems for formally parabolic equations of the form P ut = -Au + f (u)[END_REF] that for a given φ sufficiently regular (i.e. with finite energy), φ ̸ ≡ 0, and λ > 0 is sufficiently large then T max (λφ) < ∞.

If α < 2/N and φ not necessarily positive but φ ∈ L 1 (R N ) ∩ C 0 (R N ) and R N φ ̸ = 0 then it is proved, in [START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF], that T max (λφ) < ∞ for λ > 0 sufficiently small. Other blow-up results for λ small are proved in [16,[START_REF] Tayachi | The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values[END_REF]50]. The above mentioned results show in particular the interest of studying the behavior of T max (λφ) for any value of λ and with or without any sign restriction on the initial data.

For example, it is proved in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF] that given any nontrivial nonnegative initial data φ ∈ C b (R N ) then T max (λφ) ∼ λ -α , as λ → ∞ and T max (λφ) ∼ λ -α , as λ → 0 provided that φ ∞ = lim inf |x|→∞ φ(x) > 0. Shortly thereafter the exact limits were given in [START_REF] Gui | Life spans of solutions of the Cauchy problem for a semilinear heat equation[END_REF], that is lim λ→∞ λ α T max (λφ) = 1 α ∥φ∥ -α ∞ and lim λ→0 λ α T max (λφ) = 1 α φ -α ∞ . An other example is the study of the asymptotic behavior of the life-span T max (λφ) when φ ∈ C b (R N ) is nonnegative nontrivial and having also a polynomial decay at infinity, that is

0 < lim inf |x|→∞ |x| γ φ(x) ≤ lim sup |x|→∞ |x| γ φ(x) < ∞, (1.3)
0 < γ < N . This is studied in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF] for small λ > 0. It is shown in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]Theorem 3.15 (ii), p. 375] and [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]Theorem 3.21 (ii), p. 376] that if α < 2/N then

0 < lim inf λ→0 λ [( 1 α -γ 2 ) -1 ] T max (λφ) ≤ lim sup λ→0 λ [( 1 α -γ 2 ) -1 ] T max (λφ) < ∞.
These results have been generalized recently in [START_REF] Hisa | Existence of solutions for a fractional semilinear parabolic equation with singular initial data[END_REF] replacing φ ∈ C b (R N ) by φ ∈ L ∞ (R N ). See [START_REF] Hisa | Existence of solutions for a fractional semilinear parabolic equation with singular initial data[END_REF]Theorem 5.1,p. 128] and [START_REF] Hisa | Existence of solutions for a fractional semilinear parabolic equation with singular initial data[END_REF]Theorem 5.2,p. 130]. We notice that refined asymptotic is given in [33] for large λ and for φ not necessarily positive but still continuous and bounded. Other estimates are obtained for the life-span for regular and slowly decaying initial data in [START_REF] Mizoguchi | Blowup and life span of solutions for a semilinear parabolic equation[END_REF][START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF]50].

A class of initial data φ where φ is not necessarily in C b (R N ) or in L ∞ (R N ) and satisfying either (1.3) or

0 < lim inf |x|→0 |x| γ φ(x) ≤ lim sup |x|→0 |x| γ φ(x) < ∞, (1.4)
has been considered in [50]. In fact, the asymptotic behavior of the life-span for initial data φ ∈ L 1 loc (R N \ {0}), |φ(x)| ≤ c|x| -γ , where 0 < γ < N and c > 0 a constant, is studied in [50]. It is shown there that for some initial data φ singular at the origin or satisfying lim inf |x|→∞ φ(x) := φ ∞ = 0 the situation is quite different from previously known life-span results. In particular, unlike [START_REF] Gui | Life spans of solutions of the Cauchy problem for a semilinear heat equation[END_REF], the limits of λ [( 1 α -γ 2 ) -1 ] T max (λφ) as λ → 0 or as λ → ∞ may not exist. Also, if φ satisfies (1.4), so that φ is singular at the origin, then T max (λφ

) ∼ λ -( 1 α -γ 2 ) -1
, as λ → ∞ instead of λ -α if φ is regular. See [50, Corollary 1.13 and Proposition 4.6]. It is also proved that if φ(x) = |x| -γ , |x| ≤ 1, φ(x) = 0, |x| > 1, 0 < γ < N, 0 < α < 2/γ, (N -2)α < 4 then

lim λ→∞ λ ( 1 α -γ 2 ) -1
T max (λφ) = C > 0. This last behavior shows the impact of the singularity of the initial data on the behavior of the life-span for large λ.

The goal of this paper is to improve and extend the above mentioned results by considering a large class of initial data, including singular, sign changing, not necessarily polynomially decaying initial data. To carry out this goal, we use three different methods. The first is based on the contraction mapping argument used to prove local existence. We recently introduced and used this method in [50]. Here, we apply it to the nonlinear heat equation and nonlinear Hardy-Hénon parabolic equations. In a forthcoming paper, it will be applied to a variety of evolution equations in order to exhibit the generality of this method ( [START_REF] Tayachi | On the life-span of solutions to nonlinear partial differential evolution equations[END_REF]). We know of some cases where the idea behind this method was previously used in other papers (see for example [START_REF] Ikeda | Some non-existence results for the semilinear Schrödinger equation without gauge invariance[END_REF]) but to our knowledge, this method has never been presented as such or exploited in a systematic way. The second method is based on a necessary condition for local existence of non-negative solutions. The third method is based on scaling properties of the equation. Details are given below later in the introduction.

We begin with the first method and we consider the case where u 0 belongs to a Lebesgue space, where we can use the contraction mapping argument done in [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF][START_REF] Weissler | Local existence and nonexistence for semilinear parabolic equations in L p[END_REF]. It is well known that the problem (1.1)-(1.2) is locally well-posed in L q (Ω) whenever q ≥ 1, q > q c where q c = N α 2 .

(1.5)

See [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF][START_REF] Weissler | Local existence and nonexistence for semilinear parabolic equations in L p[END_REF][START_REF] Quittner | Superlinear parabolic problems. Blow-up, global existence and steady states[END_REF] and references therein. For any u 0 ∈ L q (Ω), we denote by T max (u 0 ) the existence time of the maximal (regular) solution to (1.1)-(1.2) in L q (Ω). Our first result on lower bound of the life-span is derived from [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF][START_REF] Weissler | Local existence and nonexistence for semilinear parabolic equations in L p[END_REF] using an argument from [50].

Theorem 1.1 (Initial data in Lebesgue spaces). Let N ≥ 1, α > 0 and q c be given by (1.5).

Let φ ∈ L q (Ω) with 1 ≤ q ≤ ∞, q > q c or φ ∈ C 0 (Ω). Let u ∈ C ([0, T max (λφ)); L q (Ω)) be the maximal classical solution of (1.1)-(1.2) with initial data u 0 = λφ, λ > 0 (we replace [0, T max (λφ))

by (0, T max (λφ)) if q = ∞). Then there exists a constant C = C(α, q) > 0 such that

T max (λφ)≥ C (λ∥φ∥ q ) ( 1 α -N 2q ) -1 , (1.6)
for all λ > 0.

Hereafter, ∥ • ∥ q denotes the norm in the Lebesgue space L q (Ω).

Remark 1.

1) If T max (φ, L q ) denotes the existence time of the maximal (regular) solution to (1.1)-(1.2)

for φ ∈ L q (Ω), it is known (see for example [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF] and Proposition 2.2 below) that if φ ∈ L p (Ω) ∩ L q (Ω) we have T max (φ, L q ) = T max (φ, L p ). It follows that if φ ∈ L q (Ω) ∩ L q (Ω), 1 ≤ q ≤ q ≤ ∞, q c < q, then the estimate (1.6) reads

T max (λφ) ≥ Cλ -1 α -N 2q -1 , if 0 < λ < 1, T max (λφ) ≥ Cλ -1 α -N 2q -1
, if λ > 1.

2) Theorem 1.1 includes many known results on lower bound of the life-span. For example, it is shown in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]Theorem 3.21 (ii), p. 376] that if φ ∈ C b (R N ), φ ≥ 0 and lim sup |x|→∞ |x| γ φ(x) < ∞, then for N < γ < 2/α, (hence q c < 1) we have T max (λφ) ≥

Cλ -( 1 α -N 2 ) -1
, as λ → 0. This result is a special case of Theorem 1.1. Indeed, it follows that φ ∈ C 0 (R N ) ∩ L q (R N ) with q ≥ 1 > N/γ. Thus, the estimate of [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF] follows by taking q = 1 in (1.6). Other examples will be given throughout the paper.

3) Theorem 1.1 is valid for the equation

∂ t u = ∆u + a(x)|u| α u, with a ∈ L ∞ (Ω). Under the additional assumption a > 0, it is shown in [40, Theorem 3 (i), p. 35] that, for φ ∈ C b (R N ), T max (λφ) ≥ Cλ -α , as λ → ∞.
For such initial data we may take q = ∞ and (1.6) recover the last estimate.

4) For positive initial data and bounded domain, estimate (1.6) is established in [46, Theorem 3.1, p. 2526] where it is also assumed that N ≥ 3, q > max(1, q c ). See also [START_REF] Payne | Lower bounds for blow-up time in parabolic problems under Dirichlet conditions[END_REF] for other estimates with N = 3.

5)

Let us consider the nonlinear heat equation with diffusivity:

∂ t u = µ∆u + |u| α u, u(0) = φ ∈ L q (Ω), q ≥ 1, q > q c , (1.7)
on (0, T µ max (φ)) × Ω, where α > 0, µ > 0 and T µ max (φ) denotes the existence time of the maximal solution of (1.7) with initial data φ. We want to find a lower estimate of T µ max (φ) with respect to µ. Let

v(t, x) = µ -1/α u(t/µ, x).
Then v satisfies the equation

∂ t v = ∆v + |v| α v, v(0) = µ -1/α φ, on (0, µT µ max (φ)) × Ω = 0, T max (µ -1/α φ) × Ω, where T max (µ -1/α φ)
is the existence time of the maximal solution v. Using (1.6) we get,

T µ max (φ) = µ -1 T max (µ -1/α φ) ≥ Cµ -1 µ -1/α ∥φ∥ q -1 α -N 2q -1
.

That is,

T µ max (φ) ≥ Cµ ( 2q N α -1) -1 ∥φ∥ -1 α -N 2q -1 q .
For q = ∞, the right-hand side term does not depend on µ and we have

T µ max (φ) ≥ C∥φ∥ -α ∞ .
See [34, 13, 14] for related estimates. Note that if q < ∞ then lim µ↗∞ T µ max (φ) = ∞.

Using the same method based on the contraction mapping argument as in [50], we also derive from [START_REF] Weissler | Local existence and nonexistence for semilinear parabolic equations in L p[END_REF] the following lower estimate for the life-span for the case of finite Borel measure. We denote by M the set of finite Borel measures on Ω.

Theorem 1.2 (Initial data measure). Let N ≥ 1, α > 0 and q c be given by (1.5). If m is a finite Borel measure on Ω, i.e. m ∈ M, and if q c < 1, i.e. α < 2 N , then the existence time of the maximal solution for (1.1)-(1.2) with initial data u 0 = λm satisfies

T max (λm)≥ C (λ∥m∥ M ) ( 1 α -N 2 ) -1 (1.8)
for all λ > 0, where C = C(α) > 0 is a constant.

We now estimate from below the life-span of solutions for the nonlinear heat equation (1.1) in R N with decaying initial data u 0 = λφ, which may be singular, without sign restriction and for any λ > 0. For γ > 0, 1 ≤ q ≤ ∞, we consider the weighted Lebesgue space

L q γ (R N ) = {f : R N → R, mesurable, | • | γ f ∈ L q (R N )}, endowed with the norm ∥f ∥ L q γ (R N ) := ∥| • | γ f ∥ q .
In Theorem 4.1 below, we give a well-posedness result in weighted Lebesgue spaces for the nonlinear heat equation. As a consequence, we obtain the following lower bound estimates of the life-span.

Theorem 1.3 (Initial data in weighted Lebesgue spaces). Let N ≥ 1 and α > 0. If φ ∈ L q γ (R N ), where 0 < γ < N, γ < 2/α, q ∈ (1, ∞] and

1 q + γ N < 1, N α 2q + αγ 2 < 1, (1.9)
then the existence time of the maximal solution of (1.1)

-(1.2) in L q γ (R N ) with u 0 = λφ satisfies T max (λφ)≥ C λ∥φ∥ L q γ ( 1 α -N 2q -γ 2 ) -1 , (1.10)
for all λ > 0, where C = C(α, q, γ, N ) > 0 is a constant.

Remark 2. Under the conditions 0 < γ < N, γ < 2/α, 1 < q ≤ ∞, (1.9) is equivalent to

q > N N -γ and q > N α 2 -γα .
Combining the results of Theorems 1.3 and 1.1, we get the following estimates of the existence time of the maximal solution to (1.1)-(1.2).

Corollary 1.4. Let N ≥ 1, α > 0, γ > 0, γ ̸ = N and q c be given by (1.5). Assume that

min N α 2 , γα 2 < 1. Let φ ∈ L q (R N ) ∩ L q γ (R N )
, where q ∈ (1, ∞], q > q c . If γ < N we assume further (1.9). Then the existence time of the maximal solution of (1.1)-(1.2) in L q (R N ) with u 0 = λφ satisfies

T max (λφ) ≥ Cλ -1 α -1 2 min N q +γ,N -1 , (1.11
)

for all 0 < λ ≤ 1, where C is a positive constant, C = C(α, q, γ, N, ∥φ∥ L q γ ∩L q ) if γ < N and C = C(α, q, γ, N, ∥φ∥ 1 ) if γ > N. Remark 3.
1) For the particular case q = ∞, Corollary 1.4 includes that of [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]Theorem 3.21,p. 376] and [START_REF] Hisa | Existence of solutions for a fractional semilinear parabolic equation with singular initial data[END_REF]Theorem 5.1,p. 128], where φ is continuous,

φ ≥ 0, φ ∈ L ∞ (R N ) ∩ L ∞ γ (R N ), γ > 0.
The novelty of our estimate is that it holds without any condition on the sign of the initial data. Unlike [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF][START_REF] Hisa | Existence of solutions for a fractional semilinear parabolic equation with singular initial data[END_REF], the case γ = N is not considered here.

2) Corollary 1.4 is totally new if q < ∞.

3) Obviously, if λ > 1, (1.6) is better than (1.11), which itself holds for all λ > 0, as shown in the proof.

The solution of (1.1)-(1.2) constructed with initial data in L q γ (R N ) is in C 0 (R N ) for t > 0, by Proposition 4.2 below. This is well-known to hold also for the solution with initial data in

L p (R N ), p < ∞. So the constructed solution for initial data in L p (R N ) ∩ L q γ (R N
) can be extended to a maximal solution of (1.1)-(1.2), u : (0, T max ) → C 0 (R N ). This maximal existence time is equal to that in L p (R N ) or in L q γ (R N ), as shown in Proposition 4.2 below. In the following result, which extends Corollary 1.4 for 0 < γ < N, we give a lower bound estimate of the life-span for initial data in L p (R N ) ∩ L q γ (R N ).

Corollary 1.5. Let N ≥ 1, α > 0 and q c be given by (1.5).

If φ ∈ L p (R N ) ∩ L q γ (R N ), where p > q c , 1 ≤ p ≤ ∞, 0 < γ < N, γ < 2/α, q ∈ (1,
∞] and satisfies (1.9), then the existence time of the maximal solution of (1.1)-(1.2) with u 0 = λφ satisfies

T max (λφ) ≥ C      λ -1 α -N 2 max 1 q + γ N , 1 p -1 , if 0 < λ ≤ 1, λ -1 α -N 2 min( 1 q + γ N , 1 p ) -1 , if λ > 1, (1.12)
where C = C(α, p, q, γ, N, ∥φ∥ L p ∩L q γ ) > 0 is a constant.

We now turn to results based on the second method, which gives upper-bounds on the life-span and which requires positivity. We distinguish the cases when λ > 0 is large or λ > 0 is small and begin with the case λ is large. By [START_REF] Levine | Some nonexistence and instability theorems for formally parabolic equations of the form P ut = -Au + f (u)[END_REF], T max (λφ) < ∞ in this case, if φ is sufficiently regular. By [START_REF] Weissler | L p -Energy and blow-up for a semilinear heat equation[END_REF]Theorem 1], it follows that if φ ≥ 0 is either a locally integrable function or a positive Borel measure on Ω, φ ̸ ≡ 0, then T max (λφ) < ∞ for all sufficiently large λ > 0. See section 5. See also [START_REF] Mueller | Single point blow-up for a general semilinear heat equation[END_REF]Theorem 2,p. 882] for φ ∈ C 0 (Ω). Our first life-span upper bound is as follows. (1.13) Remark 4.

1) Theorem 1.1 and Theorem 1.6 together show that

T max (λφ) ∼ (λ∥φ∥ ∞ ) -α , λ → ∞ (1.14) whenever φ ≥ 0, φ ̸ ≡ 0, φ ∈ L ∞ (Ω)
. This extends the result of [27, Theorem 3.2 (ii), p.

372] to L ∞ initial data. The lower estimate is valid even if φ is not necessarily positive.

2) With the notation of Part 5) of Remark 1 and using Theorem 1.6, we have that for φ ≥ 0,

φ ̸ ≡ 0, φ ∈ L ∞ (Ω), the maximal existence time of (1.7) satisfies lim sup µ↘0 T µ max (φ) ≤ 1 α ∥φ∥ -α ∞
and hence combined with Part 5) of Remark 1 we have

T µ max (φ) ∼ 1 α ∥φ∥ -α ∞ , as µ → 0. It is shown in [34, Theorem 1, p. 351] that lim µ↘0 T µ max (φ) = 1 α ∥φ∥ -α
∞ , without sign restriction on φ but, unlike our case, only for Ω a bounded domain and assuming also φ a continuous function on Ω.

3) Theorem 1.6 is known for bounded domain and regular initial data, see [START_REF] Sperb | Growth estimates in diffusion-reaction problems[END_REF][START_REF] Sato | Life span of solutions with large initial data for a superlinear heat equation[END_REF]. See also [START_REF] Quittner | Superlinear parabolic problems. Blow-up, global existence and steady states[END_REF]Remark 17.2(i), p. 92] for other estimates in bounded domain.

We now consider positive initial data which are singular near the origin, where we restrict ourselves to the case Ω = R N . We have obtained the following.

Theorem 1.7. Let N ≥ 1 and α > 0. Let 0 < γ < N , γ < 2 α and let ω ∈ L ∞ (R N ) be homogeneous of degree 0, ω ≥ 0 and ω ̸ ≡ 0. Suppose that φ ∈ L 1 loc (R N ), φ ≥ 0 is such that φ(x) ≥ ω(x)|x| -γ for |x| ≤ ϵ, and some ϵ > 0. It follows that T max (λφ) < ∞ for λ > 0 sufficiently large and

lim sup λ→∞ λ ( 1 α -γ 2 ) -1 T max (λφ) ≤ 1 (α 1/α ∥e ∆ (ω| • | -γ )∥ ∞ ) ( 1 α -γ 2 ) -1 .
(1.15)

Remark 5. 1) If φ is as in Theorem 1.7 such that φ ∈ L ∞ γ (R N ), 0 < γ < N , γ < 2 α then Theorem 1.3 and Theorem 1.7 together show that T max (λφ) ∼ λ -( 1 α -γ 2 ) -1 , as λ → ∞.
This extends the result of [50, Proposition 4.5] by removing the condition (N -2)α < 4, as well as the condition lim inf |x|→0 |x| γ φ(x) > 0.

2) If N < γ < 2 α , then there is no local nonnegative solution to (1.1) with initial value λ φ for all λ > 0, where

φ(x) =    ω(x)|x| -γ , |x| ≤ ϵ 0, |x| > ϵ.
(1.16)

See [START_REF] Weissler | L p -Energy and blow-up for a semilinear heat equation[END_REF][START_REF] Cazenave | Perturbations of self-similar solutions[END_REF].

We now turn to upper estimates on T max (λφ) as λ → 0. For this we need to assume that Ω is not bounded, and for simplicity we consider Ω = R N . Our first result of this type is for measures.

Consider u 0 = λm, where λ > 0 and m ∈ M, the set of finite Borel measures on R N . We suppose that m is a positive measure. To insure that (1.1) is locally well-posed on M we assume α < 2 N , and this implies (by Fujita's result) that T max (λm) < ∞ for all λ > 0.

Theorem 1.8. Let N ≥ 1 and α > 0. Suppose α < 2 N and let m ∈ M be a positive finite Borel measure on R N . It follows that T max (λm) < ∞ for all λ > 0 and

lim sup λ→0 λ ( 1 α -N 2 ) -1 T max (λm) ≤ 1 α 1/α (4π) -N/2 ∥m∥ M ( 1 α -N 2 ) -1 .
(1.17) Remark 6. Theorem 1.8 includes the case u 0 = λφ where φ ∈ L 1 (R N ), φ ≥ 0, φ ̸ ≡ 0 and α < 2 N . Indeed, consider the measure dm = φdx where dx denotes Lebesgue measure. It follows then that Theorem 1.1 and Theorem 1.8 together show that

T max (λφ) ∼ (λ∥φ∥ 1 ) -( 1 α -N 2 ) -1 , λ → 0, (1.18) whenever φ ≥ 0, φ ̸ ≡ 0, φ ∈ L 1 (R N ).
The lower estimate is valid even if φ is not necessarily positive.

We have obtained the following for positive initial data having some decay at infinity. Theorem 1.9. Let N ≥ 1 and α > 0. Let φ ∈ L 1 loc (R N ), φ ≥ 0 and suppose that φ(x) ≥ ω(x)|x| -γ for |x| ≥ R, for some R > 0, where ω ∈ L ∞ (R N ) is homogeneous of degree 0, ω ≥ 0 and ω ̸ ≡ 0. If 0 < γ < N and γ < 2 α , then T max (λφ) < ∞ for all λ > 0 and

lim sup λ→0 λ ( 1 α -γ 2 ) -1 T max (λφ) ≤ 1 (α 1/α ∥e ∆ (ω| • | -γ )∥ ∞ ) ( 1 α -γ 2 ) -1 .
(1.19)

Remark 7.

1) If φ is as in Theorem 1.9 such that φ ∈ L ∞ γ (R N ), 0 < γ < N , γ < 2 α then Theorem 1.3 and Theorem 1.9 together show that T max (λφ) ∼ λ -( 1 α -γ 2 ) -1 , as λ → 0. This extends the result of [50, Proposition 4.5] by removing the condition (N -2)α < 4, as well as the condition

lim inf |x|→∞ |x| γ φ(x) > 0. 2) If N < γ < 2 α , then φ ∈ L 1 (R N ), where φ(x) =    0, |x| < R ω(x)|x| -γ , |x| ≥ R, (1.20)
for some R > 0, and so Theorem 1.8 gives an upper life-span bound as λ → 0. So for φ as in Theorem 1.9 with γ > 0, γ ̸ = N, and by comparison argument, Remark 6 and the above one together show that T max (λφ) ∼ λ -( 1 α -1 2 min(γ,N )) -1 , as λ → 0. 3) In the particular case where φ is continuous and bounded such that lim inf |x|→∞ |x| γ φ(x) > 0, a similar result is obtained in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]Theorem 3.15

(ii)]. If φ ∈ L ∞ (R N ) and is nonnegative satisfying φ(x) ≥ (1 + |x|) -γ for almost all x ∈ R N a similar result is also obtained in [23, Theorem 5.2 (ii)]. Here φ is only L 1 loc (R N )
, and so the condition on lower bound on φ is imposed only near infinity and we do not require lim inf |x|→∞ |x| γ φ(x) > 0. In fact, by taking for example ω(x) = |x 1 |/|x|, we have lim inf |x|→∞ |x| γ φ(x) = 0. We also give an explicit upper bound.

We now consider upper-bounds of the life-span for sign changing initial data. We define the sector

Ω m = x = (x 1 , x 2 , • • • , x N ) ∈ R N ; x 1 > 0, x 2 > 0, • • • , x m > 0 , (1.21)
where 1 ≤ m ≤ N is an integer. For 0 < γ < N and integer 1 ≤ m ≤ N , we let ψ 0 : Ω m → R be given by

ψ 0 (x) = c m,γ x 1 • • • x m |x| -γ-2m , x ∈ Ω m , (1.22)
where

c m,γ = γ(γ + 2) • • • (γ + 2m -2). (1.23)
In [50] local well-posedness for φ ∈ L 1 loc (R N \ {0}), anti-symmetric with respect to x 1 , x 2 , • • • , x m , and φ |Ωm is in the Banach space

X = ψ ∈ L 1 loc (Ω m ); ψ ψ 0 ∈ L ∞ (Ω m ) , (1.24)
have been shown for 0 < α < 2/(γ + m). The solution can be extended to maximal solution u : (0, T max (φ)) → C 0 (R N ). Furthermore, there exits a constant C > 0, such that

λ [( 1 α -γ+m 2 ) -1 ] T max (λφ) ≥ C, (1.25)
for all λ > 0. We denote by e t∆m the heat semigroup on Ω m . We have obtained the following for large λ.

Theorem 1.10. Let the positive integer m and the real numbers α, γ be such that

1 ≤ m ≤ N, 0 < γ < N, 0 < α < 2 γ + m . Suppose that φ ∈ L 1 loc (R N \ {0}), anti-symmetric with respect to x 1 , x 2 , • • • , x m , φ |Ωm ∈ X , φ ≥ 0 in Ω m is such that φ(x) ≥ ω(x)ψ 0 (x) for x ∈ Ω m ∩ {|x| ≤ ϵ}, for some ϵ > 0, where ω ∈ L ∞ (R N )
is homogeneous of degree 0, anti-symmetric with respect to x 1 , x 2 , • • • , x m , ω ≥ 0 on Ω m and ω ̸ ≡ 0. It follows that T max (λφ) < ∞ for λ > 0 sufficiently large and

lim sup λ→∞ λ ( 1 α -γ+m 2 ) -1 T max (λφ) ≤ 1 (α 1/α ∥e ∆m (ωψ 0 )∥ ∞ ) ( 1 α -γ+m 2 ) -1 .
We have obtained the following for small λ.

Theorem 1.11. Let the positive integer m and the real numbers α, γ be such that

1 ≤ m ≤ N, 0 < γ < N, 0 < α < 2 γ + m . Suppose that φ ∈ L 1 loc (R N \ {0}), anti-symmetric with respect to x 1 , x 2 , • • • , x m , φ |Ωm ∈ X , φ ≥ 0 in Ω m is such that φ(x) ≥ ω(x)ψ 0 (x) for x ∈ Ω m ∩ {|x| ≥ R}, for some R > 0, where ω ∈ L ∞ (R N )
is homogeneous of degree 0, anti-symmetric with respect to x 1 , x 2 , • • • , x m , ω ≥ 0 on Ω m and ω ̸ ≡ 0. It follows that T max (λφ) < ∞ for all λ > 0 and

lim sup λ→0 λ ( 1 α -γ+m 2 ) -1 T max (λφ) ≤ 1 (α 1/α ∥e ∆m (ωψ 0 )∥ ∞ ) ( 1 α -γ+m 2 ) -1 .
Remark 8.

T max (λφ) ∼ λ -( 1 α -γ+m 2 ) -1 , as λ → ∞ (respectively as λ → 0).
The proofs of the known results cited above are based on careful constructions of super and sub-solutions, comparison and Kaplan's arguments. See, for example, [START_REF] Sato | Life span of solutions with large initial data for a superlinear heat equation[END_REF][START_REF] Li | Lifespan and a new critical exponent for a nonlocal parabolic equation with slowly decay initial values[END_REF][START_REF] Yamaguchi | Life span of positive solutions for a semilinear heat equation with non-decaying initial data[END_REF][START_REF] Yamauchi | Life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity[END_REF][START_REF] Ozawa | Life span of positive solutions for a semilinear heat equation with general nondecaying initial data[END_REF] and some references therein. In the case of decaying initial data, the results are derived via a careful analysis of the asymptotic in the L ∞ -norm of the solutions to the linear heat equation on R N with initial data having specific orders of decay at space infinity as well as Kaplan's arguments and comparison principles, see [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]. This method, [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF], has been used in many papers in the last three decades, see for example [START_REF] Li | Lifespan and a new critical exponent for a nonlocal parabolic equation with slowly decay initial values[END_REF][START_REF] Li | Life span and secondary critical exponent for degenerate and singular parabolic equations[END_REF][START_REF] Li | Cauchy problems of pseudo-parabolic equations with inhomogeneous terms[END_REF][START_REF] Cao | A note on the life span of semilinear pseudo-parabolic equation[END_REF][START_REF] Yang | Second critical exponent and life span for pseudo-parabolic equation[END_REF]] and references therein. Most of the results require that λ be either sufficiently large or sufficiently small and initial data are positive and regular. Also, some scaling arguments are applied to derive life-span estimates, such as in [START_REF] Gui | Life spans of solutions of the Cauchy problem for a semilinear heat equation[END_REF][START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF].

It interesting to compare the two methods used to prove our results above. The proof of lower bounds as already mentioned, is based on the contraction mapping argument which gives local well-posedness of solutions (as in [50]). Consequently, it does not require any positivity condition or maximum principle. To prove the upper estimates, we use a necessary condition for local existence of non-negative solutions established in [START_REF] Weissler | L p -Energy and blow-up for a semilinear heat equation[END_REF] (see Proposition 5.1 below), combined with the maximum principle, continuity properties of the heat semigroup and scaling argument. For these estimates, positivity is required. There is a certain unity in these two methods. On the one hand, the contraction mapping argument gives a sufficient condition on T > 0 for the existence of a solution on the interval [0, T ] for some initial value u 0 . This condition takes the form of an inequality involving both T and u 0 . This condition must fail for T = T max , which implies that the opposite inequality must hold. When this inequality is applied to initial values of the form u 0 = λφ, this results in a lower life-span estimate. On the other hand, inequality in [56, Theorem 1] gives a necessary condition on T > 0 for the existence of a (positive) solution on the interval [0, T ], for some initial value u 0 ≥ 0. This condition must hold for all T < T max . Moreover, this condition is stable under limits, and so must hold in the case T = T max . When the resulting inequality is applied to initial values of the form u 0 = λφ, an upper life-span estimate is obtained. We note that the lower estimates for T max (λφ) do not in and of themselves prove finite time blowup, while the upper estimates do so.

Our results based on scaling, the third approach in this paper, on the one hand use ideas introduced in [START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF], and on the other hand comparison arguments. In particular, we give life-span estimates for an initial value of the form

Φ(x) =    ω(x)|x| -γ 1 , |x| ≤ 1 ω(x)|x| -γ 2 , |x| > 1. (1.26) where 0 < γ 1 , γ 2 < N and γ 1 , γ 2 < 2 α (γ 1 ̸ = γ 2 ) and ω ∈ L ∞ (R N
) is homogeneous of degree 0, ω ≥ 0, ω ̸ ≡ 0. See Corollary 6.6 below. We show, in particular, the impact of the singularity on the life-span for λ large and the impact of the decay at infinity on the life-span for λ small. The rest of this paper is organized as follows. In Section 2, we consider the standard nonlinear heat equation and prove Theorems 1.1 and 1.2. In Section 3, we prove new estimates for the heat kernel in weighted Lebesgue spaces, see Proposition 3.1 below. Section 4 is devoted to the case of slowly decaying initial data and the proofs of Theorem 1.3 and Corollaries 1.4 and 1.5. The upper estimates, Theorems 1.6-1.11, are proved in Section 5. In Section 6, we establish life-span estimates via nonlinear scaling. In the appendix, we give some estimates of the life-span for Hardy-Hénon parabolic equations.

Throughout the paper, C will be a positive constant which may vary from line to line. For positive functions f and g, we say that f (x) ∼ g(x) as x → x 0 if there exists two positive constants

C 1 and C 2 such that C 1 g(x) ≤ f (x) ≤ C 2 g(x) in a neighborhood of x 0 .

Lower bounds for initial data in Lebesgue spaces

We consider the integral equation corresponding to the problem (1.1)-(1.2)

u(t) = e t∆ u 0 + t 0 e (t-σ)∆ |u(σ)| α u(σ) dσ, (2.1)
where e t∆ is the heat semigroup on Ω. It is known that the integral kernel corresponding to e t∆ is bounded by the Gauss kernel for the heat semigroup on R N . Hence the L q -L r smoothing inequalities are independent of Ω, i.e.

∥e t∆ u 0 ∥ L r (Ω) ≤ (4πt) -N 2 ( 1 q -1 r ) ∥u 0 ∥ L q (Ω) (2.2) whenever 1 ≤ q ≤ r ≤ ∞.
We recall for future use that in the case of

Ω = R N D τ e t∆ = e (t/τ 2 )∆ D τ (2.3)
where D τ is the dilation operator D τ f (x) = f (τ x). In particular

D √ t e t∆ = e ∆ D √ t . (2.4)
In this section the goal is to establish lower bounds for the life-span of solutions as an immediate consequence of the fixed point argument used to prove well-posedness of (2.1) in certain Banach spaces. While this argument is well-known, in order to show the applications to life-span, it is more convenient to recall some of the details.

To this end we recall the value

q c = N α 2 ,
and we require that q and r satisfy the following conditions:

q > q c (2.5) and

1 ≤ r α + 1 ≤ q ≤ r. (2.6)
Note that given q > q c and q ≥ 1, one can always choose r = q(α + 1). Also, if q > q c and q ≥ α + 1, one can choose r = q. Furthermore, in all cases above, we have r ≥ α + 1. Finally, if q = ∞, then necessarily r = ∞. We set

β = N 2 1 q - 1 r . (2.7)
We next define the space of curves in which we seek a solution to (2.1), i.e. the space in which we carry out the contraction mapping argument. For a fixed M > 0 and T > 0, (and q, r and β as above), we set

Y q,r M,T = {u ∈ C((0, T ]; L r (Ω)) : sup t∈(0,T ] t β ∥u(t)∥ r ≤ M }. (2.8)
With the distance

d(u, v) = d q,r M,T (u, v) = sup t∈(0,T ] t β ∥u(t) -v(t)∥ r (2.9)
the space Y q,r M,T is a complete metric space. To carry out the fixed point argument, we let u 0 ∈ D ′ (Ω) and suppose that there exists K > 0 such that sup t∈(0,T ] t β ∥e t∆ u 0 ∥ r ≤ K.

(2.10)

This condition includes implicitly the condition that e t∆ u 0 be well-defined and in L r (Ω) for t > 0.

Recall that if u 0 ≥ 0, then e t∆ u 0 indeed is well-defined, but perhaps infinite. In order for (2.10) to hold, it suffices for example that sup t∈(0,T ] t β ∥e t∆ |u 0 |∥ r ≤ K, since |e t∆ u 0 | ≤ e t∆ |u 0 |. We define the iterative operator by

F u 0 u(t) = e t∆ u 0 + t 0 e (t-σ)∆ |u(σ)| α u(σ) dσ. (2.11)
The following theorem is well-known. Since we are particularly interested here in the contraction mapping property, we sketch that part of the proof.

Theorem 2.1. Let N ≥ 1, α > 0, 1 ≤ q ≤ ∞ and q > q c . There is a constant C = C(α, q) > 0 such that if K > 0, M > 0 and T > 0 satisfy K + CT 1-N α 2q M α+1 ≤ M, (2.12)
and if u 0 ∈ D ′ (Ω) satisfies (2.10) for some r ≥ q with 1 ≤ r α+1 ≤ q ≤ r, then F u 0 is a strict contraction on Y q,r M,T and so has a unique fixed point

u = F u 0 u ∈ Y q,r M,T . This solution u of (2.1) is a classical solution of (1.1) on (0, T ].
Furthermore, if u 0 ∈ L q (Ω) and q < ∞, then this fixed point has the property that u ∈ C([0, T ]; L q (Ω)) with u(0) = u 0 .

Remark 9. Of course the sufficient condition (2.12) can be taken as

sup t∈(0,T ] t β ∥e t∆ u 0 ∥ r + CT 1-N α 2q M α+1 ≤ M, (2.13)
i.e. taking equality in (2.10)

Proof. We first consider when the space Y q,r M,T is preserved by the iterative operator F u 0 . Thus we suppose u ∈ Y q,r M,T , and we estimate F u 0 u(t) as follows.

t β ∥F u 0 u(t)∥ r ≤ t β ∥e t∆ u 0 ∥ r + t β t 0 ∥e (t-σ)∆ |u(σ)| α u(σ) ∥ r dσ ≤ K + t β t 0 [4π(t -σ)] -N α 2r ∥|u(σ)| α u(σ)∥ r α+1 dσ = K + (4π) -N α 2r t β t 0 (t -σ) -N α 2r ∥u(σ)∥ α+1 r dσ ≤ K + (4π) -N α 2r t β t 0 (t -σ) -N α 2r σ -β(α+1) dσ M α+1 ≤ K + (4π) -N α 2r t 1-N α 2q 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ M α+1 ≤ K + (4π) -N α 2r 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ T 1-N α 2q M α+1 ≤ K + 2(α + 1)(4π) -N α 2r 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ T 1-N α 2q M α+1 .
It follows that if (2.12) holds, where

C = C(α, q, r) = 2(α + 1)(4π) -N α 2r 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ, then Y q,r M,T is stable by F u 0 .
Next we show that F u 0 is a strict contraction on Y q,r M,T . We estimate as follows.

t β ∥F u 0 u(t) -F u 0 v(t)∥ r ≤ t β t 0 ∥e (t-σ)∆ |u(σ)| α u(σ) -|v(σ)| α v(σ) ∥ r dσ ≤ (4π) -N α 2r t β t 0 (t -σ) -N α 2r ∥ |u(σ)| α u(σ) -|v(σ)| α v(σ) ∥ r α+1 dσ ≤ (α + 1)(4π) -N α 2r t β t 0 (t -σ) -N α 2r ∥[u(σ) -v(σ)] |u(σ)| α + |v(σ)| α ∥ r α+1 dσ ≤ (α + 1)(4π) -N α 2r t β t 0 (t -σ) -N α 2r ∥u(σ) -v(σ)∥ r ∥u(σ)∥ α r + ∥v(σ)∥ α r dσ ≤ 2(α + 1)(4π) -N α 2r t β t 0 (t -σ) -N α 2r σ -β(α+1) dσ M α d q,r M,T (u, v) ≤ 2(α + 1)(4π) -N α 2r t 1-N α 2q 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ M α d q,r M,T (u, v) ≤ 2(α + 1)(4π) -N α 2r 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ T 1-N α 2q M α d q,r M,T (u, v). Thus d q,r M,T (F u 0 u, F u 0 v) ≤ 2(α + 1)(4π) -N α 2r 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ T 1-N α 2q M α d q,r M,T (u, v).
It follows that if (2.12) holds then F u 0 is a strict contraction on Y q,r M,T . The only difficulty is that C potentially depends on r as well as q and α. To rectify this, one can replace C(α, q, r) by C(α, q) = max q≤r≤q(α+1) C(α, q, r) and the result holds with C = C(α, q). □ As is well-known, Theorem 2.1 is used to show that the integral equation (2.1) is locally wellposed on L q (Ω). In particular, if u 0 ∈ L q (Ω) the resulting solution given by the fixed point argument can be extended to a unique maximal solution on an interval [0, T max (u 0 )). We will not belabor this point further.

We have also the following. Proposition 2.2. Let N ≥ 1, α > 0, 1 ≤ q < ∞ and q > q c . Let T max (φ, L q ) denotes the existence time of the maximal solution of (2.1) with initial data φ ∈ L q (Ω). Then the following hold.

(i) u(t) ∈ C 0 (Ω) for t ∈ (0, T max (φ, L q )) . (ii) If φ ∈ L q (Ω) ∩ C 0 (Ω) then T max (φ, L q ) = T max (φ, C 0 (Ω)), the existence time of the maximal solution of (2.1) with initial data φ ∈ C 0 (Ω). (iii) If φ ∈ L q (Ω) ∩ L p (Ω) with q c < p ≤ ∞ then T max (φ, L q ) = T max (φ, L p ), the existence time of the maximal solution of (2.1) with initial data φ ∈ L p (Ω). Proof. (i) By iterative argument, as in [2], u(t) ∈ L r (Ω) for q ≤ r ≤ ∞. It is known that e t∆ : L q (Ω) → C 0 (Ω), is bounded for t > 0 and 1 ≤ q < ∞. See [42, 9]. Hence, by (2.1), u(t) ∈ C 0 (Ω).
(ii) By (i) we have T max (φ, L q ) ≤ T max (φ, C 0 (Ω)). Using (2.1) and (2.2), we have

∥u(t)∥ q ≤ ∥e t∆ φ∥ q + t 0 ∥|u(σ)| α u(σ)∥ q dσ ≤ ∥φ∥ q + t 0 ∥u(σ)∥ α ∞ ∥u(σ)∥ q dσ.
By Gronwall's inequality, we get

∥u(t)∥ q ≤ ∥φ∥ q e t 0 ∥u(σ)∥ α ∞ dσ .
Hence u can not blow up in L q (Ω) before it blows up in C 0 (Ω). That is

T max (φ, C 0 (Ω)) ≤ T max (φ, L q ). (iii) Let ε ∈ (0, min(T max (φ, L q ), T max (φ, L p ))). By (i) we have u(ε) ∈ C 0 (Ω). Using (ii) we have if p < ∞, T max (u(ε), L q ) = T max (u(ε), C 0 (Ω)) = T max (u(ε), L p ). That is T max (φ, L q )-ε = T max (φ, L p )-ε. If p = ∞, then q < p hence (i)-(ii) hold and T max (φ, L q )- ε = T max (u(ε), C 0 (Ω)) = T max (u(ε), L ∞ ) = T max (φ, L ∞ ) -ε.
Hence we get the result. □

As a first application of Theorem 2.1 to life-span estimates, we prove Theorem 1.1.

Proof of Theorem 1.1. Consider u 0 = λφ, where λ > 0 and φ ∈ L q (Ω). The key observation is that if T max (λφ) < ∞, it is impossible to carry out the fixed point argument on the interval [0, T max (λφ)]

with initial value u 0 = λφ. Hence, by (2.13) sup t∈(0,Tmax(λφ)]

t β ∥e t∆ u 0 ∥ r + CT max (λφ) 1-N α 2q M α+1 > M,
for all M > 0. Recall that (4πt) β ∥e t∆ u 0 ∥ r ≤ ∥u 0 ∥ q by the L q -L r smoothing properties of the heat semigroup (2.2), so that (4π) β λ∥φ∥ q + CT max (λφ)

1-N α 2q M α+1 > M,
for all M > 0. In particular, if we set M = 2(4π) β λ∥φ∥ q , this gives

CT max (λφ) 1-N α 2q [λ∥φ∥ q ] α > 1.
Thus we have proved Theorem 1.1. □

As a second application, consider u 0 = λm, where λ > 0 and m ∈ M, the set of finite Borel measures on Ω. For example, m could be a point mass. In order to apply Theorem 2.1, we observe first that

|e t∆ m| ≤ (4πt) -N 2 R N e -|x-y| 2 4t d|m|(y),
where, by abuse of notation, m denotes both the measure on Ω and its natural extension to R N , and |m| is the total variation of m. Hence

∥e t∆ m∥ ∞ ≤ (4πt) -N 2 ∥m∥ M , ∥e t∆ m∥ 1 ≤ ∥m∥ M ;
and so by interpolation

∥e t∆ m∥ r ≤ (4πt) -N 2 (1-1 r ) ∥m∥ M , (2.14)
for all 1 ≤ r ≤ ∞.

Theorem 2.1 thus implies that (2.1) is locally well-posed on M if q c < 1. Simply take q = 1 and

r = α + 1. (This of course is well-known.)
Proof of Theorem 1.2. To obtain a life-span estimate, we again note that if the maximal existence time is finite, i.e. T max (λm) < ∞, then (2.13) can not hold with u 0 = λm and T = T max (λm).

Hence, also using (2.14), for

β = N 2 (1 -1 α+1 ), we must have (4π) β λ∥m∥ M + CT max (λm) 1-N α 2 M α+1 > M,
for all M > 0. As above, we take M = 2(4π) β λ∥m∥ M , which gives the lower estimate

CT max (λm) 1 α -N 2 λ∥m∥ M > 1.
This completes the proof of Theorem 1.

□

As a third application of Theorem 2.1 to life-span estimates we consider u 0 = λφ where λ > 0,

φ ∈ L 1 loc (R N ) and |φ| ≤ | • | -γ for some 0 < γ < N . We recall that if N γ < r, then ∥e t∆ | • | -γ ∥ r = t -γ 2 + N 2r ∥e ∆ | • | -γ ∥ r (2.15)
for all t > 0. This follows from a scaling argument. For convenience, we set

L = ∥e ∆ | • | -γ ∥ r . (2.16) Hence if |φ| ≤ | • | -γ , then ∥e t∆ (λφ)∥ r ≤ λ∥e t∆ | • | -γ ∥ r = Lλt -γ 2 + N 2r .
(2.17)

We next set q = N γ , so that 1 < q < r, and we may choose r so that (2.6) holds. Also,

β = N 2 ( 1 q -1 r ) = γ 2 -N 2r . Theorem 2.
1 clearly shows that (2.1) is locally well-posed for initial values bounded by a multiple of |x| -γ with 0 < γ < N and N γ > q c , i.e. γ < 2 α . This of course is known. See [START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF]Theorem 2.8], and also [50, Theorem 2.3].

As for a life-span estimate, if u 0 = λφ where |φ(x)| ≤ |x| -γ , then the existence time of the solution, T max (λφ), if it is finite, must verify

λL + CT max (λφ) 1-γα 2 M α+1 > M
for all M > 0, where L is given by (2.16). For M = 2λL, this gives

CT max (λφ) 1-γα 2 (λL) α > 1.
In other words, we have the following result.

Corollary 2.3. Let 0 < γ < N and γ < 2 α . Suppose φ ∈ L 1 loc (R N ) is such that |φ(x)| ≤ |x| -γ . It follows that T max (λφ)≥ C (λL) ( 1 α -γ 2 ) -1 , λ > 0, (2.18)
where L is given by (2.16) and C depends only on α and γ.

This last result recovers [50, Theorem 2.6(ii)] in the case m = 0, by a different but related method: the contraction mapping argument is formulated differently. It does not seem possible that the contraction mapping argument used in the proof of Theorem 2.1 can be used to recover [50, Theorems 2.3 and 2.6] in the case 1 ≤ m ≤ N . Indeed, that is the point of the paper [50]. Note also that Theorem 1.3 gives also the result but here the constant at the right-hand side is explicit.

Estimates for the heat kernel in weighted spaces

In this section we prove the following heat kernel estimates. For simplicity, the space L p (R N ), will be denoted by L p . We recall that the norm in

L p (R N ), ∥ • ∥ L p (R N ) is denoted by ∥ • ∥ p . Proposition 3.1. Let N ≥ 1, 0 ≤ γ ≤ µ < N, q 1 ∈ (1, ∞] and q 2 ∈ (1, ∞] satisfy 0 ≤ 1 q 2 < µ -γ N + 1 q 1 ≤ µ N + 1 q 1 < 1.
Then there exists a constant C > 0 depending on N, γ, µ, q 1 and q 2 such that

|.| γ e t∆ u q 2 ≤ Ct -N 2 1 q 1 -1 q 2 -µ-γ 2 ∥|.| µ u∥ q 1 , t > 0, |.| µ u ∈ L q 1 . (3.1)
Remark 10.

1) The estimate (3.1) is well-known for µ = γ = 0, that is (2.2) (see for example [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF]). For the case γ = 0, 0 < µ < N, (3.1) is established in [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. Estimate (3.1) is known for [START_REF] Tsutsui | The Navier-Stokes equations and weak Herz spaces[END_REF][START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF]. See also [25] for the case q 1 = q 2 = ∞. It follows by [START_REF] Tsutsui | The Navier-Stokes equations and weak Herz spaces[END_REF][START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF]

0 < γ ≤ µ < N, 0 ≤ 1 q 2 ≤ 1 q 1 < 1 q 1 + µ N < 1 in
that (3.1) holds for 0 < γ = µ < N, q 1 = q 2 = q ∈ (1, ∞], 1 q + µ N < 1.
2) The power of t in (3.1) is optimal. This can be shown by scaling argument as in [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. In fact, for t > 0, we have

e ∆ u = D √ t e t∆ D 1 √ t u for all u ∈ S ′ (R N ). Also ∥| • | γ D √ t f ∥ r = t -N 2r -γ 2 ∥| • | γ f ∥ r for all | • | γ f ∈ L r , r ≥ 1. Writing (3.1) for t = 1 as follows |.| γ e ∆ u q 2 = |.| γ D √ t e t∆ D 1/ √ t u q 2 ≤ C ∥|.| µ u∥ q 1 . Setting D 1/ √ t u = v that is u = D √ t v, we get |.| γ D √ t e t∆ v q 2 ≤ C |.| µ D √ t v q 1 . That is t -N 2q 2 -γ 2 |.| γ e t∆ v q 2 ≤ Ct -N 2q 1 -µ 2 ∥|.| µ v∥ q 1 .
This gives (3.1) for all t > 0.

3) The fact that γ ≤ µ is necessary. This follows by translation argument. See also [START_REF] De Nápoli | Weighted convolution inequalities for radial functions[END_REF]. We take, in (3.1

), t = 1, u = G 1 (• -τ x 0 ), τ > 0, x 0 ∈ R N , |x 0 | = 1.
In fact, we have that

e ∆ G 1 (• -τ x 0 ) = G 1 ⋆ G 1 (• -τ x 0 ) = G 2 (• -τ x 0 ).
On the other hand,

∥| • | γ G 2 (• -τ x 0 )∥ q 2 = ∥| • +τ x 0 | γ G 2 ∥ q 2 = τ γ • τ + x 0 γ G 2 q 2 and ∥| • | µ G 1 (• -τ x 0 )∥ q 1 = τ µ • τ + x 0 µ G 1 q 1 .
Hence, (3.1) reads,

τ -(µ-γ) • τ + x 0 γ G 2 q 2 ≤ C • τ + x 0 µ G 1 q 1 .
Then we let τ → ∞, since

• τ + x 0 µ G 1 q 1 → ∥G 1 ∥ q 1 < ∞ and • τ + x 0 γ G 2 q 2 → ∥G 2 ∥ q 2 > 0, to deduce that γ ≤ µ if q 2 , q 1 < ∞.
4) Our estimate is different from that of [START_REF] Tsutsui | The Navier-Stokes equations and weak Herz spaces[END_REF][START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF] since we do not require q 1 ≤ q 2 if γ < µ. In fact, since the condition γ ≤ µ, is necessary by the above remark, all that we require is that the power of t in (3.1) is negative.

To prove Proposition 3.1, we establish the following estimates for the heat kernel in weighted Lorentz spaces. Since the cases 0 = γ < µ < N, ( [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]) and 0 < γ = µ < N ( [START_REF] Tsutsui | The Navier-Stokes equations and weak Herz spaces[END_REF][START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF]) are known, we only give the proof for 0 < γ < µ < N.

Proposition 3.2. Let N ≥ 1, 0<γ<µ < N, 1 ≤ q ≤ ∞, q 1 ∈ (1, ∞] and q 2 ∈ (1, ∞] satisfy 0 ≤ 1 q 2 < µ -γ N + 1 q 1 < µ N + 1 q 1 < 1.
Then there exists a constant C > 0 depending on N, γ, µ, q, q 1 and q 2 such that

|.| γ e t∆ u L q 2 ,q ≤ Ct -N 2 1 q 1 -1 q 2 -µ-γ 2 ∥|.| µ u∥ L q 1 ,∞ , t > 0, |.| µ u ∈ L q 1 ,∞ , (3.2)
with if q 2 = ∞ then q = ∞.
Remark 11. A Young's inequality is proved in [26, Theorem 3.1, p. 201] for weighted Lebesgue spaces where it is assumed also q 1 , q 2 < ∞. We do not use this here and we provide a simpler proof for our case as a convolution with a Gaussian. See [47] for (3.2) with γ = 0 < µ < N,

0 ≤ 1 q 2 < µ N + 1 q 1 < 1.
Proof of Proposition 3.2. From the embedding L q 2 ,1 → L q 2 ,q , q ≥ 1, q 2 < ∞, it is sufficient to give the proof for q = 1. Since γ > 0, then by the inequality |x| γ ≤ C(|y| γ + |x -y| γ ), we write

||.| γ e t∆ u| = ||.| γ (G t ⋆ u)| ≤ CG t ⋆ (|.| γ |u|) + C(|.| γ G t ) ⋆ |u|, t > 0. (3.3) Let γ < µ < N, q 1 ∈ (1, ∞) and q 2 ∈ (1, ∞) be such that 1 q 2 < µ -γ N + 1 q 1 < µ N + 1 q 1 < 1. Set 1 p 1 = 1 + 1 q 2 - 1 p 2 , with 1 p 2 = µ -γ N + 1 q 1 . Since γ < N, then p 1 ∈ (1, ∞) and satisfies γ N < 1 - µ -γ N + 1 q 1 < 1 p 1 < 1.
Let us introduce the numbers p1 , p2 defined by

1 p1 = 1 p 1 - γ N , 1 p2 = µ N + 1 q 1 .
We have

1 p1 = 1 + 1 q 2 - 1 p2 ,
Since 0 < γ < µ, and by the conditions on q 1 , q 2 , we have that

0 < 1 p 2 < 1 p2 < 1, 0 < 1 p1 < 1, 0 < 1 p 1 + 1 p 2 -1 = 1 p 1 -1 + 1 p 2 < 1 p 2 < 1.
Using generalized Young's inequality, see [37, Theorem 2.6, p. 137] or [17, 28], we deduce that

∥|.| γ e t∆ u∥ L q 2 ,1 ≤ C∥G t ⋆ |.| γ |u|∥ L q 2 ,1 + C∥|.| γ G t ⋆ |u|∥ L q 2 ,1 ≤ C∥G t ∥ L p 1 ,1 ∥|.| γ |u|∥ L p 2 ,∞ + C∥|.| γ G t ∥ L p1 ,1 ∥|u|∥ L p2 ,∞ := CI 1 + CI 2 , (3.4)
where

1 + 1 q 2 = 1 p 1 + 1 p 2 = 1 p1 + 1 p2 , 1 < q 2 , p 2 , p2 < ∞, 1 < p 1 , p1 < ∞.
We first estimate I 1 . Using the generalized Hölder inequality, see [37, Theorem 3.4, p. 141] or [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]17]), we get

I 1 ≤ C∥G t ∥ L p 1 ,1 ∥|.| -(µ-γ) ∥ L N µ-γ ,∞ ∥|.| µ u∥ L q 1 ,∞ , 0 < µ -γ < N, 1 p 2 = µ -γ N + 1 q 1 < 1. Since G t (x) = t -N 2 G 1 (x/ √ t) = t -N 2 (4π) -N 2 e -|x| 2 4t ∈ L p 1 ,1 , we deduce from [17] that ∥G t ∥ L p 1 ,1 = t -N 2 ∥D 1/ √ t G 1 ∥ L p 1 ,1 = t -N 2 t N 2p 1 ∥G 1 ∥ L p 1 ,1 = Ct -N 2 1-1 p 1 , with 1 - 1 p 1 = 1 p 2 - 1 q 2 = 1 q 1 - 1 q 2 + µ -γ N > 0.
Then, we deduce that

I 1 ≤ Ct -N 2 1 q 1 -1 q 2 -µ-γ 2 ∥|.| µ u∥ L q 1 ,∞ . (3.5)
We now estimate I 2 . Using the generalized Hölder inequality, we get

I 2 ≤ C∥|.| γ G t ∥ L p1 ,1 ∥|.| -µ ∥ L N µ ,∞ ∥|.| µ u∥ L q 1 ,∞ , 0 < µ < N, 1 p2 = µ N + 1 q 1 < 1. Since |x| γ G t (x) = t -N 2 + γ 2 |x/ √ t| γ G 1 (x/ √ t) = t -N 2 (4π) -N 2 |x| γ e -|x| 2 4t ∈ L p1 ,1 , we deduce from [17] that ∥|.| γ G t ∥ L p1 ,1 = t -N 2 + γ 2 ∥D 1/ √ t (|.| γ G 1 )∥ L p1 ,1 = t -N 2 + γ 2 t N 2 p1 ∥|.| γ G 1 ∥ L p1 ,1 = Ct -N 2 1-1 p 1 .
Then, we deduce that If q 2 ∈ (1, ∞) and q 1 = ∞, hence q 2 > N/(µ -γ), the above calculations for estimating I 1 , I If q 2 = q = ∞, the proof follows by using the generalized Young inequality, [37, Theorem 3.6, p.

I 2 ≤ Ct -N 2 1 q 1 -1 q 2 -µ-γ 2 ∥|.| µ u∥ L q 1 ,∞ . ( 3
141] (see also [28, Proposition 2.4 b), p. 20]) as follows

∥|.| γ e t∆ u∥ ∞ ≤ C∥G t ∥ L p 1 ,1 ∥|.| γ |u|∥ L p 2 ,∞ + C∥|.| γ G t ∥ L p1 ,1 ∥|u|∥ L p2 ,∞ , with 1 - µ -γ N + 1 q 1 = 1 p 1 ∈ (γ/N, 1)
and by similar calculations as above. This completes the proof. □

We now give the proof of Proposition 3.1.

Proof of Proposition 3.1. The proof follows by taking q = q 2 in (3.2) and using the fact that

∥|.| µ u∥ L q 1 ,∞ ≤ C ∥|.| µ u∥ L q 1 ,q 1 = C ∥|.| µ u∥ L q 1 . □ 4.
Lower bounds for slowly decaying initial data

In this section we apply Proposition 3.1 in order to show local well-posedness in weighted

Lebesgue spaces for the nonlinear heat equation (2.1). This allows us to obtain more precise estimates for the lower bound of the life-span in relation with the weight. For γ ≥ 0, 1 ≤ q ≤ ∞, we consider the weighted Lebesgue space

L q γ (R N ) = {f : R N → R, mesurable, | • | γ f ∈ L q (R N )}.
Endowed with the norm

∥f ∥ L q γ := ∥| • | γ f ∥ L q , L q γ (R N
) is a Banach space. Clearly, if 0 < γ < N, 1 ≤ q ≤ ∞, and 1 q + γ N < 1, using the Hölder inequality, we have L q γ (R N ) ⊂ S ′ (R N ). Also, for u 0 ∈ L q γ (R N ), 0 < γ < N, N N -γ < q < ∞, we have lim t→0 ∥e t∆ u 0 -u 0 ∥ L q γ (R N ) = 0. This follows as for the standard L q (R N ) case, that is γ = 0. We are interested in the local well-posedness for the nonlinear heat equation (2.1) in L q γ (R N ). We consider initial data u 0 ∈ L q γ (R N ) where q, γ satisfy 0 < γ < N, γ < 2 α .

1 q + γ N < 1, N α 2q + αγ 2 < 1.
The critical exponent in the weighted Lebesgue space L q γ (R N ) is given by

q c (γ) = N α 2 -γα . (4.1)
The value of the critical exponent q c (γ) can be explained by scaling argument. In fact, if u is a solution of the equation (1.1), with Ω = R N , then for any µ > 0, u µ is also a solution of (1.1), where

u µ (t, x) = µ 2 α u(µ 2 t, µx).
We have

∥u µ (t)∥ L q γ = µ 2 α -γ-N
q ∥u(t)∥ L q γ , and on initial data u(0) = u 0 we have

∥µ 2 α u 0 (µ•)∥ L q γ = µ 2 α -γ-N q ∥u 0 ∥ L q γ .
The only weighted Lebesgue exponent (obviously if its exponent is greater than 1) for which the norm is invariant under these dilations is

N q c (γ) = 2 α -γ.
Hence q c (γ) is given by (4.1). We have the following local well-posedness result.

Theorem 4.1 (Local well-posedness in L q γ ). Let N ≥ 1 be an integer, α > 0 and γ such that

0 < γ < N, γ < 2/α. (4.2)
Let q c (γ) be given by (4.1). Then we have the following.

(i) If γ(α + 1) < N and q is such that

q > N (α + 1) N -γ(α + 1)
, q > q c (γ) and q ≤ ∞, then equation (2.1) is locally well-posed in L q γ (R N ). More precisely, given u 0 ∈ L q γ (R N ), then there exist T > 0 and a unique solution u ∈ C [0, T ]; L q γ (R N )) of (2.1) (we replace [0, T ] by (0, T ] if q = ∞ and u satisfies lim t→0 ∥u(t) -

e t∆ u 0 ∥ L ∞ γ (R N ) = 0). Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax ∥u(t)∥ L q γ = ∞.
(ii) Assume that q > q c (γ) with N N -γ < q ≤ ∞. It follows that equation (2.1) is locally wellposed in L q γ (R N ) as in part (i) except that uniqueness is guaranteed only among functions u ∈ C [0, T ]; L q γ (R N )) which also verify t

N 2 ( 1 q -1 r )+ να 2 ∥u(t)∥ L r ν is bounded on (0, T ],
where r = (α+1)q, ν(α+1) = γ, (we replace [0, T ] by (0, T ] if q = ∞ and u satisfies lim t→0 ∥u(t)-

e t∆ u 0 ∥ L ∞ γ (R N ) = 0). Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax ∥u(t)∥ L q γ = ∞. Furthermore, ∥u(t)∥ L q γ ≥ C (T max -t) N 2q -2-γα 2α , ∀ t ∈ [0, T max ), (4.3)
where C is a positive constant.

Proof. (i) Let us define the maps

K t (u) = e t∆ (|u| α u) , t > 0.
Using the following inequality, which follows by the Hölder inequality,

|.| ν(α+1) |u| α u -|v| α v p α+1 ≤ C ∥u∥ α L p ν + ∥v∥ α L p ν ∥u -v∥ L p ν , p ≥ α + 1, ν ≥ 0, (4.4)
and Proposition 3.1 that is e t∆ : L q α+1 (α+1)γ → L q γ is bounded for each t > 0, we have that K t : L q γ -→ L q γ is locally Lipschitz with

∥K t (u) -K t (v)∥ L q γ ≤ Ct -N 2 ( α+1 q -1 q )-αγ 2 ∥|u| α u -|v| α v∥ L q α+1 (α+1)γ ≤ Ct -N α 2q -αγ 2 (∥u∥ α L q γ + ∥v∥ α L q γ )∥u -v∥ L q γ ≤ 2CM α t -N α 2q -αγ 2 ∥u -v∥ L q γ ,
for ∥u∥ L q γ ≤ M and ∥v∥ L q γ ≤ M. We have also, that t

-N α 2q -αγ 2 ∈ L 1 loc (0, ∞), since q > q c (γ). Obviously t → ∥K t (0)∥ ∞ = 0 ∈ L 1 loc (0, ∞)
, also e s∆ K t = K t+s for s, t > 0. Then the proof follows by [53, Theorem 1, p. 279].

(ii) We consider r and ν > 0 such that ν < γ, ν(α + 1) < N, r > q. Hence we have

1 r < α + 1 r + να N < α + 1 r + ν(α + 1) N < 1, 1 r < 1 q + γ -ν N < 1 q + γ N < 1.
The choice of r, ν is to guaranties that the maps e t∆ : L r α+1 (α+1)ν → L r ν and e t∆ : L q γ → L r ν are bounded so that we may apply Proposition 3.1. In order that e t∆ : L r α+1 (α+1)ν → L q γ is bounded, we choose for simplicity, r = (α + 1)q, ν(α + 1) = γ, (If q = ∞ we have r = ∞), and we may apply [8, Lemma 2.1] to get that e t∆ : L r α+1 (α+1)ν = L q γ → L q γ is bounded. With this choice, the conditions on r and ν are satisfied, since

1 q + γ N < 1.
Define

β(ν) = N 2q - N 2r + αν 2 . (4.5)
We choose K > 0, T > 0, M > 0 such that

K + CM α+1 T 1-N α 2q -γα 2 ≤ M, (4.6)
where C is a positive constant. We will show that there exists a unique solution u of (2.1) such

that u ∈ C [0, T ]; L q γ (R N ) ∩ C (0, T ]; L r ν (R N ) with ∥u∥ = max sup t∈[0,T ] ∥u(t)∥ L q γ , sup t∈(0,T ] t β(ν) ∥u(t)∥ L r ν ≤ M.
The proof is based on a contraction mapping argument in the set

Y q,γ M,T = {u ∈ C [0, T ]; L q γ (R N ) ∩ C((0, T ]; L r ν ) : ∥u∥ ≤ M }.
Endowed with the metric d(u, v) = ∥u -v∥, Y q,γ M,T is a nonempty complete metric space. We note that for u 0 ∈ L q γ we have

∥e t∆ u 0 ∥ L r ν ≤ Ct -N 2 ( 1 q -1 r )-γ-ν 2 ∥u 0 ∥ L q γ = Ct -N 2 ( 1 q -1 r )-να 2 ∥u 0 ∥ L q γ = Ct -β(ν) ∥u 0 ∥ L q γ .
We will show that F u 0 defined in (2.11) is a strict contraction on Y q,γ M,T . The condition on the initial data ∥u 0 ∥ L q γ ≤ K will implies that t β ∥e t∆ u 0 ∥ L r ν ≤ K. We have

t β(ν) ∥F u 0 u(t)∥ L r ν ≤ t β(ν) ∥e t∆ u 0 ∥ L r ν + t β(ν) t 0 ∥e (t-σ)∆ |u(σ)| α u(σ) ∥ L r ν dσ ≤ K + Ct β(ν) t 0 (t -σ) -N α 2r - ν(α+1)-ν 2 ∥| • | ν(α+1) |u(σ)| α u(σ)∥ r/(α+1) dσ = K + Ct β(ν) t 0 (t -σ) -N α 2r -να 2 ∥u(σ)∥ α+1 L r ν dσ ≤ K + CM α+1 t β(ν) t 0 (t -σ) -N α 2r -να 2 σ -β(ν)(α+1) dσ ≤ K + CM α+1 t 1-N α 2q -γα 2 1 0 (1 -σ) -N α 2r -να 2 σ -β(ν)(α+1) dσ ≤ K + CM α+1 T 1-N α 2q -γα 2 1 0 (1 -σ) -N α 2r -να 2 σ -β(ν)(α+1) dσ.
By the hypotheses and the fact that q < r and ν < γ we have

N α 2r + να 2 < N α 2q + γα 2 < 1, β(ν)(α + 1) = N α 2q + αγ 2 < 1.
We estimate in L q γ as follows,

∥F u 0 u(t)∥ L q γ ≤ ∥e t∆ u 0 ∥ L q γ + t 0 ∥e (t-σ)∆ |u(σ)| α u(σ) ∥ L q γ dσ ≤ K + C t 0 ∥| • | ν(α+1) |u(σ)| α u(σ)∥ r/(α+1) dσ = K + C t 0 ∥u(σ)∥ α+1 L r ν dσ ≤ K + CM α+1 t 0 σ -β(ν)(α+1) dσ ≤ K + CM α+1 t 1-N α 2q -γα 2 1 0 σ -β(ν)(α+1) dσ ≤ K + CM α+1 T 1-N α 2q -γα 2 1 0 σ -β(ν)(α+1) dσ.
The condition (4.6) implies that the space Y q,r M,T is preserved by the iterative operator F u 0 . We show similarly the contraction. The proof of the other parts follows as in [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. So we omit the details. This completes the proof of the theorem. □

We note that uniqueness in Part (ii) of Theorem 4.1 holds in u ∈ C [0, T ]; L q γ (R N ))∩C (0, T ]; L r ν (R N )). This follows by similar argument as in [START_REF] Slimene | Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems[END_REF]. We will not belabor this point further.

Proof of Theorem 1.3. Consider u 0 = λφ, where λ > 0 and φ ∈ L q γ . If T max (λφ) < ∞, it is impossible to carry out the fixed point argument on the interval [0, T max (λφ)] with initial value u 0 = λφ. Hence, by (4.6)

K + CT max (λφ) 1-N α 2q -γα 2 M α+1 > M,
for all M > K. Letting K = ∥u 0 ∥ L q γ = λ∥φ∥ L q γ , so that

λ∥φ∥ L q γ + CT max (λφ) 1-N α 2q -γα 2 M α+1 > M,
for all M > λ∥φ∥ L q γ . In particular, if we set M = 2λ∥φ∥ L q γ , this gives

CT max (λφ) 1-N α 2q -γα 2 [λ∥φ∥ L q γ ] α > 1.
Thus we have proved Theorem 1.3. □ Remark 12. If u 0 ∈ L q γ with γ > 0, q ≤ ∞ are as in Theorem 4.1 then writing

|u 0 | = |u 0 1 {|x|≤1} + u 0 1 {|x|>1} | ≤ |x| -γ 1 {|x|≤1} |x| γ |u 0 |1 {|x|≤1} + |x| -γ 1 {|x|>1} (|x| γ |u 0 |) ≤ |x| -γ 1 {|x|≤1} |x| γ |u 0 |1 {|x|≤1} + |x| γ |u 0 |,
where 1 A is the indicator function of a subset A of R N , we see by the Hölder inequality that

u 0 ∈ L r + L s with 1 r = 1 q + γ N , s = q if q < ∞ and r ≥ 1, N α 2 < r < N γ < s < ∞ if q = ∞.
Then the local well-posedness is proved in [11, Theorem 2.8]. The fixed point argument used in [START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF] seems not to give an explicit lower bound estimate of life span, as T < 1, the minimal local existence time, is required in the proof there and the constants in particular in [11, Inequality (2.10)], seems to depend on T .

The construction of solutions to (2.1) with initial data in the intersection of two metric spaces follows by well-known argument. See also the proof of [2, Proposition 3.2, p. 126]. We have the following result for the existence time of the maximal solution.

Proposition 4.2. Let N ≥ 1 be an integer, α > 0 and 0 < γ < N, γ < 2/α. Let q c (γ) be given by (4.1). Let q > q c (γ), N N -γ < q < ∞ and T max (φ, L q γ ) denotes the existence time of the maximal solution of (2.1) with initial data φ ∈ L q γ . Then the following hold.

(i) u(t) ∈ C 0 (R N ) ∩ L ∞ γ for t ∈ (0, T max (φ, L q γ )) . (ii) If φ ∈ L q γ ∩ C 0 (R N ) then T max (φ, L q γ ) = T max (φ, C 0 (R N ))
, the existence time of the maximal solution of (2.1) with initial data φ ∈ C 0 (R N ).

(iii) If φ ∈ L q γ ∩ L p γ with p > q c (γ), N N -γ < p < ∞ then T max (φ, L q γ ) = T max (φ, L p γ )
, the existence time of the maximal solution of (2.1) with initial data φ ∈ L p γ . (iv) If φ ∈ L q γ ∩ L p with q c < p ≤ ∞, then T max (φ, L q γ ) = T max (φ, L p ), the existence time of the maximal solution of (2.1) with initial data φ ∈ L p .

Proof. (i) Let φ ∈ L q γ (R N ), q > q c (γ) and q > N N -γ . Let r = (α + 1)q, ν(α + 1) = γ and β(ν) be given by (4.5). Let p be such that r < p ≤ ∞. Hence p > q and 0 ≤

1 p < α + 1 r < γ N + α + 1 r < 1, 1 p < 1 q < γ N + 1 q < 1.
For 0 < T < T max (φ, L q γ ), we have

∥u(t)∥ L p γ ≤ ∥e t∆ φ∥ L p γ + C t 0 (t -σ) -N 2 ( α+1 r -1 p ) ∥u(σ)∥ α+1 L r ν dσ ≤ Ct -N 2 ( 1 q -1 p ) ∥φ∥ L q γ + Ct 1-N 2 ( α+1 r -1 p )-β(ν)(α+1) sup s∈(0,T ] s β(ν)(α+1) ∥u(s)∥ α+1 L r ν × 1 0 (1 -σ) -N 2 ( α+1 r -1 p ) σ -β(ν)(α+1) dσ ≤ Ct -N 2 ( 1 q -1 p ) ∥φ∥ L q γ + M α+1 Ct 1-N 2 ( α+1 q -1 p )-γα 2 1 0 (1 -σ) -N 2 ( α+1 r -1 p ) σ -β(ν)(α+1) dσ.
Since r > q > q c (γ), it follows that if

α + 1 r - 2 N < 1 p < 1 r ,
then u(t) is in L p γ for all t ∈ 0, T max (φ, L q γ ) . The result for general p > q follows by iteration. Hence

u(t) is in L ∞ γ , for t ∈ 0, T max (φ, L q γ ) . Then u(t) ∈ L r + L s for r ≥ 1, N α 2 < r < N γ < s < ∞. Hence by [11, Theorem 2.8] u(t) ∈ L p for s ≤ p ≤ ∞. Then it follows that u(t) ∈ C 0 (R N ), for t ∈ 0, T max (φ, L q γ ) .
(ii) By (i) we have T max (φ, L q γ ) ≤ T max (φ, C 0 (R N )). Using (2.1), we have

∥u(t)∥ L q γ ≤ ∥e t∆ φ∥ L q γ + C t 0 ∥|u(σ)| α u(σ)∥ L q γ dσ ≤ C∥φ∥ L q γ + C t 0 ∥u(σ)∥ α ∞ ∥u(σ)∥ L q γ dσ
By Gronwall's inequality, we get

∥u(t)∥ L q γ ≤ C∥φ∥ L q γ e C t 0 ∥u(σ)∥ α ∞ dσ .
Hence u can not blow up in L q γ before it blows up in C 0 (R N ). That is

T max (φ, C 0 (R N )) ≤ T max (φ, L q γ ). (iii) Let ε ∈ (0, min(T max (φ, L q γ ), T max (φ, L p γ )))
. By (i) we have u(ε) ∈ C 0 (R N ). Using (ii) we have

T max (u(ε), L q γ ) = T max (u(ε), C 0 (R N )) = T max (u(ε), L p γ ).
That is T max (φ, L q γ ) -ε = T max (φ, L p γ ) -ε, hence we get the result. (iv) Follows similarly as (iii). This completes the proof of Proposition 4.2. □ Proof of Corollary 1.4. Since T max (φ, L q ), the maximal existence time in L q , is equal to T max (φ, L q ∩ L q γ ) the maximal existence time in L q ∩ L q γ , we deduce that 1) If N α < 2 we discuss the two cases (i) γ < N hence γ < 2/α and we have

1 q + γ N < 1, N α 2q + γα 2 = N α 2 1 q + γ N < N α 2 < 1,
hence, we apply Theorem 1.3 to get

T max (λφ) ≥ Cλ -1 α -N 2q -γ 2 -1
.

(ii) γ > N then φ ∈ L 1 . In fact, we write φ = φ1 {|x|≤1} + φ1 {|x|>1} . On one hand, since φ ∈ L q , q > 1, hence φ1 {|x|≤1} ∈ L 1 . On the other hand, by the Hölder inequality,

∥φ1 {|x|>1} ∥ 1 = ∥φ|x| γ |x| -γ 1 {|x|>1} ∥ 1 ≤ ∥φ|x| γ ∥ q ∥∥|x| -γ 1 {|x|>1} ∥ q ′ = ∥φ∥ L q γ ∥|x| -γ 1 {|x|>1} ∥ q ′ < ∞,
since γq ′ ≥ γ > N and since φ ∈ L q γ , that is φ1 {|x|>1} ∈ L 1 . Hence both results give that φ ∈ L 1 . We may then apply Theorem 1.1 in L 1 , using 1

> N α 2 to get T max (λφ) ≥ Cλ -( 1 α -N 2 ) -1 .
2) If N α > 2 then we assume γ < 2/α and we have only one case, γ < N. Hence since q > q c (γ) we apply Theorem 1.3 to get T max (λφ) ≥ Cλ

-1 α -N 2q -γ 2 -1
.

In the all cases we have T max (λφ) ≥ Cλ

-1 α -1 2 min( N q +γ,N ) -1
. This completes the proof of Corollary 1.4. □

Proof of Corollary 1.5. Since φ ∈ L p ∩ L q γ , then the existence time of the maximal solution is the same to that in L p and to that in L q γ . By the Hölder inequality φ ∈ L r τ for 1 r = θ p + 1-θ q , τ = (1 -θ)γ, θ ∈ [0, 1]. The existence time of the maximal solution is also the same in L r τ . The function

x ∈ (0, 2/(N α)) → λ -( 1 α -N 2 x) -1
is increasing for λ ∈ (0, 1) and decreasing for λ ∈ (1, ∞). Letting

x = x θ = 1 r + τ N = θ 1 p -1 q -γ + 1 q + γ N , we have that max θ∈[0,1] x θ = max( 1 q + γ N , 1 p ) = max(x 0 , x 1 ) and min θ∈[0,1] x θ = min( 1 q + γ N , 1 p ) = min(x 0 , x 1 ). Using Theorem 1.3 if θ = 0 or Theorem 1.1 if θ = 1, we have that T max (λφ) ≥ C λ∥φ∥ L r τ -( 1 α -N 2 [ 1 r + τ N ]) -1 ≥ C λ∥φ∥ L p ∩L q γ -( 1 α -N 2 [ 1 r + τ N ]) -1
.

The result follows then by taking in the last inequality max θ∈[0,1] x θ , for λ ∈ (0, 1) and min θ∈[0,1] x θ for λ ∈ (1, ∞). This completes the proof of the Corollary. □ Example 4.3. Let 0 < γ < N and γ < 2/α. Let φ be given by (1.16). Then φ

∈ L p ∩ L ∞ γ with N α 2 < p < N γ .
By Corollary 1.5 we have

T max (λφ) ≥ C    λ -1 α -N 2p -1 , if 0 < λ ≤ 1, λ -( 1 α -γ 2 ) -1
, if λ > 1.

Upper bounds for nonnegative solutions

In this section we exploit a well-known necessary condition for the existence of a nonnegative solution to (2.1). More precisely, if u is a nonnegative solution of the integral equation (2.1) on (0, T ) × Ω then αt(e t∆ u 0 ) α ≤ 1, (5.1) for all t ∈ (0, T ], where u 0 ≥ 0 can be either a locally integrable function or a positive Borel measure on Ω. See [START_REF] Weissler | L p -Energy and blow-up for a semilinear heat equation[END_REF]Theorem 1].

We let T max (u 0 ) denote the maximal existence time of a nonnegative solution of (2.1), and so 0 ≤ T max (u 0 ) ≤ ∞. Indeed, there are three possibilities, all of which can be realized: there is no local nonnegative solution with initial value u 0 , there is at least one local solution on some interval (0, T ), but no global solution, i.e. on (0, ∞), or there is indeed a global solution. In the case u 0 = λφ, then (5.1) becomes

αλ α t(e t∆ φ) α ≤ 1, (5.2)
for all t ∈ (0, T ]. If φ ≥ 0, φ ̸ ≡ 0, this implies that T max (λφ) < ∞ for all sufficiently large λ > 0 and that lim λ→∞ T max (λφ) = 0.

(5.3) Indeed, given any t > 0, (5.2) can not be true for sufficiently large λ > 0, and so t ≥ T max (λφ) for sufficiently large λ > 0. This shows the first statements of Theorems 1.6, 1.7 and 1.10.

It is important to realize that T max (u 0 ) as just defined, i.e. the maximal existence time of a nonnegative solution, is not necessarily the same as the maximal existence time of a regular nonnegative solution. Indeed, in some cases, a nonnegative solution can be continued after blowup.

See [START_REF] Baras | Complete blow-up after Tmax for the solution of a semilinear heat equation[END_REF] for example. However, any upper bound on T max (u 0 ) is also an upper bound on the maximal existence time of a regular nonnegative solution.

Proposition 5.1. Let u 0 ≥ 0 be either a locally integrable function or a positive Borel measure on Ω, and let T max (u 0 ) denote the maximal existence time of a nonnegative solution of (2.1). If

0 < T max (u 0 ) < ∞, then αT max (u 0 )(e Tmax(u 0 )∆ u 0 ) α ≤ 1.
(5.4)

In particular, if u 0 = λφ, then

αλ α T max (λφ)∥e Tmax(λφ)∆ φ∥ α ∞ ≤ 1.
(5.5)

Proof. Inequality (5.1) is true for all 0 < t < T max (u 0 ). Hence it is true for t = T max (u 0 ). □

We now give the proofs of the upper bounds.

Proof of Theorem 1.6. Since T max (λφ) → 0 as λ → ∞, it suffices by (5.5)

to observe that if φ ∈ L ∞ (Ω), then ∥e t∆ φ∥ ∞ → ∥φ∥ ∞ as t → 0. Indeed, ∥e t∆ φ∥ ∞ ≤ ∥φ∥ ∞ so lim sup t→0 ∥e t∆ φ∥ ∞ ≤ ∥φ∥ ∞ .
On the other hand, e t∆ φ → φ weak* as t → 0, so ∥φ∥ ∞ ≤ lim inf t→0 ∥e t∆ φ∥ ∞ . □

Proof of Theorem 1.7. In order to estimate T max (λφ) from above, it suffices to estimate T max (λ φ) from above, where φ is defined in (1.16). Indeed, since 0 ≤ φ(x) ≤ φ(x) it follows that T max (λφ) ≤ T max (λ φ).

To find an upper estimate on T max (λ φ) as λ → ∞, it suffices by Proposition 5.1 to determine the behavior of ∥e t∆ φ∥ ∞ as t → 0. Let D τ be the dilation operator D τ f (x) = f (τ x). We have

∥e t∆ φ∥ ∞ = ∥D √ t e t∆ φ∥ ∞ = ∥e ∆ D √ t φ∥ ∞ = t -γ 2 ∥e ∆ [t γ 2 D √ t φ]∥ ∞ . (5.6) Since t γ 2 D √ t φ → ω| • | -γ in D ′ (R N ) as t → 0, it follows by [7, Proposition 3.8 (i), page 1123] that t γ 2 ∥e t∆ φ∥ ∞ → ∥e ∆ (ω| • | -γ )∥ ∞ ,
as t → 0. Since by (5.3), T max (λφ) → 0 as λ → ∞, this along with (5.5) implies lim sup

λ→∞ λ α T max (λφ) 1-αγ 2 ≤ 1 α∥e ∆ (ω| • | -γ )∥ α ∞ ,
which is the desired result. □ Proof of Theorem 1.8. Applying (5.5) we see that

αλ α T max (λm)∥e Tmax(λm)∆ m∥ α ∞ ≤ 1.
Furthermore,

∥e t∆ m∥ ∞ = ∥D √ t e t∆ m∥ ∞ = ∥e ∆ D √ t m∥ ∞ = t -N 2 ∥e ∆ [t N 2 D √ t m]∥ ∞ so that αλ α T max (λm) 1-N α 2 ∥e ∆ [T max (λm) N 2 D √ Tmax(λm) m]∥ α ∞ ≤ 1.
(5.7)

The result follows since T max (λm) → ∞ as λ → 0 (by continuous dependence or Theorem 1.2) and

t N 2 D √ t m → ∥m∥ M δ as t → ∞. In fact, µ t = t N/2 D √ t m is the measure defined by R N f (x)dµ t (x)dx = R N D 1/ √ t f (x)dm(x), f ∈ C 0 (R N ).
We have

D 1/ √ t f → f (0) as t → ∞ m a.e. Since m is finite, then by the dominated convergence theorem R N f (x)dµ t (x)dx → ∥m∥ M δ as t → ∞ for every f in C 0 (R N ). Then t N 2 D √ t m → ∥m∥ M δ as t → ∞ in the dual space (C 0 (R N )) ′ . We know that e ∆ : L 1 (R N ) → C 0 (R N ) is a continuous operator then, by duality, e ∆ : (C 0 (R N )) ′ → (L 1 (R N )) ′ = L ∞ (R N ), is a continuous operator. Hence ∥e ∆ [T max (λm) N 2 D √ Tmax(λm) m]∥ ∞ converges to ∥m∥ M ∥e ∆ δ∥ ∞ = ∥m∥ M (4π) -N/2
as λ → 0. This along with (5.7) implies lim sup

λ→∞ λ α T max (λm) 1-N α 2 ≤ 1 α 1/α (4π) -N/2 ∥m∥ M α .
This gives the desired result. □

Proof of Theorem 1.9. If φ is too singular, it may happen that T max (λφ) = 0, i.e. there is no local nonnegative solution with initial value λφ. This is not a problem, since we will be obtaining upper bounds.

Since φ ≥ φ, where φ is defined in (1.20), it suffices to estimate T max (λ φ). The calculation in (5.6) gives

∥e t∆ φ∥ ∞ = t -γ 2 ∥e ∆ [t γ 2 D √ t φ]∥ ∞ . Moreover, t γ 2 D √ t φ → ω| • | -γ as t → ∞ in D ′ (R N ). It follows, by [7, Proposition 3.8 (i), page 1123] that t γ 2 ∥e t∆ φ∥ ∞ = ∥e ∆ [t γ 2 D √ t φ]∥ ∞ → ∥e ∆ [ω| • | -γ ]∥ ∞ , as t → ∞. If 0 < t < T max (λ φ
), then by (5.2) we must have

αλ α t 1-αγ 2 [t γ 2 ∥e t∆ φ∥ ∞ ] α ≤ 1. (5.8)
It follows that if γ < 2 α , then T max (λ φ) < ∞ for all λ > 0. This is of course a consequence of Fujita's result (including the limiting case) if α ≤ 2 N . (See for example [START_REF] Weissler | Existence and non-existence of global solutions for a semilinear heat equation Israel[END_REF].) If α > 2 N , i.e. q c > 1, this is a consequence of the more general result [50, Theorem 1.7] in the case m = 0. (See also [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]Theorem 3.2(i)] and [START_REF] Ph | Self-similar subsolutions and blow-up for nonlinear parabolic equations[END_REF]Theorem 2].) In these cases, putting t = T max (λ φ) in (5.8) and letting λ → 0, we obtain

lim sup λ→0 λ ( 1 α -γ 2 ) -1 T max (λ φ) ≤ 1 (α 1/α ∥e ∆ (ω| • | -γ )∥ ∞ ) ( 1 α -γ 2 ) -1 . □ Proof of
(i) If α < 2 N , then λ ( 1 α -N 2 ) -1 T max (λδ) = T max (δ)
for all λ > 0.

(ii) If 0 < γ < N and γ < 2 α , and if ψ(x) = ω(x)|x| -γ where ω ∈ L ∞ (R N ) is homogeneous of degree 0, then

λ ( 1 α -γ 2 ) -1 T max (λψ) = T max (ψ)
for all λ > 0.

There are two other possibilities which allow us to obtain life-span estimates. On the one hand, it could be that µ γ D µ φ has a limit as µ → 0 or as µ → ∞, possibly along a subsequence. If one can control T max (µ γ D µ φ) as this limit is attained, one obtains a corresponding life-span estimate from (6.3). This procedure was introduced in the paper [START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF]. For results of this type, we refer the reader to [START_REF] Dickstein | Blowup stability of solutions of the nonlinear heat equation with a large life span[END_REF]Theorems 1.3,1.4,1.5] and [50, Theorems 1.9, 1.10, 1.12, Corollary 1.13, Propositions 4.5, 4.6]. As these latter results show, one can have different life-span behaviors along different subsequences, either as λ → 0 or as λ → ∞. We recall that all of these results depend on delicate continuity properties of the blowup time.

The other approach uses comparison. As a first, and simple, example, we have the following immediate consequence of Corollary 6.2.

Corollary 6.3. If 0 < γ < N and γ < 2 α , and if φ ∈ L 1 loc (R N ), |φ(x)| ≤ ω(x)|x| -γ where ω ∈ L ∞ (R N ), ω ≥ 0 is homogeneous of degree 0, then λ ( 1 α -γ 2 ) -1 T max (λφ) ≥ T max (ω| • | -γ )
for all λ > 0. This is essentially the same as Corollary 2.3. Theorem 1.3 gives also the result but here the constant at the right-hand side is explicit.

Proof. The absolute value of the solution with initial value λφ is bounded above by the solution with initial value λω| • | -γ . We then apply the second assertion of Corollary 6.2. □

We have the following for the function φ given by (1.16).

Corollary 6.4. Let N ≥ 1, α > 0, 0 < γ < N, γ < 2 α , ω ∈ L ∞ (R N
) is homogeneous of degree 0, ω ≥ 0, ω ̸ ≡ 0 and φ be given by (1.16). Then the following hold.

(i) There exists T 1 ∈ [T max (ω| • | -γ ), T max ( φ)) such that lim λ→∞ λ ( 1 α -γ 2 ) -1 T max (λ φ) = T 1 . (ii) lim λ→0 λ ( 1 α -γ 2 ) -1 T max (λ φ) = ∞.
Proof. (i) The function µ → µ γ D µ φ is decreasing on (0, ∞), and

lim µ→0 µ γ D µ φ = ω| • | -γ , lim µ→∞ µ γ D µ φ = 0,
where the first limit is realized in L q 1 (R N ) + L q 2 (R N ) and the second in L q 1 (R N ) whenever 0

≤ N q 2 < γ < N q 1 ≤ N . Consequently φ ≤ µ γ D µ φ ≤ ω| • | -γ , ∀µ ≤ 1.
Applying (6.4), and since γ < 2/α, we conclude that λ → λ

( 1 α -γ 2 ) -1 T max (λ φ) is decreasing on (0, ∞) and T max (ω| • | -γ ) ≤ λ ( 1 α -γ 2 ) -1 T max (λ φ) ≤ T max ( φ), ∀λ ≥ 1.
The existence of the limit T 1 follows by monotonicity.

(ii) We have that φ ∈ L q for q ≥ 1, N α 2 < q < N γ . Hence, by Theorem 1.1

T max (λ φ) ≥ Cλ -1 α -N 2q -1
.

Then

λ ( 1 α -γ 2 ) -1 T max (λ φ) ≥ Cλ ( 1 α -γ 2 ) -1 -1 α -N 2q -1 , that is lim λ→0 λ ( 1 α -γ 2 ) -1 T max (λ φ) = ∞.
For the second assertion, we may also use the continuous dependence in L q (R N ), where γ < N q < 2 α . In fact, since lim µ→∞ µ γ D µ φ = 0, we know that T max (µ γ D µ φ) → ∞ as µ → ∞; so that by (6.4) we have

λ ( 1 α -γ 2 ) -1 T max (λ φ) → ∞, as λ → 0. □ Remark 13. 1) It is natural to conjection that λ ( 1 α -γ 2 ) -1 T max (λ φ) → T max (ω| • | -γ ), as λ → ∞.
This holds in particular for (N -2)α < 4, by continuous dependence of the maximal time of existence.

2) We remark that the upper bound on λ ( 1 α -γ 2 ) -1 T max (λ φ) for large λ > 0 is of the same order as given in Theorem 1.7. As for small λ > 0, if α < 2 N , then Theorem 1.2 gives the stronger estimate λ ( 1 α -N 2 ) -1 T max (λ φ) ≥ c > 0 for all λ > 0, and is of the same order as given in Theorem 1.8 and Remark 6. However, Part (ii) improves the estimate of Corollary 2.3 in [START_REF] Weissler | Existence and non-existence of global solutions for a semilinear heat equation Israel[END_REF].

the case α = 2 N , where T max (λ φ) < ∞ for all λ > 0. If α > 2 N , then T max (λ φ) = ∞ for sufficiently small λ > 0, since φ ∈ L qc (R N ), see
For the function φ, we have the following.

Corollary 6.5. Let N ≥ 1, α > 0, 0 < γ < N, γ < 2 α , ω ∈ L ∞ (R N
) is homogeneous of degree 0, ω ≥ 0, ω ̸ ≡ 0 and φ be given by (1.20). Then the following hold.

(i) There exists T 2 ∈ [T max (ω| • | -γ ), T max ( φ)) such that lim λ→0 λ ( 1 α -γ 2 ) -1 T max (λ φ) = T 2 . (ii) lim λ→∞ λ ( 1 α -γ 2 ) -1 T max (λ φ) = ∞.
Proof. (i) The function µ → µ γ D µ φ is increasing on (0, ∞), and

lim µ→0 µ γ D µ φ = 0, lim µ→∞ µ γ D µ φ = ω| • | -γ ,
where the first limit is realized in L q 2 (R N ) and the second in

L q 1 (R N ) + L q 2 (R N ) whenever 0 ≤ N q 2 < γ < N q 1 ≤ N . Consequently φ ≤ µ γ D µ φ ≤ ω| • | -γ , ∀µ ≥ 1.
Applying (6.4) we conclude that λ → λ

( 1 α -γ 2 ) -1 T max (λ φ) is increasing on (0, ∞) and T max (ω| • | -γ ) ≤ λ ( 1 α -γ 2 ) -1 T max (λ φ) ≤ T max ( φ), ∀λ ≤ 1.
(ii) By continuous dependence in L ∞ (R N ), we know that T max (µ γ D µ φ) → ∞ as µ → 0; so that by (6.4) we have

λ ( 1 α -γ 2 ) -1 T max (λ φ) → ∞, as λ → ∞.
For the second assertion, we may also use the following argument. We have that φ

∈ L ∞ γ ∩ L ∞ . That is, by Corollary 1.4 T max (λ φ) ≥ Cλ -( 1 α -γ 2 ) -1
and by Theorem 1.1 Next, we consider the function Φ given by (1.26). We have the following.

T max (λ φ) ≥ Cλ -( 1 α ) -1 . Then λ ( 1 α -γ 2 ) -1 T max (λ φ) ≥ max C, Cλ ( 1 α -γ 2 ) -1 -( 1 α ) -1 . Hence lim λ→∞ λ ( 1 α -γ 2 ) -1 T max (λ φ) = ∞. □ Remark 14. 1) It is natural to conjection that λ ( 1 α -γ 2 ) -1 T max (λ φ) → T max (ω| • | -γ ), as λ → 0. 2) We remark that φ ∈ L ∞ (R N ),
Corollary 6.6. Let N ≥ 1, α > 0, 0 < γ 1 , γ 2 < N and γ 1 , γ 2 < 2 α (γ 1 ̸ = γ 2 ). Let ω ∈ L ∞ (R N
) is homogeneous of degree 0, ω ≥ 0, ω ̸ ≡ 0 and let Φ be defined by (1.26). Then we have the following.

(i) There exists T 3 ∈ (T max (Φ), T max (ω| • | -γ 1 )] such that lim λ→∞ λ ( 1 α - γ 1 2 ) -1 T max (λΦ) = T 3 and if α < 4 N -2 , or γ 1 > γ 2 then T 3 = T max (ω| • | -γ 1 ).
Also,

µ γ 2 Φ(µx) =          µ γ 2 -γ 1 ω(x)|x| -γ 1 , |x| ≤ 1 ν µ γ 2 -γ 1 ω(x)|x| -γ 1 , 1 ν ≤ |x| ≤ 1 µ ω(x)|x| -γ 2 , |x| ≥ 1 µ . =          µ γ 2 -γ 1 ω(x)|x| -γ 1 , |x| ≤ 1 ν µ γ 2 -γ 1 |x| γ 2 -γ 1 ω(x)|x| -γ 2 , 1 ν ≤ |x| ≤ 1 µ ω(x)|x| -γ 2 , |x| ≥ 1 µ . ≤          ν γ 2 -γ 1 ω(x)|x| -γ 1 , |x| ≤ 1 ν ω(x)|x| -γ 2 , 1 ν ≤ |x| ≤ 1 µ ω(x)|x| -γ 2 , |x| ≥ 1 µ . = ν γ 1 Φ(νx).
It follows that

• The function µ → T max (µ γ 1 D µ Φ) is increasing on (0, ∞), and
T max (Φ) ≥ lim µ→0 T max (µ γ 1 D µ Φ) ≥ T max (ω| • | -γ 1 ), and if α < 4 N -2 , then lim µ→0 T max (µ γ 1 D µ Φ) = T max (ω| • | -γ 1 ). Also lim µ→∞ T max (µ γ 1 D µ Φ) = ∞.
• The function µ → T max (µ γ 2 D µ Φ) is decreasing on (0, ∞), and

T max (Φ) ≥ lim µ→∞ T max (µ γ 2 D µ Φ) ≥ T max (ω| • | -γ 2 ), and if α < 4 N -2 , then lim µ→∞ T max (µ γ 2 D µ Φ) = T max (ω| • | -γ 2 ). Also lim µ→0 T max (µ γ 2 D µ Φ) = ∞.
Next, applying (6.4), we have

λ ( 1 α - γ 1 2 ) -1 T max (λΦ) = T max (µ γ 1 D µ Φ), λ = µ γ 1 -2 α , (6.7)
λ ( 1 α - γ 2 2 ) -1 T max (λΦ) = T max (µ γ 2 D µ Φ), λ = µ γ 2 -2 α . (6.8)
This completes the proof of (i)-(ii) if γ 1 < γ 2 and (iii).

2) Analysis of T max (λΦ) in the case γ 1 > γ 2 .

In this case Φ = ω max[| • | -γ 1 , | • | -γ 2 ], so Φ ≥ ω| • | -γ 1 , Φ ≥ ω| • | -γ 2 and Φ ≤ ω(| • | -γ 1 + | • | -γ 2 ).
Hence

ω| • | -γ 1 ≤ µ γ 1 D µ Φ ≤ ω(| • | -γ 1 + µ γ 1 -γ 2 | • | -γ 2 ) (6.9) and ω| • | -γ 2 ≤ µ γ 2 D µ Φ ≤ ω(µ γ 2 -γ 1 | • | -γ 1 + | • | -γ 2 ) (6.10)
for all µ > 0.

We claim that

• The function µ → µ γ 1 D µ Φ is increasing on (0, ∞), and

lim µ→0 µ γ 1 D µ Φ = ω| • | -γ 1 , lim µ→∞ µ γ 1 D µ Φ = ∞,
where the limits are in L q 1 (R N ) + L q 2 (R N ) whenever 0 ≤ N q 2 < γ 2 < γ 1 < N q 1 ≤ N , by (6.9). • The function µ → µ γ 2 D µ Φ is decreasing on (0, ∞), and

lim µ→0 µ γ 2 D µ Φ = ∞, lim µ→∞ µ γ 2 D µ Φ = ω| • | -γ 2 ,
where the limits are in

L q 1 (R N ) + L q 2 (R N ) whenever 0 ≤ N q 2 < γ 2 < γ 1 < N q 1 ≤ N , by (6.10). Proof of the claim. Let 0 < µ < ν < ∞, so that µ γ 2 -γ 1 > ν γ 2 -γ 1 . We have µ γ 1 Φ(µx) =          ω(x)|x| -γ 1 , |x| ≤ 1 ν ω(x)|x| -γ 1 , 1 ν ≤ |x| ≤ 1 µ µ γ 1 -γ 2 ω(x)|x| -γ 2 , |x| ≥ 1 µ . =          ω(x)|x| -γ 1 , |x| ≤ 1 ν ( 1 |x| ) γ 1 -γ 2 ω(x)|x| -γ 2 , 1 ν ≤ |x| ≤ 1 µ µ γ 1 -γ 2 ω(x)|x| -γ 2 , |x| ≥ 1 µ . ≤          ω(x)|x| -γ 1 , |x| ≤ 1 ν ν γ 1 -γ 2 ω(x)|x| -γ 2 , 1 ν ≤ |x| ≤ 1 µ ν γ 1 -γ 2 ω(x)|x| -γ 2 , |x| ≥ 1 µ . = ν γ 1 Φ(νx).
Also,

µ γ 2 Φ(µx) =          µ γ 2 -γ 1 ω(x)|x| -γ 1 , |x| ≤ 1 ν µ γ 2 -γ 1 ω(x)|x| -γ 1 , 1 ν ≤ |x| ≤ 1 µ ω(x)|x| -γ 2 , |x| ≥ 1 µ . =          µ γ 2 -γ 1 ω(x)|x| -γ 1 , |x| ≤ 1 ν µ γ 2 -γ 1 |x| γ 2 -γ 1 ω(x)|x| -γ 2 , 1 ν ≤ |x| ≤ 1 µ ω(x)|x| -γ 2 , |x| ≥ 1 µ . ≥          ν γ 2 -γ 1 ω(x)|x| -γ 1 , |x| ≤ 1 ν ω(x)|x| -γ 2 , 1 ν ≤ |x| ≤ 1 µ ω(x)|x| -γ 2 , |x| ≥ 1 µ . = ν γ 1 Φ(νx).
It follows that

• The function µ → T max (µ γ 1 D µ Φ) is decreasing on (0, ∞), and
lim µ→0 T max (µ γ 1 D µ Φ) = T max (ω| • | -γ 1 ).
• The function µ → T max (µ γ 2 D µ Φ) is increasing on (0, ∞), and

lim µ→∞ T max (µ γ 2 D µ Φ) = T max (ω| • | -γ 2 ).
Next, applying (6.4), we have

λ ( 1 α - γ 1 2 ) -1 T max (λΦ) = T max (µ γ 1 D µ Φ), λ = µ γ 1 -2 α , λ ( 1 α - γ 2 2 ) -1 T max (λΦ) = T max (µ γ 2 D µ Φ), λ = µ γ 2 -2 α .
To show that we reach the above exact limits we use the following.

Observation. If ϕ k → ϕ and assume we are in a situation of continuous dependence, then we know lim inf k→∞ T max (ϕ k ) ≥ T max (ϕ). Suppose also that ϕ ≤ ϕ k for all k, hence T max (ϕ k ) ≤ T max (ϕ) for all k, so lim sup k→∞ T max (ϕ k ) ≤ T max (ϕ). Hence lim k→∞ T max (ϕ k ) = T max (ϕ).

This allows us to obtain the above exact limits for the case γ 1 > γ 2 . This completes the proof of

(i)-(ii) if γ 1 > γ 2 .
We may also show (iii) as follows. The function Φ verifies: if

γ 1 < γ 2 then Φ ∈ L q , γ 1 N < 1 q < γ 2 N < min(1, 2 N α ).
Hence, by Theorem 1.1

T max (λΦ) ≥ Cλ -1 α -N 2q -1
.

Then lim λ→0 λ ( 1 α - γ 1
2 )

-1 T max (λΦ) = ∞ = lim λ→∞ λ ( 1 α - γ 2
2 )

-1

T max (λΦ).

This completes the proof of Corollary 6.6. □ maximal solution of (A.3) with initial data u 0 = λφ, λ > 0. Then there exists a constant C = C(N, l, α, q) > 0 such that

T max (λφ)≥C (λ∥φ∥ q ) -2+l 2α -N 2q -1 , (A.6)
for all λ > 0.

Proof of Theorem A.1. For q > q c (l), let r > q satisfying (A.5). Let us define

β(l) = N 2 1 q - 1 r .
We note that r depends on l, hence β also. The well-posedness results for (A.1) has been obtained in [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF][START_REF] Slimene | Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems[END_REF]. We now give the proof of (A.6). Let -min(2, N ) < l < 0, α > 0, λ > 0, K > 0 and φ ∈ C 0 (R N ), ∥φ∥ ∞ ≤ K or φ ∈ L q , q > 1, q > q c (l) such that ∥φ∥ q ≤ K. Let u ∈ C [0, T max (λφ)); L q (R N ) ∩ C (0, T max (λφ)); L r (R N ) with r > q satisfying (A.5), be the maximal solution of (A.1) on [0, T max (λφ)). It is proved in [2, Inequalities (3.5), (3.6), p. 124] that for K, T, M > 0 such that

K + CT 1-N α 2q + l 2 M α+1 ≤ M
the solution u of (A.1) is defined on [0, T ] and verifies max sup t∈(0,T ] t β(l) ∥u(t)∥ r , sup [0,T ] ∥u(t)∥ q ≤ M. Here C is a positive constant. Then for T max (φ) we should have

K + C T max (φ) 1-N α 2q + l 2 M α+1 > M,
for all M > K. That is it must be λK + C T max (λφ)

1-N α 2q + l 2 M α+1 > M,
for all M > λK. If we set M = 2λK, we get

2 α+1 K α CT max (λφ) 1-N α 2q + l 2 λ α > 1.
Then taking K = ∥φ∥ q , we get that there exists C = C(N, α, l, q) > 0 (since r itself depends on q) such that (A.6) holds. This completes the proof of the Theorem. □

Using similar argument developed to prove Theorem A.1, we derive the same result for the equation

∂ t u = ∆u + a(x)|u| α u, (A.7) where a(x)| • | -l is in L ∞ (R N ).
In particular, we may take a regular near the origin. Then we have the following.

Corollary A.2. Let N ≥ 1, -min(2, N ) < l < 0, α > 0, and q c (l) be given by (A.4). Let φ ∈ L q (R N ) with 1 < q ≤ ∞, q > q c (l) or φ ∈ C 0 (R N ) and r > q satisfies (A.5). Let u ∈ C [0, T max (λφ)); L q (R N ) ∩ C((0, T max (λφ)); L r (R N )) be the maximal mild solution of (A.

T max (λφ) ≥ Cλ -( 1 α -N 2 ) -1
, as λ → 0.

For this class φ ∈ L q (R N ) for all q ≥ 1. Since a ∈ L ∞ (R N ) because l < 0 and a is regular, then we may use Theorem 1.1 (which is valid for such a as noted before) and apply (1.6) with q = 1 since q c < 1, and then recover the result of [40].

The second class considered in [40] is for φ such that there exist constants c 1 , c 2 > 0 and

c 1 ≤ φ ≤ c 2 .
T max (λφ) ≥ Cλ -2α 2+l , as λ → 0.
Here φ ∈ L ∞ (R N ), the previous estimate is the same as (A.6) with q = ∞. We then recover the results of [40].

Appendix B. Nonlinear Hénon parabolic equations

In this part of the appendix we study a nonlinear heat equation with a spatially growing variable coefficient. We consider the equation (A.1) for l > 0 and with the initial condition (A.2). Local well-posedness is known in C(R N ) ∩ L ∞ ∩ L ∞ l/α , (see [START_REF] Wang | On the Cauchy problem for reaction-diffusion equations[END_REF][START_REF] Majdoub | Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation[END_REF]). Recently local well-posedness is established in L q s for some q ≥ α + 1 and s satisfying some conditions (see [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF]). Not much is known about this equation in comparison with the standard nonlinear heat equation, that is the case l = 0. In particular, the life-span is only known for small lambda and rapidly decaying positive initial data, see [40]. Note that the blowup may hold at the origin as it may also not hold at the origin. See [START_REF] Guo | Excluding blowup at zero points of the potential by means of Liouville-type theorems[END_REF][START_REF] Guo | Blow-up behavior for a parabolic equation with spatially dependent coefficient[END_REF][START_REF] Guo | Blow-up for a reaction-diffusion equation with variable coefficient[END_REF][START_REF] Guo | Blowing up at zero points of potential for an initial boundary value problem[END_REF][START_REF] Filippas | On similarity solutions of a heat equation with a nonhomogeneous nonlinearity[END_REF]. To show lower-bound estimates of the life-span, we establish local well-posedness results. Using Proposition 3.1, we prove local existence for (A.1) in L q γ for γ = l α < N and q is such that

q > q c = N α 2 , N N -γ < q ≤ ∞. (B.1)
This value of γ is inspired by [START_REF] Wang | On the Cauchy problem for reaction-diffusion equations[END_REF].

We note that for 0 < γ = l/α < N,

q c > N N -γ ⇔ q F := N α 2 + l > 1,
where q F is the Fujita exponent for the equation (A.1) (see [START_REF] Qi | The critical exponents of parabolic equations and blow-up in R n[END_REF]). We have the following local well-posedness result.

Theorem B.1 (Local well-posedness in L q γ ). Let N ≥ 1 be an integer, α > 0 and l > 0 be such that

γ := l α < N. (B.2)
Let q c be given by (B.1). Then we have the following.

(i) If q is such that q > N (α + 1) N -γ , q > q c and q ≤ ∞, then equation (A.3) is locally well-posed in L q γ (R N ). More precisely, given u 0 ∈ L q γ (R N ), then there exist T > 0 and a unique solution of

(A.3) u ∈ C [0, T ]; L q γ (R N ) if q < ∞ and u ∈ C (0, T ]; L ∞ γ (R N ) , lim t→0 ∥u(t) -e t∆ u 0 ∥ L ∞ γ (R N ) = 0 if q = ∞.
Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax ∥u(t)∥ L q γ = ∞. (ii) Assume that q > q c with N N -γ < q ≤ ∞. It follows that equation (A.3) is locally well-posed in L q γ (R N ) as in part (i) except that uniqueness is guaranteed only among functions u ∈ C [0, T ]; L q γ (R N ) which also verify t

N 2 ( 1 q -1 r ) ∥u(t)∥ L r γ , is bounded on (0, T ],
where r is given below, (we replace [0, T ] by (0, T ] if q = ∞ and u satisfies lim t→0 ∥u(t)-e t∆ u 0 ∥ L ∞ γ (R N ) = 0). Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax ∥u(t)∥ L q γ = ∞. Furthermore,

∥u(t)∥ L q γ ≥ C (T max -t) N 2q -1 α , ∀ t ∈ [0, T max ), (B.3)
where C is a positive constant.

Remark 16. Unlike in [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF], here we dot not impose q ≥ α + 1. Also, our strategy is different from that of [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF]. In fact, we use a method of [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF][START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. Precisely, to prove the local well-posedness in L q γ , we use an auxiliary space L r γ for some r as an auxiliary parameter, while in [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF] the weight γ = l/α is replaced by a real number s that is considered as an auxiliary parameter.

Proof of Theorem B.1. (i) Let us define the maps

K t,l (u) = e t∆ | • | l |u| α u , t > 0.
By the Hölder inequality and Proposition 3.1 with γ = l α = µ, q 1 = q/(α + 1), q 2 = q, for each t > 0 and if q > N (α+1) N -γ , q ≤ ∞, K t,l : L q γ -→ L q γ is locally Lipschitz with

∥K t,l (u) -K t,l (v)∥ L q γ ≤ Ct -N 2 ( α+1 q -1 q ) | • | l (|u| α u -|v| α v) L q α+1 γ = Ct -N α 2q ∥| • | αγ (|u| α u -|v| α v)∥ L q α+1 γ ≤ Ct -N α 2q (∥u∥ α L q γ + ∥v∥ α L q γ )∥u -v∥ L q γ ≤ 2CM α t -N α 2q ∥u -v∥ L q γ ,
for ∥u∥ L q γ ≤ M and ∥v∥ L q γ ≤ M. We have also, that t

-N α 2q ∈ L 1 loc (0, ∞), since q > q c = N α 2 . Obviously t → ∥K t,l (0)∥ ∞ = 0 ∈ L 1 loc (0, ∞)
, also e s∆ K t,l = K t+s,l for s, t > 0. Then the proof follows by [START_REF] Weissler | Semilinear evolution equations in Banach spaces[END_REF]Theorem 1,p. 279].

(ii) We begin with the observation that, since q > N N -γ , there exists r > q satisfying 1 q(α + 1)

< 1 r < N -γ N (α + 1) . (B.4)
We then observe that, since q > q c , we have

1 q - 2 N (α + 1) < 1 q(α + 1)
.

Hence any r > q satisfying (B.4) verifies

1 q - 2 N (α + 1) < 1 r .
This last inequality implies that

β(α + 1) < 1,
where

β = N 2q - N 2r . (B.5)
We choose K > 0, T > 0, M > 0 such that

K + CM α+1 T 1-N α 2q ≤ M, (B.6)
where C is a positive constant given below. We will show that there exists a unique solution u of

(A.3) such that u ∈ C [0, T ], L q γ (R N ) and u ∈ C (0, T ], L r γ (R N ) with ∥u∥ = max sup t∈[0,T ] ∥u(t)∥ L q γ , sup t∈(0,T ] t β ∥u(t)∥ L r γ ≤ M.
The proof is based on a contraction mapping argument in the set

Y q,γ M,T = {u ∈ C [0, T ], L q γ (R N ) ∩ C((0, T ], L r γ ) : ∥u∥ ≤ M }.
Endowed with the metric d(u, v) = ∥u -v∥, Y q,γ M,T is a nonempty complete metric space. We note that for u 0 ∈ L q γ we have

∥e t∆ u 0 ∥ L r γ ≤ Ct -N 2 ( 1 q -1 r ) ∥u 0 ∥ L q γ = Ct -β ∥u 0 ∥ L q γ .
The condition on initial data ∥u 0 ∥ L q γ ≤ K implies that t β ∥e t∆ u 0 ∥ L r γ ≤ K. We will show that

F u 0 u(t) = e t∆ u 0 + t 0 e (t-σ)∆ | • | l |u(σ)| α u(σ) dσ. (B.7)
is a strict contraction on Y q,γ M,T .

Using Proposition 3.1, that is the boundedness of the map e t∆ : L q γ → L r γ , for the first term and the boundedness of the map e t∆ : L r α+1 γ → L r γ , for the second term, we have

t β ∥F φ (u)(t) -F ψ (v)(t)∥ L r γ ≤ t β ∥e t∆ (φ -ψ)∥ L r γ + t β t 0 ∥e (t-s)∆ [|.| l |u| α u(s) -|v| α v(s) ]∥ L r γ ds ≤ ∥φ -ψ∥ L q γ + Ct β t 0 (t -s) -N 2 ( α+1 r -1 r ) |.| γα |u| α u(s) -|v| α v(s) L r α+1 γ ds ≤ ∥φ -ψ∥ L q γ + 2(α + 1)CM α t β t 0 (t -s) -N α 2r s -β(α+1) ds d(u, v) ≤ ∥φ -ψ∥ L q γ + 2(α + 1)CM α t 1-N α 2q 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ d(u, v) ≤ ∥φ -ψ∥ L q γ + C 1 M α T 1-N α 2q d(u, v),
where

C 1 = 2(α + 1)C 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ < ∞.
Using [8, Lemma 2.1] that is the boundedness of the map e t∆ : L q γ → L q γ , for the first term and Proposition 3.1, the boundedness of the map e t∆ : L r α+1 γ → L q γ , for the second term, we have

∥F φ (u)(t) -F ψ (v)(t)∥ L q γ ≤ ∥e t∆ (φ -ψ)∥ L q γ + t 0 ∥e (t-s)∆ [|.| l |u| α u(s) -|v| α v(s) ]∥ L q γ ds ≤ ∥φ -ψ∥ L q γ + C t 0 (t -s) -N 2 ( α+1 r -1 q ) |.| γα |u| α u(s) -|v| α v(s) L r α+1 γ ds ≤ ∥φ -ψ∥ L q γ + 2(α + 1)CM α t 0 (t -s) -N 2 ( α+1 r -1 q ) s -β(α+1) ds d(u, v) ≤ ∥φ -ψ∥ q + 2(α + 1)CM α t 1-N α 2q 1 0 (1 -σ) -N 2 ( α+1 r -1 q ) σ -β(α+1) dσ d(u, v) ≤ ∥φ -ψ∥ L q γ + C 2 M α T 1-N α 2q d(u, v).
where

C 2 = 2(α + 1)C 1 0 (1 -σ) -N 2 ( α+1 r -1 q ) σ -β(α+1) dσ < ∞.
From the above estimates, it follows that

d(F φ (u), F ψ (v)) ≤ ∥φ -ψ∥ L q γ + CM α T 1-N α 2q d(u, v), (B.8)
where C = max(C 1 , C 2 ). The rest of the proof follows similarly as that of Theorem 4.1 and as in [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. This completes the proof. □ Theorem B.1 allows us to obtain the following.

Corollary B.2 (Hénon parabolic equations). Let

N ≥ 1, α > 0, 0 < l < N α. If φ ∈ L q γ (R N ), where γ = l/α < N, q > N α 2 and N N -γ < q ≤ ∞,
then the life-span of (A.3) with initial data λφ satisfies

T max (λφ)≥C(λ∥φ∥ L q γ ) -( 1 α -N 2q ) -1 , (B.9)
for all λ > 0, where C = C(α, q, l, N ).

Remark 17.

1) We see that l has no effect on the lower bound of the life span. This is because blow up may not occurs at the origin nor at |x| infinite.

2) Corollary B.2 is totally new for q < ∞.

Proof of Corollary B.2. The proof follows using (B.6) and is similar to that of Theorem 1.1. □

In the case of initial data in L q (R N ) ∩ L q γ (R N ) we have the following result which generalizes that of [START_REF] Wang | On the Cauchy problem for reaction-diffusion equations[END_REF] known for q = ∞.

Theorem B.3 (Local well-posedness in L q ∩ L q γ ). Let N ≥ 1 be an integer, α > 0 and l > 0 be such that

0 < γ := l α < N.
Let q c be given by (B.1). Then we have the following.

(i) Equation (A.3) is locally well-posed in L ∞ (R N ) ∩ L ∞ γ (R N ). More precisely, given u 0 ∈ L ∞ (R N ) ∩ L ∞ γ (R N ), then there exist T > 0 and a unique solution u ∈ C (0, T ]; L ∞ (R N ) ∩ L ∞ γ (R N )) of (A.3) and u satisfies lim t→0 ∥u(t) -e t∆ u 0 ∥ L ∞ ∩L ∞ γ (R N ) = 0. Moreover, u can be extended to a maximal interval (0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax (∥u(t)∥ ∞ + ∥u(t)∥ L ∞ γ ) = ∞. (ii) If q is such that q > N (α + 1)
N -γ , q > q c and q < ∞, then equation (A.3) is locally well-posed in L q (R N ) ∩ L q γ (R N ). More precisely, given u 0 ∈ L q (R N ) ∩ L q γ (R N ), then there exist T > 0 and a unique solution u ∈ C [0, T ]; L q (R N ) ∩ L q γ (R N ) of (A.3). Moreover, u can be extended to a maximal interval [0, T max ) such that either

T max = ∞ or T max < ∞ and lim t→Tmax (∥u(t)∥ q + ∥u(t)∥ L q γ ) = ∞. (iii) Assume that q > q c with N N -γ < q ≤ ∞. It follows that equation (A.3) is locally well-posed in L q (R N ) ∩ L q γ (R N ) as in part (ii) except that uniqueness is guaranteed only among functions u ∈ C [0, T ]; L q (R N ) ∩ L q γ (R N ) which also verify t N 2 ( 1 q -1 r ) ∥u(t)∥ L r γ , t N 2 ( 1 q -1
r ) ∥u(t)∥ r are bounded on (0, T ], where r is as above (we replace [0, T ] by (0, T ] if q = ∞ and u satisfies

lim t→0 ∥u(t) -e t∆ u 0 ∥ L ∞ ∩L ∞ γ (R N ) = 0). Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax (∥u(t)∥ q + ∥u(t)∥ L q γ ) = ∞. Furthermore, ∥u(t)∥ L q ∩L q γ ≥ C (T max -t) N 2q -1 α , ∀ t ∈ [0, T max ), (B. 10
)
where C is a positive constant.

Proof of Theorem B.3. We will just give the new elements of the proof.

(i)-(ii) By the Hölder inequality and Proposition 3.1 with γ = l α = µ, q 1 = q/(α + 1), q 2 = q, for each t > 0 and if q > N (α+1)

N -γ , q ≤ ∞, K t,l : L q ∩ L q γ -→ L q ∩ L q γ is locally Lipschitz and, since

q > N (α+1) N -γ > α + 1, ∥K t,l (u) -K t,l (v)∥ q ≤ Ct -N 2 ( α+1 q -1 q ) | • | l (|u| α u -|v| α v) q/(α+1) ≤ Ct -N α 2q (∥u∥ α L q γ + ∥v∥ α L q γ )∥u -v∥ q ≤ 2CM α t -N α 2q ∥u -v∥ q ,
for max(∥u∥ q , ∥u∥ L q γ ) ≤ M and max(∥v∥ q , ∥v∥ L q γ ) ≤ M. We have also, that t

-N α 2q ∈ L 1 loc (0, ∞), since q > q c = N α 2 .
Then the proof follows by [53, Theorem 1, p. 279]. (iii) We choose r > q satisfying (B.4). We consider K > 0, T > 0, M > 0 such that

K + CM α+1 T 1-N α 2q ≤ M, (B.11)
where C is a positive constant given below. We will show that there exists a unique solution u of

(A.3) such that u ∈ C [0, T ]; L q (R N ) ∩ L q γ (R N ) and u ∈ C (0, T ]; L r (R N ) ∩ L r γ (R N ) with ∥u∥ = max sup t∈[0,T ] ∥u(t)∥ q , sup t∈[0,T ] ∥u(t)∥ L q γ , sup t∈(0,T ] t β ∥u(t)∥ L r γ , sup t∈(0,T ] t β ∥u(t)∥ r ≤ M.
The proof is based on a contraction mapping argument in the set

Y q,γ M,T = {u ∈ C [0, T ]; L q (R N ) ∩ L q γ (R N ) ∩ C((0, T ]; L r ∩ L r γ ) : ∥u∥ ≤ M }.
Endowed with the metric d(u, v) = ∥u -v∥, Y q,γ M,T is a nonempty complete metric space. We note that for u

0 ∈ L q , ∥e t∆ u 0 ∥ r ≤ Ct -N 2 ( 1 q -1 r ) ∥u 0 ∥ q = Ct -β ∥u 0 ∥ q .
The condition on initial data max(∥u 0 ∥ q , ∥u 0 ∥ L q γ ) ≤ K implies that t β ∥e t∆ u 0 ∥ L r γ ≤ K, t β ∥e t∆ u 0 ∥ L r ≤ K. We will show that F u 0 defined in (2.11) is a strict contraction on Y q,γ M,T . Using Proposition 3.1, that is e t∆ : L q → L r , for the first term and e t∆ : L r α+1 → L r , for the second term, we have

t β ∥F φ (u)(t) -F ψ (v)(t)∥ r ≤ t β ∥e t∆ (φ -ψ)∥ r + t β t 0 ∥e (t-s)∆ [|.| l |u| α u(s) -|v| α v(s) ]∥ r ds ≤ ∥φ -ψ∥ q + Ct β t 0 (t -s) -N 2 ( α+1 r -1 r ) |.| αγ |u| α u(s) -|v| α v(s) r α+1 ds ≤ ∥φ -ψ∥ q + 2(α + 1)CM α t β t 0 (t -s) -N α 2r s -β(α+1) ds d(u, v) ≤ ∥φ -ψ∥ q + 2(α + 1)CM α t 1-N α 2q 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ d(u, v) ≤ ∥φ -ψ∥ L q + C 3 M α T 1-N α 2q d(u, v),
where

C 3 = 2(α + 1)C 1 0 (1 -σ) -N α 2r σ -β(α+1) dσ < ∞.
Using Proposition 3.1, that is e t∆ : L q → L q , for the first term and e t∆ : L r α+1 → L q , for the second term, we have

∥F φ (u)(t) -F ψ (v)(t)∥ q ≤ ∥e t∆ (φ -ψ)∥ q + t 0 ∥e (t-s)∆ [|.| l |u| α u(s) -|v| α v(s) ]∥ q ds ≤ ∥φ -ψ∥ q + C t 0 (t -s) -N 2 ( α+1 r -1 q ) |.| αγ |u| α u(s) -|v| α v(s) r α+1 ds ≤ ∥φ -ψ∥ q + 2(α + 1)CM α t 0 (t -s) -N 2 ( α+1 r -1 q ) s -β(α+1) ds d(u, v) ≤ ∥φ -ψ∥ q + 2(α + 1)CM α t 1-N α 2q 1 0 (1 -σ) -N 2 ( α+1 r -1 q ) σ -β(α+1) dσ d(u, v) ≤ ∥φ -ψ∥ L q + C 4 M α T 1-N α 2q d(u, v).
where

C 4 = 2(α + 1)C 1 0 (1 -σ) -N 2 ( α+1 r -1 q ) σ -β(α+1) dσ < ∞. From the above estimates, it follows that d(F φ (u), F ψ (v)) ≤ ∥φ -ψ∥ L q ∩L q γ + CM α T 1-N α 2q d(u, v), (B.12)
where C = max(C 1 , C 2 , C 3 , C 4 ). The rest of the proof follows similarly as above and as in [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. □

We have also the following result. Let T max (φ, L q ∩ L q γ ) denotes the existence time of the maximal solution of (A.3) with initial data φ ∈ L q ∩ L q γ . Then we have the following.

(i) If φ ∈ L q ∩ L q γ , then for t ∈ (0, T max (φ, q)) , u(t) ∈ L ∞ ∩ L ∞ γ . (ii) If φ ∈ L p ∩ L p γ ∩ L q ∩ L q γ , N N -γ < q < p ≤ ∞ and q > q c . Then T max (φ, L p ∩ L p γ ) = T max (φ, L q ∩ L q γ ).

Proof. (i) Let φ ∈ L q (R N ), q > q c and q > N N -γ . Let r and β be as above and (B.5). Let p be such that r < p ≤ ∞. Hence p > q,

0 ≤ 1 p < γ N + α + 1 r < 1, 1 p < γ N + 1 q < 1,
and for 0 < T < T max (φ, q), we have ∥u(t)∥ p ≤ ∥e t∆ φ∥ p + C t 0 (t -σ) (1 -σ)

-N 2 ( α+1 r -1 p ) ∥u(σ)∥ α L r γ ∥u(σ)∥ r dσ ≤ (4πt) -N 2 ( 1 q -1 p ) ∥φ∥ q + Ct
-N 2 ( α+1 r -1 p ) σ -β(α+1) dσ ≤ (4πt) -N 2 ( 1 q -1 p ) ∥φ∥ q + M α+1 Ct 1-N 2 ( α+1 q -1 p ) 1 0
(1 -σ)

-N 2 ( α+1 r -1 p ) σ -β(α+1) dσ.

Hence, using (C.1) and (C.2), we have l α < ν < γ, 0 < ν < ν(α + 1) -l < N, 0 < γ < ν(α + 1) -l < N.

Let now r > q be such that 1 q(α + 1) -ν(α + 1) -l -γ N (α + 1) < 1 r < N -ν(α + 1) + l N (α + 1) . This is possible by (C.3). Hence, we have

1 r < α + 1 r + ν(α + 1) -l -ν N < α + 1 r + ν(α + 1) -l N < 1, 1 q < α + 1 r + ν(α + 1) -l -γ N < α + 1 r + ν(α + 1) -l N < 1.
That is, by [8, Lemma 2.1] e t∆ : L q γ → L q γ , is bounded and we may apply Proposition 3.1, so that the maps e t∆ : L q γ → L r ν , e t∆ : L r/(α+1) ν(α+1)-l → L q γ and e t∆ : L r/(α+1) ν(α+1)-l → L r ν are bounded. Let us introduce βl = N 2

1 q - 1 r + γ -ν 2 .
Hence βl > γ -ν 2 > 0.

We have, βl (α + 1) = N 2 α + 1 q -α + 1 r + (α + 1) γ -ν 2

≤ N 2 α + 1 q - 1 q + ν(α + 1) -l -γ N + (α + 1) γ -ν 2 = N α 2q + αγ 2 - l 2 < 1.
We also have

N α 2r + να 2 - l 2 < N α 2q + γα 2 - l 2 < 1, N 2 α + 1 r - 1 q + ν(α + 1) -l -γ 2 < N α 2r + να 2 - l 2 < 1.
These last three estimates are crucial to the local existence argument below. We note that if l > 0, we may take ν(α + 1) -l = γ, r = (α + 1)q. With the above choice of the parameters, we can show the following local well-posedness result. Let γ be satisfying (C.2) and q c (γ, l) be given by (C.3). Then we have the following.

(i) If γ(α + 1) < N + l and q is such that q > N (α + 1) N + l -γ(α + 1)

, q > q c (γ, l) and q ≤ ∞, then equation (A.3) is locally well-posed in L q γ (R N ). More precisely, given u 0 ∈ L q γ (R N ), then there exist T > 0 and a unique solution u ∈ C [0, T ]; L q γ (R N ) of (A.3) (we replace [0, T ] by (0, T ] if q = ∞ and u satisfies lim t→0 ∥u(t) -e t∆ u 0 ∥ L ∞ γ (R N ) = 0). Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax ∥u(t)∥ L q γ = ∞. (ii) Assume that q > q c (γ, l) with N N -γ < q ≤ ∞. It follows that equation (A.3) is locally wellposed in L q γ (R N ) as in part (i) except that uniqueness is guaranteed only among functions u ∈ C [0, T ]; L q γ (R N ) which also verify t βl ∥u(t)∥ L r ν , is bounded on (0, T ], where r and ν are as above (we replace [0, T ] by (0, T ] if q = ∞ and u satisfies lim t→0 ∥u(t) -e t∆ u 0 ∥ L ∞ γ (R N ) = 0). Moreover, u can be extended to a maximal interval [0, T max ) such that either T max = ∞ or T max < ∞ and lim t→Tmax ∥u(t)∥ L q γ = ∞. Furthermore,

∥u(t)∥ L q γ ≥ C (T max -t) N 2q + γ 2 -2+l 2α , ∀ t ∈ [0, T max ), (C.5)
where C is a positive constant.

Proof. (i) Using the inequality (4.4), and Proposition 3.1 that is e t∆ : L q α+1 (α+1)γ-l → L q γ is bounded, for each t > 0 we have that K t,l : L q γ -→ L q γ is locally Lipschitz with

∥K t,l (u) -K t,l (v)∥ L q γ ≤ Ct -N 2 ( α+1 q -1 q )-αγ-l 2 ∥|u| α u -|v| α v∥ L q α+1 (α+1)γ ≤ Ct -N α 2q -αγ 2 + l 2 (∥u∥ α L q γ + ∥v∥ α L q γ )∥u -v∥ L q γ ≤ 2CM α t -N α 2q -αγ 2 + l 2 ∥u -v∥ L q γ ,
for ∥u∥ L q γ ≤ M and ∥v∥ L q γ ≤ M. The rest of the proof is similar to that of Theorems 4.1 and B.1. (ii) For u 0 ∈ L q γ we have ∥e t∆ u 0 ∥ L r ν ≤ Ct

-N 2 ( 1 q -1 r )-γ-ν
2 ∥u 0 ∥ L q γ = Ct -βl ∥u 0 ∥ L q γ . We choose K > 0, T > 0, M > 0 such that

K + CM α+1 T 1-N α 2q -γα 2 + l 2 ≤ M, (C.6)
where C is a positive constant. We will show that there exists a unique solution u of (A. 2) Similar results, using scaling argument, seems to be proved in [START_REF] Li | Life span and secondary critical exponent for degenerate and singular parabolic equations[END_REF][START_REF] Li | Cauchy problems of pseudo-parabolic equations with inhomogeneous terms[END_REF] for related equations, but only for small λ, q = ∞ and positive initial data.

3) If φ ∈ L q ∩ L q γ , or φ ∈ L q l + /α ∩ L q γ , where l + = max(l, 0) then (C.7) is better than (A.6) and (B.9) for 0 < λ < 1. Let ω ∈ L ∞ (R N ) be homogeneous of degree 0, ω ≥ 0 and ω ̸ ≡ 0, φ be given by (1.16) and φ be given by (1.20). Let 0 < γ < N be such that l α < γ, and φ ∈ L q γ (R N ), where N N -γ < q ≤ ∞. Then we have the following. , 0 < λ < 1.

To prove Proposition C.3, we use a scaling argument. We recall the definition of the dilation operators D µ φ = φ(µ •), µ > 0. It is clear that if u is a solution of the equation (A.1) then for any µ > 0, u µ is also a solution of (A.1), where u µ (t, x) = µ 2+l α u(µ 2 t, µx). Hence, σ in (6.1) is given by σ = 2 + l α .

So that, for λ = µ γ-2+l α , (6. T max (φ).

Similarly, if µ γ D µ φ ≥ φ, for some µ > 0, and T max (φ) < ∞, we have that

T max (λφ) ≤ λ -( 2+l 2α -γ 2 ) -1
T max (φ).

Proof of Proposition C.3. The condition (C.8) implies that T max (λφ) < ∞ as well as T max (λ φ) < ∞, and T max (λ φ) < ∞, for any λ > 0. See [START_REF] Qi | The critical exponents of parabolic equations and blow-up in R n[END_REF].

(i) By comparison argument it suffices to give the proof for T max (λ φ). We have that

µ γ D µ φ ≥ φ, µ < 1.
Since γ < (2 + l)/α and λ = µ γ-2+l α then µ < 1 is equivalent to λ > 1. By the above calculations,

T max (λφ 1 ) ≤ Cλ -( 2+l 2α -γ 2 )
-1

, λ > 1.

(ii) By comparison argument it suffices to give the proof for T max (λ φ). We have that

µ γ D µ φ ≥ φ, µ > 1.
Since γ < (2 + l)/α and λ = µ γ-2+l α then µ > 1 is equivalent to λ < 1. Then by the above calculations,

T max (λ φ) ≤ Cλ -( 2+l 2α -γ 2 ) -1 , λ < 1.
This completes the proof of the proposition. □ Remark 20. We may take q = ∞ in Proposition C. , as λ → 0. This shows that, for large initial data the life-span increases as the power l increases, while, for small initial data the life-span decreases as the power l increases.

Theorem 1 . 6 .

 16 Let N ≥ 1 and α > 0. Let φ ∈ L ∞ (Ω), φ ≥ 0 and φ ̸ ≡ 0. It follows that the existence time for the maximal solution of (1.1)-(1.2) with u 0 = λφ satisfies T max (λφ) < ∞ for λ > 0 sufficiently large and lim sup λ→∞ λ α T max (λφ) ≤ 1 α∥φ∥ α ∞ .

1 )

 1 Theorems 1.10 and 1.11 improve the results of [50, Theorem 1.10, Proposition 4.5] by removing the condition (N -2)α < 4. Also the conditions on the lim inf |x|→∞ |x| γ+m x 1 x 2 •••xm φ(x) > 0 or on the lim inf |x|→0 |x| γ+m x 1 x 2 •••xm φ(x) > 0 are not required here. 2) Theorem 1.10 (respectively Theorem 1.11) together with (1.25) show that

. 6 )

 6 Plugging (3.5) and (3.6) in (3.4) we get (3.2).

  Theorem 1.10. The proof follows similarly as that of Theorem 1.7, replacing R N by Ω m , |x| -γ by ψ 0 hence γ by γ + m, and using [35, Proposition 4.1 (ii), p. 359], for the convergence. □ Proof of Theorem 1.11. The proof follows similarly as that of Theorem 1.9 by replacing R N by Ω m , |x| -γ by ψ 0 , hence γ by γ + m, using [50, Theorem 1.7] for the blow up of the solution and [35, Proposition 4.1 (ii), p. 359], for the convergence.□

  and Theorem 1.1 and Theorem 1.6 give the precise order of magnitude of T max (λ φ) as λ → ∞.

  7) such that a(x)| • | -l ∈ L ∞ (R N ) and with initial data u 0 = λφ, λ > 0, constructed by [2, Theorem 1.1, p.117 ] and [2, p. 142] (we replace [0, T max (λφ)) by (0, T max (λφ)) if q = ∞). Then (A.6) holds for all λ > 0.Corollary A.2 includes many known results. We will compare our results with those of[40]. For this we restrict ourselves to the case where a is positive and Hölder continuous as assumed in[40].Also, it is supposed in [40] that φ ∈ C b (R N ), φ ≥ 0. For λ > 0 small,two classes of initial data are considered in [40]. The first class is for φ dominated by a Gaussian. It is shown in [40, Theorem 1 (i), p. 33] that if 0 < α < (2 + l)/N, then

  The estimates, as stated in [40, Theorem 2 (i)-(a), (ii)-(a), (iii)-(a), pp. 33-34], reads

Proposition B. 4 .

 4 Let α > 0 and let 0 < γ := l/α < N . Assume the hypotheses of Theorem B.3.

1 -N 2 ( α+1 r - 1 p

 121 )-β(α+1) sup s∈(0,T ] s βα ∥u(s)∥ α L r γ × sup s∈(0,T ] s β ∥u(s)∥ r 1 0

Theorem C. 1 .

 1 Let N ≥ 1 be an integer, α > 0, -min(2, N ) < l and l N < α. (C.4)

19 . 1 )

 191 3) such that u ∈ C [0, T ]; L q γ (R N ) and u ∈ C (0, T ]; L r ν (R N ) with ∥u∥ = max sup t∈[0,T ] ∥u(t)∥ L q γ , sup t∈(0,T ] t βl ∥u(t)∥ L r ν ≤ M.The proof is based on a contraction mapping argument in the setY q,γ M,T = {u ∈ C [0, T ]; L q γ (R N ) ∩ C((0, T ]; L r ν ) : ∥u∥ ≤ M }.Remark Corollary C.2 answerers a problem left open in [40]. In fact, when l > 0 only exponentially decaying initial data are considered in [40].

Proof of Corollary C. 2 .

 2 The proof follows using (C.6) and is similar to that of Theorem 1.1, so we omit the details. □ We complement Corollary C.2 by the following upper bound estimates. Proposition C.3 (Upper bounds of life-span for Hardy-Hénon equations). Let N ≥ 1 be an integer α > 0 and -min(2, N ) < l. Assume that

( i )

 i If φ ≥ φ then T max (λφ) ≤ Cλ -( If φ ≥ φ then T max (λφ) ≤ Cλ -( 2+l 2α -

2 ) - 1 ]

 21 T max (λφ) = T max (λµ 2+l α D µ φ) = T max (µ γ D µ φ).(C.9)Let 0 < γ < (2 + l)/α. Let φ be a nonnegative function, satisfying µ γ D µ φ ≤ φ, for some µ > 0.Then, since λ = µ γ-2+l α , we have by comparison argument (see[START_REF] Wang | On the Cauchy problem for reaction-diffusion equations[END_REF] Theorem 2.4, p. 564]) and (C.9) thatT max (λφ) ≥ λ -( 2+l 2α -γ 2 ) -1

3 .-γ 2 ) - 1 ,

 321 In particular, combining Corollary C.2 and Proposition C.8, we have T max (λ φ) ∼ λ -( 2+l 2α as λ → ∞, and T max (λ φ) ∼ λ -(

Life-span estimates via nonlinear scaling

In this section we show how certain scaling arguments can give upper (and lower) life-span bounds for solutions of (2.1) on R N . Similar arguments can be used on sectors of R N . The previous section likewise used scaling arguments, but only in regard to properties of e t∆ φ. In this section, we use nonlinear scaling arguments, which can then be adapted to other equations which are scale invariant. Some of the results in this section are the same as in the previous section, but obtained by a different method.

We begin with some observations in a general context. We consider an evolution partial differential equation defined either on R N or on some domain Ω which is a cone, i.e. if x ∈ Ω then µx ∈ Ω for all µ > 0. We also suppose that the set of solutions of the evolution equation is invariant under the transformation

In other words, u is a solution if and only if u µ is a solution for all µ > 0. If u has initial value

Now let us suppose that φ has certain properties with respect to a scaling different from that of the equation, for example µ γ D µ φ where γ ̸ = σ. If so, we may set

In the simplest case, µ γ D µ φ ≡ φ, i.e. φ is homogeneous of degree -γ, we have therefore the following formal proposition.

Proposition 6.1. Let Ω ⊂ R N be a domain which is also a cone. Suppose that the solutions of an evolution equation (the set of trajectories of a dynamical system over Ω) are invariant under the transformation (6.1).

for all λ > 0.

In the case of the nonlinear heat equation, σ = 2 α and so (6.2) and (6.3) become

We immediately deduce the following.

Corollary 6.2. Let T max (u 0 ) denote the maximal solution to (2.1) on R N with initial value u 0 .

(ii) There exists T3 ∈ (T max (Φ), T max (ω|

2

2

Remark 15. Corollary 6.6 shows that the asymptotic behavior of the life-span as λ → ∞ is determined by the singularity of the initial data and when λ → 0 it is determined by the decay rate at infinity of the initial value.

Proof of Corollary 6.6.

1) Analysis of

and

for all µ > 0.

We claim that

where the limits are in

where the limits are in

Our purpose in the appendix is to estimate the life-span of solutions for the nonlinear Hardy-Hénon parabolic equations

) < l and with initial value

A mild solution of the problem (A.1)-(A.2) is a solution of the integral equation

and it is in this form that we consider problem (A.1)-(A.2).

In this first part of the appendix we consider the case l < 0, that is the Hardy case. The problem

and r > q satisfies 1 q(α + 1)

See [2, Theorem 1.1, p. 117] and [START_REF] Slimene | Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems[END_REF]. This solution can be extended to a maximal solution defined on [0, T max (u 0 )). We have obtained the following.

Theorem A.1 (The nonlinear Hardy parabolic equations). Let N ≥ 1, -min(2, N ) < l < 0, α > 0, and q c (l) be given by (A.4).

γ for all t ∈ 0, T max (φ, q) . The result for general p > q follows by iteration. Hence

) then the life-span of (A.1) with initial data λφ satisfies

for all λ > 0, which gives a power of λ not depending on l, unlike the case l < 0.

Appendix C. The Hardy-Hénon equations with decaying initial data

In this part of the appendix, we investigate lower bound estimates for life-span for the solutions of the equation (A.1) with initial data having some decay. As in Section 4, we work in L q γ with γ > 0, γ > l/α. This allows us to obtain a lower bound of the life span for initial data having more decay than l/α, if l > 0. We consider the Duhamel formulation of (A.1)-(A.2), that is the equation (A.3) and suppose that

Let γ be such that

q c (γ, l) is the critical exponent of (A.1) for initial data in L q γ . The condition (C.3) can be reformulated as follows:

Let 0 < ν < γ be such that

Endowed with the metric d(u, v) = ∥u-v∥, Y q,γ M,T is a nonempty complete metric space. We consider u 0 such that ∥u 0 ∥ L q γ ≤ K and we estimate as follows:

and similarly for the contraction. The other estimates can be handled similarly as above, see also [START_REF] Ben Slimene | Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations[END_REF]. So we omit the details. This completes the proof of Theorem C.1. □ Remark 18.

1) We can take γ = max(0, l α ) in Theorem C.1 as well as l = 0, it is then a generalization of Theorems 4.1 and B.1.

2) See [8, Theorem 1.13] for related results. The range of the values of q in (ii) are larger than in [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF], while (i) is essentially contained in [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF] which we give for completeness. Also the methods are different. In fact, we work in an auxiliary space L r ν , for some r and ν while in [START_REF] Chikami | Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces[END_REF] some auxiliary spaces L q ν for some ν but q is fixed are considered. 3) If q = ∞ we may replace L ∞ γ by the space obtained by the closure, with respect to the L ∞ γ -topology, of D(R N ), the space of compactly supported C ∞ (R N ) functions. For initial data in this sub-space of L ∞ γ the result holds on [0, T ] instead of (0, T ]. 4) Using argument of [START_REF] Slimene | Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems[END_REF], we can show that uniqueness in the part (ii) of Theorem C.1 holds in u ∈ C [0, T ]; L q γ (R N ) ∩ C (0, T ]; L r ν (R N ) .

Theorem C.1 gives the following.

Corollary C.2 (Hénon parabolic equations with decaying initial data). Let N ≥ 1 be an integer, α > 0 and -min(2, N ) < l < N α. If φ ∈ L q γ (R N ), where

Then the life-span of (A.3) with initial data λφ satisfies

for all λ > 0, where C = C(α, q, l, γ, N ) > 0 is a constant.