Elissar Nasreddine
email: elissar.nasreddine@math.univ-toulouse.fr

Global existence of solutions to a parabolic-elliptic chemotaxis system with critical degenerate diffusion

Keywords: Chemotaxis, Keller-Segel model, Parabolic equation, Elliptic equation, Global existence, Uniqueness

This paper is devoted to the analysis of non-negative solutions for a degenerate parabolic-elliptic Patlak-Keller-Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, there by completing previous results on finite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.

Introduction

Chemotaxis is the movement of biological organisms oriented towards the gradient of some substance, called the chemoattractant. The Patlak-Keller-Segel (PKS) model (see [START_REF] Keller | Initiation of slide mold aggregation viewed as an instability[END_REF], [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] and [START_REF] Patlak | Random walk with persistence and external bias[END_REF]) has been introduced in order to explain chemotaxis cell aggregation by means of a coupled system of two equations: a drift-diffusion type equation for the cell density u, and a reaction diffusion equation for the chemoattractant concentration ϕ. It reads

(P KS)            ∂ t u = div(∇u m -u • ∇ϕ)
x ∈ Ω, t > 0, -∆ϕ = u-< u >

x ∈ Ω, t > 0, < ϕ(t) > = 0 t > 0, ∂ ν u = ∂ ν ϕ = 0

x ∈ ∂Ω, t > 0, u(0, x) = u 0 (x)

x ∈ Ω,

where Ω ⊂ R N is an open bounded domain, ν the outward unit normal vector to the boundary ∂Ω and m ≥ 1. An important parameter in this model is the total mass M of cells, which is formally conserved through the evolution:

M =< u >= 1 |Ω| Ω u(t, x) dx = 1 |Ω| Ω u 0 (x) dx. (2)
Several studies have revealed that the dynamics of (1) depend sensitively on the parameters N , m and M . More precisely, if N = 2 and m = 1, it is well-known that the solutions of (1) may blow up in finite time if M is sufficiently large (see [START_REF] Patlak | Random walk with persistence and external bias[END_REF][START_REF] Nagai | Blow-up of radially symmetric solutions to a chemotaxis system[END_REF]) while solutions are global in time for M sufficiently small [START_REF] Patlak | Random walk with persistence and external bias[END_REF], see also the survey articles [START_REF] Blanchet | On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher[END_REF][START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF].

The situation is very different when m = 1 and N = 2. In fact, if N = 1, there is global existence of solutions of (1) whatever the value of the mass of initial data M , see [START_REF] Cieślak | Finite-time blow-up in a quasilinear system of chemotaxis[END_REF] and the references therein. If N ≥ 3, for all M > 0, there are initial data u 0 with mass M for which the corresponding solutions of (1) explode in finite time (see [START_REF] Nagai | Blow-up of radially symmetric solutions to a chemotaxis system[END_REF]). Thus, in dimension N ≥ 3 and m = 1, the threshold phenomenon does not take place as in dimension 2, but we expect the same phenomenon when N ≥ 3 and m is equal to the critical value m = m c = 2(N -1)

N

. More precisely, we consider a more general version of [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF] where the first equation of ( 1) is replaced by ∂ t u = div(φ(u) ∇uu ∇ϕ), t > 0, x ∈ Ω, and the diffusitivity φ is a positive function in C 1 ([0, ∞[) which does not grow to fast at infinity. In [START_REF] Cieślak | Finite-time blow-up in a quasilinear system of chemotaxis[END_REF], the authors proved that there is a critical exponent such that, if the diffusion has a faster growth than the one given by this exponent, solutions to (1) (with φ(u) instead of mu m-1 ) exist globally and are uniformly bounded, see also [START_REF] Calvez | Volume effect in the Keller-Segel model: energy estimates preventing blow-up[END_REF][START_REF] Kowalczyk | Preventing blow-up in a chemotaxis model[END_REF] for N = 2. More precisely, the main results in [START_REF] Cieślak | Finite-time blow-up in a quasilinear system of chemotaxis[END_REF] read as follows:

• If φ(u) ≥ c(1 + u) p for all u ≥ 0 and some c > 0 and p > 1 -2 N then all solutions of (1) are global and bounded.

• If φ(u) ≤ c(1 + u) p for all u ≥ 0 and some c > 0 and p < 1 -2

N then there exist initial data u 0 such that lim t→T ||u(., t)|| ∞ = ∞, for some finite T > 0.

Except for N = 2, the critical case m = 2(N -1) N is not covered by the analysis of [START_REF] Cieślak | Finite-time blow-up in a quasilinear system of chemotaxis[END_REF]. Recently, Cieślak and Laurençot in [START_REF] Cieślak | Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system[END_REF] show that if φ(u) ≤ c(1 + u) 1-2 N and N ≥ 3, there are solutions of (1) blowing up in finite time when M exceeds an explicit threshold. In order to prove that, when N ≥ 3 and m = 2 (N -1)

N

, we have a threshold phenomenon similar to dimension N = 2 with m = 1, it remains to show that solutions of (1) are global when M is small enough. The goal of this paper is to show that this is indeed true, see Theorem 2.2 below.

By combining Theorem 2.2 with the blow-up result obtained in [START_REF] Cieślak | Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system[END_REF], we conclude that, for N ≥ 3 and m = 2(N -1)

N

, there exists 0 < M 1 ≤ M 2 < ∞ such that the solutions of (1) are global if the mass M of the initial data u 0 is in [0, M 1 ), and may explode in finite time if M > M 2 . An important open question is whether M 1 = M 2 when Ω is a ball in R N and u 0 is a radially symmetric function. Notice that, in the radial case, this result is true when N = 2 and m = 1, and the threshold value of the mass for blow-up is M 1 = M 2 = 8π, see [START_REF] Calvez | Volume effect in the Keller-Segel model: energy estimates preventing blow-up[END_REF][START_REF] Nagai | Blow-up of radially symmetric solutions to a chemotaxis system[END_REF][START_REF] Nagai | Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains[END_REF][START_REF] Perthame | PDE models for chemotactic movements. Parabolic, hyperbolic and kinetic[END_REF]. Again, for N = 2 and m = 1, but for regular, connected and bounded domain, it has been shown that

M 1 = 4π = M 2
2 (see [START_REF] Nagai | Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains[END_REF][START_REF] Nagai | Blow-up of radially symmetric solutions to a chemotaxis system[END_REF] and the references therein). Such a result does not seem to be known for N ≥ 3 and m = 2(N -1) N .

Still, in the whole space Ω = R N when the equation for ϕ in (1) is replaced by the Poisson equation ϕ = E N * u, with E N being the Poisson kernel, it has been shown in [START_REF] Dolbeault | Optimal critical mass in the two-dimensional Keller-Segel model in R 2[END_REF][START_REF] Blanchet | Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions[END_REF][START_REF] Bedrossian | Local global well-possedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion[END_REF][START_REF] Sugiyama | Global existence in sub-critical cases and finite time blow-up in supercritical cases to degenerate Keller-Segel systems[END_REF][START_REF] Sugiyama | Time global existence and asymptotic behavior for solutions to degenerate quasi-linear parabolic systems of chemotaxis[END_REF][START_REF] Bedrossian | Global Existence and Finite Time Blow-Up for Critical Patlak-Keller-Segel Models with Inhomogeneous Diffusion[END_REF] that:

• When N ≥ 3 and 1 ≤ m < 2 -2
N , this modified version of (1) has a global weak solution if M = u 0 1 is sufficiently small, while finite time blow-up occurs for some initial data with sufficiently large mass.

• When N ≥ 2 and m > 2 -2
N , this modified version of (1) has a global weak solution whatever the value of M .

• When N ≥ 2 and m = 2 -2 N , there is a threshold mass M c > 0 such that solutions to this modified version of (1) exist globally if M = u 0 1 ≤ M c , and might blow up in finite time if M > M c .

From now on, we assume that N ≥ 3 and m = 2(N -1) N .

Main Theorem

Throughout this paper , we deal with weak solutions of (1). Our definition of weak solutions now reads:

Definition 2.1. Let T ∈ (0; ∞]. A pair (u, ϕ) of functions u : Ω × [0, T ) -→ [0, ∞), ϕ : Ω × [0, T ) -→ R is called a weak solution of (1) in Ω × [0, T ) if • u ∈ L ∞ ((0, T ); L ∞ (Ω)); u m ∈ L 2 ((0, T ); H 1 (Ω)) and < u >= M .
• ϕ ∈ L 2 ((0, T ); H 1 (Ω)) and < ϕ >= 0.

• (u, ϕ) satisfies the equation in the sense of distributions ; i.e,

- T 0 Ω (∇u m • ∇ψ -u∇ϕ • ∇ψ -u ∂ t ψ) dxdt = Ω u 0 (x) ψ(0, x) dx, T 0 Ω ∇ϕ • ∇ψ dxdt = T 0 Ω (u -M ) ψ dxdt,
for any continuously differentiable function ψ ∈ C 1 ([0, T ] × Ω) with ψ(T ) = 0 and T > 0.

For ϕ ∈ H 1 (Ω) satisfying < ϕ >= 0, we denote by C s the Sobolev constant where

||∇ϕ|| 2 ≥ C s ||ϕ|| 2 * , where 2 * = 2N N -2 . ( 3
)
The main theorem gives the existence and uniqueness of a time global weak solution to (1) which corresponds to a degenerate version of the "Nagai model" for the semi-linear Keller-Segel system, when u 0 ∈ L ∞ (Ω) and the initial data is assumed to be small. Theorem 2.2. Define

M * := 2 C 2 s (m -1) |Ω| 2 N N 2 , ( 4
)
where C s is the Sobolev constant in (3).

Assume that u 0 is nonnegative function in L ∞ (Ω), which satisfies

||u 0 || 1 < M * . (5)
Then the equation (1) has a global weak solution (u, ϕ) in the sense of Definition 2.1. Moreover, if we assume that

ϕ ∈ L ∞ ((0, T ); W 2,∞ (Ω)) (6)
for all T > 0 then this solution is unique.

In order to prove the previous theorem, we introduce the following approximated equations

(KS) δ        ∂ t u δ = div (∇(u δ + δ) m -u δ ∇ϕ δ ) x ∈ Ω, t > 0, -∆ϕ δ = u δ -< u δ > x ∈ Ω, t > 0, ∂ ν u δ = ∂ ν ϕ δ = 0 x ∈ ∂Ω, t > 0, u δ (0, x) = u 0 (x) x ∈ Ω,
where δ ∈ (0, 1), and we show that under a smallness condition on the mass of initial data, the Liapunov function

L δ (u, ϕ) = Ω (b δ (u) + 1 2 |∇ϕ δ | 2 -u δ ϕ δ ) dx,
yields the L m bound of u δ (t) independent of δ. Then using Gagliardo-Nirenberg and Poincaré inequalities, we obtain for p > m, the L p bound for u δ (t) independent of δ.

As a consequence of Sobolev embedding theorem, we improve the regularity of ϕ δ . And thus, under the same assumptions on the initial data, Moser's iteration technique yields the uniform bound of u δ . Then, thanks to the local well-posedness result [8, Theorem 3.1] we obtain the existence of a global solution of (KS) δ . The existence of solutions stated in Theorem 2.2 is then proved using a compactness method; for that purpose we show an additional estimate on ∂ t u m δ which, together with the already derived estimates, guarantees the compactness in space and time of the family (u δ ) δ∈(0,1) . Finally, in the presence of nonlinear diffusion and under some additional regularity assumption on ϕ δ , we prove the uniqueness using a classical duality technique.

Approximated Equations

The first equation of (1) is a quasilinear parabolic equation of degenerate type. Therefore, we cannot expect the system (1) to have a classical solution at the point where u vanishes. In order to prove Theorem 2.2, we use a compactness method and introduce the following approximated equations of (KS):

(KS) δ        ∂ t u δ = div (∇(u δ + δ) m -u δ ∇ϕ δ ) x ∈ Ω, t > 0, -∆ϕ δ = u δ -< u δ > x ∈ Ω, t > 0, ∂ ν u δ = ∂ ν ϕ δ = 0 x ∈ ∂Ω, t > 0, u δ (0, x) = u 0 (x) x ∈ Ω, (7)
where δ ∈ (0, 1).

The main purpose of this section is to construct the time global strong solution of (7).

Existence of global strong solution of (KS) δ

Theorem 3.1. For δ ∈ (0, 1) and T > 0, we consider an initial condition u 0 ∈ L ∞ (Ω), u 0 ≥ 0 and such that ||u 0 || 1 < M * where M * is defined in (4). Then (KS) δ has a global strong solution (u δ , ϕ δ ) which is bounded in L ∞ ((0, T ) × Ω) for all T > 0 uniformly with respect to δ ∈ (0, 1).

The starting point of the proof of Theorem 3.1 is the following local well-posedness result [8, Theorem 1.3]: Lemma 3.2. Let the same assumptions as that in Theorem 3.1 hold. There exists a maximal existence time T δ max ∈ (0, ∞] and a unique solution

(u δ , ϕ δ ) of (KS) δ in [0, T δ max ) × Ω. Moreover, if T δ max < ∞ then lim t→T δ max ||u δ (t, .)|| ∞ = ∞.
In addition < u δ (t) >=< u 0 >= M for all t ∈ [0, T δ max ). To prove Theorem 3.1 we need to prove some lemmas which control L m norm, L p norm and L ∞ norm of the solution u δ of (7).

L

p -estimates, 1 ≤ p ≤ ∞.
Our goal is to show that if ||u 0 || 1 is small enough then all solutions are global in time and uniformly bounded.

Let us first prove the L m bound for u δ . Lemma 3.3. Let the same assumptions as that in Theorem 3.1 hold and (u δ , ϕ δ ) be the nonnegative maximal solution of (KS) δ . Then, u δ satisfies the following estimate

||u δ (t)|| m ≤ C 0 , for all t ∈ [0, T δ max )
and ||u δ (t)|| 1 = ||u 0 || 1 where C 0 is a constant independent of T δ max and δ. Proof. In this proof, the solution to equation ( 7) should be denoted by (u δ , ϕ δ ) but for simplicity we drop the index. Let us define the functional L δ by

L δ (u, ϕ) = Ω (b δ (u) + 1 2 |∇ϕ| 2 -u ϕ) dx, where b δ (u) := u 1 z 1 m(σ + δ) m-1 σ dσ dz, such that b δ (1) = b ′ δ (1) = 0 and b(u) ≥ 0.
According to [START_REF] Horstmann | Lyapunov functions and L p -estimates for a class of reaction-diffusion systems[END_REF] it is a Liapunov functional for (KS) δ . Indeed,

d dt L δ (u(t), ϕ(t)) = Ω b ′ δ (u) ∂ t u dx - Ω ∆ϕ ∂ t ϕ dx - Ω ∂ t u ϕ dx - Ω u ∂ t ϕ dx = Ω ∂ t u (b ′ δ (u) -ϕ) dx - Ω (∆ϕ + u) ∂ t ϕ dx = Ω div m (u + δ) m-1 ∇u -u ∇ϕ (b ′ δ (u) -ϕ) dx - Ω < u(t) > ∂ t ϕ dx = - Ω (m (u + δ) m-1 ∇u -u ∇ϕ) (b ′′ δ (u) ∇u -∇ϕ) dx -M d dt Ω ϕ dx = - Ω u (b ′′ δ (u) ∇u -∇ϕ) 2 dx ≤ 0.
Then, we can conclude that for all t ∈ [0, T δ max ) we have L δ (u(t), ϕ(t)) ≤ L δ (u 0 , ϕ 0 ). Using Sobolev inequality (3), Hölder inequality, and Young inequality we obtain

Ω u ϕ dx ≤ ||ϕ|| 2 * ||u|| 2N N+2 ≤ C -1 s ||∇ϕ|| 2 ||u|| 2N N+2 ≤ 1 2 ||∇ϕ|| 2 2 + C -2 s 2 ||u|| 2 2N N+2
.

Since 2 N N +2 < m, and using interpolation inequality we get,

||u|| 2N N+2 ≤ ||u|| 1 N 1 ||u|| N-1 N m ≤ M 1 N |Ω| 1 N ||u|| m 2 m .
Then,

Ω u ϕ dx ≤ 1 2 ||∇ϕ|| 2 2 + C -2 s 2 M 2 N |Ω| 2 N ||u|| m m .
Substituting this into the Liapunov functional, we find:

L δ (u, ϕ) ≥ Ω (b δ (u) + 1 2 |∇ϕ| 2 ) dx - 1 2 ||∇ϕ|| 2 2 - C -2 s 2 M 2 N |Ω| 2 N ||u|| m m ≥ Ω b δ (u) dx - C -2 s 2 M 2 N |Ω| 2 N ||u|| m m .
We next observe that:

b δ (u) = m u 1 z 1 (δ + s) m-1 s dsdz ≥ m u 1 z 1 s m-2 dsdz ≥ u m m -1 - m m -1 u + 1 ≥ u m m -1 - m m -1 u.
Then:

L δ (u, ϕ) ≥ 1 m -1 ||u|| m m - C -2 s 2 |Ω| 2 N M 2 N ||u|| m m - m m -1 M |Ω| = 1 m -1 - C -2 s 2 M 2 N |Ω| 2 N ||u|| m m - m m -1 M |Ω|.
Let us define ω M by

ω M := 1 m -1 - C -2 s 2 M 2 N |Ω| 2 N = |Ω| 2 N 2 C 2 s (M 2 N * -M 2 N ). Since M = ||u 0 || 1 < M * , then ω M is positive. Finally we get, L δ (u 0 , ϕ 0 )+ m m -1 M |Ω| ≥ L δ (u(t), ϕ(t))+ m m -1 M |Ω| ≥ ω M ||u(t)|| m m for t ∈ [0, T δ max ).
In addition, we can see that L δ (u 0 , ϕ 0 ) ≤ C where C is independent of δ ∈ (0, 1). In fact,

L δ (u 0 , ϕ 0 ) = Ω (b δ (u 0 ) + 1 2 |∇ϕ 0 | 2 -u 0 ϕ 0 ) dx, and, since (δ + s) m-1 ≤ δ m-1 + s m-1 ≤ 1 + s m-1 we obtain b δ (u 0 ) = m u 0 1 z 1 (δ + s) m-1 s dsdz ≤ m u 0 1 z 1 1 + s m-1 s dsdz ≤ m(u 0 ln u 0 -u 0 + 1) + m m -1 u m 0 m -u 0 + 1 .
Using Young inequality we get

L δ (u 0 , ϕ 0 ) ≤ m ||u 0 || 2 2 + m |Ω| + ||u 0 || m m m -1 + m |Ω| m -1 + 1 2 ||∇ϕ 0 || 2 2 + 1 2 ||u 0 || 2 2 + 1 2 ||ϕ 0 || 2 2 .
since u 0 ∈ L ∞ (Ω) and ϕ 0 ∈ H 1 (Ω) we get L δ (u 0 , ϕ 0 ) ≤ C where C is independent of δ and the proof of the lemma is complete.

Thanks to Lemma 3.3, let us now show that for all p > m the L p bound for u δ .

Lemma 3.4. Let the same assumptions as that in Theorem 3.1 hold. Then for all T > 0 and all p ∈ (1, ∞) there exists C(p, T ) independent on δ such that, for all t ∈ [0,

T δ max ) ∩ [0, T ], the solution (u δ , ϕ δ ) to (KS) δ satisfies ||u δ (t)|| p ≤ C(p, T ), ( 8
)
and t 0 Ω (δ + u δ ) m-1 u p-2 δ |∇u δ | 2 dxds ≤ C(p, T ). ( 9
)
To prove the previous lemma we need the following preliminary result [START_REF] Sugiyama | Global existence in sub-critical cases and finite time blow-up in supercritical cases to degenerate Keller-Segel systems[END_REF].

Lemma 3.5. Consider 0 < q 1 < q 2 ≤ 2 * . There is C 1 depending only on N such that

||u|| q 2 ≤ C θ 1 ||u|| θ H 1 (Ω) ||u|| 1-θ q 1 , for u ∈ H 1 (Ω), ( 10
)
with θ = 2N (q 2 -q 1 ) q 2 [(N + 2)q 1 + 2N (1 -q 1 )] ∈ [0, 1].
Proof. For u ∈ H 1 (Ω) we have by Sobolev inequality

||u|| 2 * ≤ C N ||u|| H 1 . (11)
By interpolation inequality we have for 0

< q 1 < q 2 ≤ 2 * ||u|| q 2 ≤ ||u|| θ 2 * ||u|| 1-θ q 1 , (12)
where 1 q 2 = θ(N -2)

2N

+ 1-θ q 1 . Hence, substitute [START_REF] Horstmann | Lyapunov functions and L p -estimates for a class of reaction-diffusion systems[END_REF] into [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] and the lemma is proved. Now, we recall the following generalized Poincaré inequality. Lemma 3.6. For u ∈ H 1 (Ω) we have for 0 < q 1 ≤ 1 the following inequality

||u|| 2 H 1 ≤ C 2 (q 1 ) (||∇u|| 2 2 + ||u|| 2 q 1 ),
where C 2 depends only on Ω and q 1 . Now using the last two lemmas, let us prove Lemma 3.4.

Proof. In this proof, the solution to equation ( 7) should be denoted by (u δ , ϕ δ ) but for simplicity we drop the index. We choose p > 1, K ≥ 0 and we multiply the first equation in ( 7) by (u -K) p-1 + and integrate by parts using the boundary conditions for u and ϕ to see that

1 p d dt ||(u -K) + || p p = -m(p -1) Ω (δ + u) m-1 (u -K) p-2 + |∇u| 2 dx + (p -1) Ω u ∇ϕ (u -K) p-2 + • ∇u dx = -m(p -1) Ω (δ + u -K + K) m-1 (u -K) p-2 + |∇u| 2 dx + (p -1) Ω (u -K + K) ∇ϕ • (u -K) p-2 + ∇u dx ≤ -m(p -1) Ω (u + δ -K) m-1 (u -K) p-2 + |∇u| 2 dx + (p -1) Ω (u -K) p-1 + ∇ϕ • ∇u dx + (p -1)K Ω ∇ϕ (u -K) p-2 + • ∇u dx ≤ -m(p -1) Ω (u + δ -K) m-1 (u -K) p-2 + |∇u| 2 dx - p -1 p Ω (u -K) p + ∆ϕ dx -K Ω (u -K) p-1 + ∆ϕ dx ≤ -m(p -1) Ω (u + δ -K) m-1 (u -K) p-2 + |∇u| 2 dx + (I),
where, thanks to the second equation in [START_REF] Cieślak | Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system[END_REF],

(I) = p -1 p Ω (u -K) p + (u -M ) dx + K Ω (u -K) p-1 + (u -M ) dx = p -1 p ||(u -K) + || p+1 p+1 + p -1 p (K -M )||(u -K) + || p p + K||(u -K) + || p p + K(K -M )||(u -K) + || p-1 p-1 ≤ K 2 ||(u -K) + || p-1 p-1 + 2K ||(u -K) + || p p + ||(u -K) + || p+1 p+1 .
Since for a > 0 and b > 0 we have

a p-1 b ≤ a p+1 + b p+1 2
and a p b ≤ a p+1 + b p+1 then,

(I) ≤ 3||(u -K) + || p+1 p+1 + (2K) p+1 + K p+1 , (13)
and we get

d dt ||(u -K) + || p p ≤ -m(p -1) Ω (u + δ -K) m-1 (u -K) p-2 + |∇u| 2 dx + 3p||(u -K) + || p+1 p+1 + C p K p+1 , (14)
for all t ∈ [0, T δ max ). The term ||(u -K) + || p+1 p+1 can be estimated with the help of Lemma 3.5 and Lemma 3.6. Assuming now that p > 2 we remark that 0

< 2 p+m-1 ≤ 1 and 1 < 2(p+1) p+m-1 = 2N N -2 1+p 1+ Np N-2 ≤ 2N N -2
, then thanks to Lemma 3.5 and Lemma 3.6 we obtain

||(u -K) p+m-1 2 + || 2(p+1) p+m-1 2(p+1) p+m-1 ≤ C(p) ||∇(u -K) p+m-1 2 + || 2(p+1) p+m-1 θ 2 ||(u -K) p+m-1 2 + || 2(p+1) p+m-1 (1-θ) 2 p+m-1 + ||(u -K) p+m-1 2 + || 2(p+1) p+m-1 2 p+m-1 , (15)
where

θ = p + m -1 p + 1 ∈ (0, 1). ( 16
)
Since

||(u -K) p+m-1 2 + || 2(p+1) p+m-1 2(p+1) p+m-1 = Ω (u -K) p+1 + dx = ||(u -K) + || p+1 p+1 , (17)
||(u -K) p+m-1 2 + || 2(p+1) p+m-1 (1-θ) 2 p+m-1 = Ω (u -K) + dx (p+1)(1-θ) = ||(u -K) + || 2 N 1 , (18)
and by Lemma 3.3

||(u -K) + || 1 = u≥K (u -K) dx ≤ 1 K m-1 u≥K K m-1 u dx ≤ ||u|| m m K m-1 ≤ C m 0 K m-1 , (19)
we substitute ( 17), ( 18) and ( 19) into ( 15) and obtain

||(u -K) + || p+1 p+1 ≤ C 3 (p) ||∇(u -K) m+p-1 2 + || 2 2 K -2(m-1) N + K -(m-1)(p+1) . (20)
We may choose K = K * large enough such that

3 p C 3 (p) K -2(m-1) N * ≤ 2 p (p -1) m (m + p -1) 2 , Hence d dt ||(u -K * ) + || p p ≤ C(p) K p+1 * , so that ||(u(t) -K * ) + || p p ≤ C(p) t + ||u 0 || p p , for t ∈ [0, T δ max ). As Ω |u| p dx ≤ u<2K * (2 K * ) p-1 |u| dx + u≥2K * |u -K * + K * | p dx ≤ (2K * ) p-1 M + u≥2K * (2 |u -K * |) p dx, ≤ (2K * ) p-1 M + 2 p ||(u -K * ) + || p p ,
the previous inequality warrants that

||u(t)|| p ≤ C(p, T ), t ∈ [0, T max ) ∩ [0, T ], (21)
where C(p, T ) is a constant independent of δ.

We next take K = 0 in ( 14), integrate with respect to time and use [START_REF] Cieślak | Finite-time blow-up in a quasilinear system of chemotaxis[END_REF] to obtain (9). Thanks to Lemma 3.4, we can improve the regularity of ϕ δ . Lemma 3.7. Let the same assumptions as that in Theorem 3.1 hold, the solution ϕ δ satisfies ||∇ϕ δ (t)|| ∞ ≤ L(T ), t ∈ [0, T δ max ) ∩ [0, T ] where T > 0 and L is a positive constant independent of δ.

Proof. Using standard elliptic regularity estimates for ϕ δ , we infer from Lemma 3.4 that given T > 0, and p ∈ (1, ∞), there is C(p, T ) such that ∈ H 1 (Ω). Then it holds that

||ϕ δ (t)|| W 2,p ≤ C(p) ||u δ (t)|| p ≤ C(p, T ), for t ∈ [0, T max ) ∩ [0, T ].
||u|| r ≤ C 2θ r+m-1 1 ||u|| 1-θ r 4 ||u r+m-1 2 || 2θ r+m-1 H 1 (22) with θ = 3 N (r + m -1) (3N + 2)r + 4N (m -1) ∈ (0, 1). ( 23
)
Proof. For r ≥ 4, we can see that

||u|| r = Ω (u r+m-1 2 ) 2r r+m-1 dx 1 r = ||u r+m-1 2 || 2 r+m-1 2r r+m-1 , and r 2(r + m -1) < 1 < 2r r + m -1 < 2 < 2N N -2 .
By Lemma 3.5,

||u|| r = ||u r+m-1 2 || 2 r+m-1 2r r+m-1 ≤ C θ 1 ||u r+m-1 2 || θ H 1 (Ω) ||u r+m-1 2 || 1-θ r 2(r+m-1) 2 r+m-1 and θ = 2N ( 2r r+m-1 - r 2 (r+m-1) ) 2r r+m-1 2N (1 - r 2(r+m-1) ) + (N + 2) r 2 (r+m-1) = 3N (r + m -1) (3N + 2) r + 4N (m -1)
∈ (0, 1).

In addition, we have

||u r+m-1 2 || r 2(r+m-1) = Ω |u| r+m-1 2 r 2(r+m-1) dx 2(r+m-1) r = ||u|| r+m-1 2 r 4
, and we obtain (22).

We are now in a position to prove the uniform L ∞ (Ω) bound for u δ . Lemma 3.9. Let the same assumptions as that in Theorem 3.1 hold, and (u δ , ϕ δ ) be the nonnegative maximal solution of [START_REF] Cieślak | Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system[END_REF]. For all T > 0, there is C ∞ (T ) such that

||u δ (t)|| ∞ ≤ C ∞ (T ), for all t ∈ [0, T δ max ) ∩ [0, T ],
where C ∞ (T ) is a positive constant independent on δ.

Proof. In this proof we omit the index δ, and we employ Moser's iteration technique developed in [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF][START_REF] Sugiyama | Time global existence and asymptotic behavior for solutions to degenerate quasi-linear parabolic systems of chemotaxis[END_REF] to show the uniform norm bound for u.

We multiply the first equation in ( 7) by u r-1 , where r ≥ 4, and integrate it over Ω. Then, we have

d dt ||u|| r r r = - Ω (∇(u + δ) m -u ∇ϕ) • ∇u r-1 dx = -m(r -1) Ω (u + δ) m-1 u r-2 |∇u| 2 dx + (r -1) Ω u r-1 ∇ϕ • ∇u dx ≤ -m(r -1) Ω u m+r-3 |∇u| 2 dx + (r -1) Ω u r-1 ∇ϕ • ∇u dx.
By Young's inequality and Lemma 3.7,

1 r d dt ||u|| r r ≤ -4m(r -1) (r + m -1) 2 Ω |∇u r+m-1 2 | 2 dx + 2(r -1) ||∇ϕ|| ∞ (r + m -1) Ω u r-m+1 2 |∇u r+m-1 2 | dx ≤ -2m(r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + r -1 2m ||∇ϕ|| 2 ∞ Ω u r-m+1 dx ≤ -2m(r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + C(T ) r Ω u r-m+1 dx.
Using Hölder and Young inequalities and Lemma 3.3 we obtain 1 r

d dt ||u|| r r ≤ -2m (r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + r C(T ) ||u|| m-1 r-1 1 ||u|| r(r-m) r-1 r ≤ -2m (r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + C r + r 2 ||u|| r r , (24)
where we have used that r r-1 r-m ≤ r 2 for r ≥ 4. By Lemma 3.8, we have for r ≥ 4

||u|| r r ≤ C 2 r θ r+m-1 1 ||u|| r(1-θ) r 4 ||u r+m-1 2 || 2 r θ r+m-1 H 1 , (25)
where

θ = 3 N (r + m -1) (3N + 2)r + 4N (m -1) < 1.
Therefore, Young inequality and (25) yield that

2 r 2 ||u|| r r ≤ 2 r 2 C 2 r θ r+m-1 1 ||u|| r(1-θ) r 4 || u r+m-1 2 || 2 r θ r+m-1 H 1 ≤ θ r r + m -1 m (r -1) (r + m -1) 2 r + m -1 θ r C 2 (1) ||u r+m-1 2 || 2 H 1 + r + m -1 -θr r + m -1 C 2 (1) θ(r + m -1)r m(r -1) θr r(1-θ)+m-1 × (2 r 2 ) (r+m-1) r(1-θ)+m-1 C 2 θr r(1-θ)+m-1 1 ||u|| (1-θ)r (r+m-1) r(1-θ)+m-1 r 4
, where C 2 (1) is the Poincaré constant defined in Lemma 3.6. Then we obtain

2 r 2 ||u|| r r ≤ m (r -1) C 2 (1) (r + m -1) 2 ||u r+m-1 2 || 2 H 1 + C θ r r(1-θ)+m-1 1 2 (r+m-1) r(1-θ)+m-1 r 2(r+m-1)+θr r(1-θ)+m-1 ||u|| (1-θ)r(r+m-1)
r(1-θ)+m-1 r 4 . Now, since N > 2, which gives 4N ≥ 3N + 2, we find the following upper bound for θ

θ ≤ 3N 3N + 2 (26) In addition, θ r r(1 -θ) + m -1 ≤ θ 1 -θ = -1 + 1 1 -θ ≤ 3N 2 , (27)
r + m -1 r(1 -θ) + m -1 ≤ r + m -1 (1 -θ)(r + m -1) ≤ 1 1 -θ ≤ 3N + 2 2 , (28) and 2
(r + m -1) + θr r(1 -θ) + m -1 ≤ 2 + θ 1 -θ ≤ 9N + 4. (29)
As C 1 ≥ 1 and r ≥ 1, we get

2 r 2 ||u|| r r ≤ m (r -1) C 2 (1) (r + m -1) 2 ||u r+m-1 2 || 2 H 1 + C r 9N +4 ||u|| (1-θ)r(r+m-1) r(1-θ)+m-1 r 4 . (30) Using Lemma 3.6 we have ||u r+m-1 2 || 2 H 1 ≤ C 2 (1) ||∇u r+m-1 2 || 2 2 + ||u r+m-1 2 || 2 1 . (31)
Using Hölder inequality, Young inequality and Lemma 3.3, we get

||u r+m-1 2 || 2 1 = ||u|| r+m-1 r+m-1 2 ≤ ||u|| r r+m-3 r-1 r ||u|| r-m+1 r-1 1 ≤ ||u|| r r+m-3 r-1 r ||u 0 || r-m+1 r-1 1 , then, m (r -1) (r + m -1) 2 ||u r+m-1 2 || 2 1 ≤ (r -1) r-1 r+m-3 r + m -3 r -1 ||u|| r r + 2 -m r -1 m (r + m -1) 2 ||u 0 || r+m-1 r-1 1 r-1 2-m ≤ r 2 ||u|| r r + m (r + m -1) 2 ||u 0 || r+m-1 r-1 1 r-1 2-m ≤ r 2 ||u|| r r + C r 4 . (32)
Now substituting (32) and ( 31) into (30) we get

2 r 2 ||u|| r r ≤ m (r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + ||u r+m-1 2 || 2 1 + C r 9N +4 ||u|| (1-θ)r(r+m-1) r(1-θ)+m-1 r 4 ≤ m (r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + r 2 ||u|| r r + C r 4 + C r 9N +4 ||u|| (1-θ)r(r+m-1) r(1-θ)+m-1 r 4 , hence r 2 ||u|| r r ≤ m (r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + C r 4 + C r 9N +4 ||u|| r (1-θ)(r+m-1) r(1-θ)+m-1 r 4 .
We apply Young inequality again to the last term of the above inequality. It is easy to see that

2 3N + 2 ≤ 1 -θ ≤ (1 -θ) (r + m -1) r(1 -θ) + m -1 = (1 -θ)r + (1 -θ)(m -1) r(1 -θ) + m -1 < 1, so that r 2 ||u|| r r ≤ m (r -1) (r + m -1) 2 ||∇u r+m-1 2 || 2 2 + C r 4 + 1 + C r 9N +4 3N +1 ||u|| r r 4 , (33)
for any r ∈ [4, ∞).

Substituting (33) into (24) we end up with

d dt ||u|| r r ≤ r C r 4 + r + r C r 9N +4 3N +1 ||u|| r r 4 ≤ C r 5 + Cr α ||u|| r r 4 , (34)
for any r ∈ [4, ∞), where α = (9N + 4)(3N + 1) + 1. After integrating (34) from 0 to t, we obtain the L r estimate for u as follows:

sup 0<t<T ||u(t)|| r r ≤ ||u 0 || r r + T C r 5 + C r α T sup 0<t<T ||u(t)|| r r 4 . ( 35
) Since ||u 0 || r ≤ ||u 0 || r-1 r ∞ ||u 0 || 1 r 1 ≤ C 6 , then sup 0<t<T ||u(t)|| r r ≤ C 7 (T ) r α max C 6 , sup 0<t<T ||u(t)|| r 4 r , (36)
and we obtain for r ≥ 4

sup 0<t<T ||u(t)|| r ≤ C 7 (T ) 1 r r α r max C 6 , sup 0<t<T ||u(t)|| r 4 . (37)
We are now in a position to derive the claimed L ∞ estimate. To this end, we set Consequently, by letting p tend to ∞, we see that u ∈ L ∞ ((0, T ) × Ω) and

sup 0<t<T ||u(t)|| ∞ ≤ C 8 (T ). ( 38
)
Since the right hand side is independent of δ, we have proved the lemma.

Lemma 3.10. Let the same assumptions as that in Theorem 3.1 hold, and (u δ , ϕ δ ) be the solution to [START_REF] Cieślak | Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system[END_REF]. Then for all T > 0 there is C 9 (T ) such that the solution u δ satisfies the following derivation estimate

T 0 ||∂ t u m δ || (W 1,N+1 ) ′ dt ≤ C 9 (T ).
Proof. Consider ψ ∈ W 1,N +1 (Ω) and t ∈ (0, T ), we have

Ω m u m-1 δ (t) ∂ t u δ (t) ψ dx = m Ω ∇(u m-1 δ ψ) • (∇u m δ -u δ ∇ϕ δ ) dx = m Ω (u m-1 δ ∇ψ + ψ ∇u m-1 δ ) • (∇u m δ -u δ ∇ϕ δ ) dx ≤ m Ω u m-1 δ |∇u m δ | |∇ψ| + u m δ |∇ψ| |∇ϕ δ | + |ψ| m(m -1) u 2m-3 δ |∇u δ | 2 + |ψ|(m -1)u m-1 δ |∇u δ | |∇ϕ δ | dx ≤ m ||u δ || m-1 ∞ ||∇u m δ || 2 ||∇ψ|| 2 + ||∇ψ|| 2 ||u δ || m ∞ ||∇ϕ δ || ∞ |Ω| 1 2 + ||ψ|| ∞ 4m(m -1) (2m -1) 2 ||∇u m-1 2 δ || 2 2 + ||ψ|| 2 m -1 m ||∇u m δ || 2 ||∇ϕ δ || ∞ .
Using Lemma 3.8, Lemma 3.9, and the embedding of W 1,N +1 (Ω) in L ∞ (Ω), we end up with

|< ∂ t u m δ (t), ψ >| ≤ C(T ) ||∇u δ (t) m || 2 + ||∇u m-1 2 δ (t)|| 2 2 + 1 ||ψ|| W 1,N+1 ,
and a duality argument gives

||∂ t u m δ (t)|| (W 1,N+1 ) ′ ≤ C(T ) ||∇u m δ (t)|| 2 + ||∇u m-1 2 δ (t)|| 2 2 + 1 .
Integrating the above inequality over (0, T ) and using Lemma 3.4 with p = 2 and p = m give Lemma 3.10.

4 Proof of Theorem 2.2

Existence

In this section, we assume that u 0 is a nonnegative function in L ∞ (Ω) satisfying [START_REF] Blanchet | Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions[END_REF]. For δ ∈ (0, 1), (u δ , ϕ δ ) denotes the solution to (KS) δ constructed in Section 3. To prove existence of a weak solution, we use a compactness method. For that purpose, we first study the compactness properties of (u δ , ϕ δ ) δ .

Lemma 4.1. There are functions u and ϕ and a sequence (δ n ) n≥1 , δ n → 0, such that, for all T > 0 and p ∈ (1, ∞),

u δn -→ u, in L p ((0, T ) × Ω) as δ n → 0, (39)
ϕ δn -→ ϕ, in L p ((0, T ); W 2,p (Ω)) as δ n → 0. ( 40
)
In addition, u ∈ L ∞ ((0, T ) × Ω) for all T > 0 is nonnegative.

Proof. Thanks to Lemma 3.4 and Lemma 3.9, (u m δ ) δ is bounded in L 2 ((0, T );

H 1 (Ω)) while (∂ t u m δ ) δ is bounded in L 1 ((0, T ); (W 1,N +1 ) ′ (Ω)) by Lemma 3.10. Since H 1 (Ω) is compactly embedded in L 2 (Ω) and L 2 (Ω) is continuously embedded in (W 1,N +1 ) ′ (Ω), it follows from [19, corollary 4] that (u m δ ) is compact in L 2 ((0, T ) × Ω) for all T > 0. Since r -→ r 1 m is 1
m -Hölder continuous, it is easy to check that the previous compactness property implies that (u δ ) is compact in L 2m ((0, T ) × Ω) for all T > 0. There are thus a function u ∈ L 2m ((0, T ) × Ω) for all T > 0 and a sequence (δ n ) n≥1 such that

u δn -→ u in L 2m ((0, T ) × Ω) as δ n → 0, (41)
for all T > 0, owing to Lemma 3.9, we may also assume that

u δn * ⇀ u in L ∞ ((0, T ) × Ω) as δ n → 0. ( 42
)
for all T > 0. It readily follows from ( 41) and (42), and Hölder inequality that (39) holds true. Since elliptic regularity ensure that

||ϕ δ k -ϕ δn || W 2,p ≤ C(p) ||u δ k -u δn || p ,
for all k ≥ 1, n ≥ 1, and p ∈ (1, ∞), a straightforward consequence of (39) is that (ϕ δn ) n≥1 is a Cauchy sequence in L p ((0, T ); W 2,p (Ω)) and thus converges to some function ϕ in that space. Finally, the nonnegativity of u follows easily from that of u δn by (39).

Proof of Theorem 2.2 (existence). It remains to identify the equations solved by the limit (u, ϕ) of (u δn , ϕ δn ) n≥1 constructed in Lemma 4.1. To this end we first note that , owing to (39) and the boundedness of (u δn ) n and u in L ∞ ((0, T ) × Ω), we have

u m δn -→ u m in L p ((0, T ) × Ω) as δ n → 0, (43)
for all T > 0. Since (∇(u δn + δ n )

2 ) n≥1 and (∇u m δn ) n≥1 are bounded in L 2 ((0, T ) × Ω) for all T > 0 by Lemma 3.4 with p = 2 and p = m + 1, we may extract a further subsequence (not relabeled) such that

∇(u δn + δ n ) m+1 2 ⇀ ∇u m+1 2 in L 2 ((0, T ) × Ω), (44)
∇u m δn ⇀ ∇u m in L 2 ((0, T ) × Ω), (45)
for all T > 0. Then if ψ ∈ L 4 ((0, T ) × Ω; R N ), T 0 Ω ψ • [∇(u δn + δ n ) m -∇u m ] dxds = 2 m + 1 T 0 Ω ψ • (u δn + δ n ) m-1 2 ∇(u δn + δ n ) m+1 2 -u m-1 2 ∇u m+1 2 dxds ≤ 2 m + 1 T 0 Ω ψ • ∇(u δn + δ n ) m+1 2 ((u δn + δ n ) m-1 2 -u m- 1
2 ) dxds Thus, we have constructed a weak solution (u, ϕ) of (KS).

+ 2 m + 1 T 0 Ω u m-1 2 ψ • ∇(u δn + δ n ) m+1 2 -∇u

Uniqueness

In this section, we prove the uniqueness statement of Theorem 2.2 under the additionnal assumption (6) on ϕ. The proof relies on a classical duality technique, and on the method presented in [START_REF] Bedrossian | Local global well-possedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion[END_REF] Proof. The proof estimates the difference of weak solutions in dual space H 1 (Ω) ′ of H 1 (Ω), motivated by the fact that the nonlinear diffusion is monotone in this norm.

Notice that ||∇ϕ(0)|| 2 = 0 which follows from (49) and the property u(0) = 0. Thus, inequality (54) implies ||∇ϕ(t)|| 2 2 ≤ e C(T ) t ||∇ϕ(0)|| 2 2 = 0. Consequently, ∇ϕ(t) = 0 for all t ∈ [0, T ] and, since < ϕ(t) >= 0, we have ϕ(t) = 0 for all t ∈ [0, T ]. Using (49), we conclude that u(t) = 0 for all t ∈ [0, T ]. Consequently (u 1 , ϕ 1 ) = (u 2 , ϕ 2 ).

Lemma 3 .Lemma 3 . 8 .

 338 7 then readily follows from Sobolev embedding theorem upon choosing p > N . Let N ≥ 3, r ≥ 4, u ∈ L r 4 (Ω), and u r+m-1 2

α 1 4 4 α 2 p C 7 (T ) 1 4 p α p- 1 since p ≤ 2 p for p ≥ 1 .

 142111 p := max C 6 , sup 0<t<T ||u(t)|| 4 pfor p ≥ 0. Then we take r = 4 p with p ≥ 0 in (37) which readsα p ≤ 4 α p 4 p C 7 (T ) p max C 6 , sup 0<t<T ||u(t)|| 4 p-1 ,≤ Arguing by induction we conclude thatα p ≤ 4 α p k=1 2 -k C 7 (T ) p k=1 4 -k α 0 .Then by using Lemma 3.3 we get sup 0<t<T ||u(t)|| 4 p ≤ 4 α C 7 (T ) α 0 ≤ C 8 (T ).

||ψ|| 4 2 ψ 2 ψ[

 422 ||∇(u δn + δ n ) m+1 2 || 2 ||(u δn + δ n ) • ∇(u δn + δ n ) ∈ L 2 ((0, T ) × Ω), we deduce from (39) and (44) that the right-hand side of the above inequality converges to zero as n -→ ∞. In other words,∇(u δn + δ n ) m ⇀ ∇u m in L 4 3 ((0, T ) × Ω),(46)for all T > 0. Now, we going to show that (u, ϕ) in Lemma 4.1 is the desired weak solution in Theorem 2.2. Let T > 0 and ψ ∈ C 1 ([0, T ] × Ω) with ψ(T ) = 0. The solution of (7) satisfiesT 0 Ω [∇(u δn + δ n ) m • ∇ψu δn ∇ϕ δn • ∇ψu δn ∂ t ψ] dxdt = Ω u 0 ψ(0, x) dx, ∇ϕ δn • ∇ψ + M ψu δn ψ] dxdt = 0. (48)From (46) we see thatT 0 Ω ∇(u δn + δ n ) m • ∇ψ dxdt -→ T 0 Ω ∇u m • ∇ψ dxdt as δ n → 0.From (39) we getT 0 Ω u δn ∂ t ψ dxdt -→ T 0 Ω u ∂ t ψ dxdt as δ n → 0.From (39) and (40) we getT 0 Ω u δn ∇ϕ δn • ∇ψ dxdt -→ T 0 Ω u ∇ϕ • ∇ψ dxdt as δ n → 0.Thus we conclude that u satisfiesT 0 Ω (∇u m • ∇ψ -u∇ϕ • ∇ψu • ∂ t ψ) dxdt = Ω u 0 (x) • ψ(0, x) dx.Similarly, from (40) we see thatT 0 Ω ∇ϕ δn • ∇ψ dxdt -→ T 0 Ω∇ϕ • ∇ψ dxdt as δ n → 0, and from (39) we see that as δ n → 0.

since -∆ϕ 2 = u 2 -(|D 2 ϕ 2 | + 1 )

 2221 < u >∈ L ∞ ((0, T ) × Ω). Together with (53) the previous inequality impliesΩ u ∇ϕ 2 • ∇ϕ dx ≤ C(T ) Ω |∇ϕ| 2 dx. ≤ C(T ) (||ϕ 2 || L ∞ ((0,T );W 2,∞ (Ω)) + 1) Ω |∇ϕ| 2 dx,provided that the L ∞ ((0, T ); W 2,∞ (Ω)) norm of the function ϕ 2 is bounded. Thus, substituting the above estimates in (52), one finally obtainsd dt Ω |∇ϕ| 2 dx ≤ C(T ) Ω |∇ϕ| 2 dx.

ACKNOWLEDGMENT

I thank professors Philippe Laurençot and Marjolaine Puel for their helpful advices and comments during this work.

Assume that we have two different weak solutions (u 1 , ϕ 1 ) (u 2 , ϕ 2 ) to equations [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF] corresponding to the same initial conditions, and fix T > 0. We put

Then ϕ is the strong solution of

Now it follows from (1) that u satisfies the equation

Substituting (51) in (50), we obtain 1 2

The first integral on the right-hand side of (52) is nonnegative due to the fact that z → z m is an increasing function. The second integral on the right-hand side of (52) can be estimated by

For the last integral, using an integration by parts we obtain

integrating by parts the second integral on the right-hand side of (53),