Serge Nicaise
email: serge.nicaise@univ-valenciennes.fr

Delio Mugnolo
email: delio.mugnolo@uni-ulm.de

DIFFUSION PROCESSES ON AN INTERVAL UNDER LINEAR MOMENT CONDITIONS DELIO MUGNOLO AND

Keywords: 2010 Mathematics Subject Classification. 47D06, 47H20, 34B10, 76S05 Nonlocal conditions for PDEs, Porous Medium Equation, Heat equation, Subdifferentials

We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n = 1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.

Introduction

In [START_REF] Vázquez | Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium[END_REF] J.L. Vázquez made the simple observation that possibly diffusion-type equations of the form (1.1) ∂u ∂t (t, x) = ∆(|u| p-2 u)(t, x), which are well-known to be associated with a well-posed Cauchy problem in H -1 (R d ) if d = 1 (and even for d > 1), enjoy conservation of mass and barycenter. Here and in the following, p is some constant strictly larger than 1: This result applies therefore to both the porous medium equation (PME) and the fast diffusion equation (FDE) along with the linear heat equation, corresponding to the cases of p ∈ (2, ∞), p ∈ (1, 2) and p = 2, respectively. Vázquez' assertion is easily explained: For a density distribution function f : R → R denote by µ n (f ) the n-th moment (say, about 1), i. e.,

µ n (f ) := R (1 -x) n f (x) dx for f ∈ L 1 (R).
Then in particular µ 0 (f ) and µ 1 (f ) represent the total mass and the barycenter of the distribution described by f , respectively. Now, differentiating with respect to time the moments of order 0 and 1 of a solution u of (1.1) with initial data u 0 and integrating by parts with respect to space a simple localization argument shows that µ 0 (u(t)) = µ 0 (u 0 ) and µ 1 (u(t)) = µ 1 (u 0 ) ∀t ≥ 0.

Choosing boundary conditions judiciously, one can see that conservation of mass and/or barycenter may also hold in the case of a PME or a FDE on a bounded interval. Observe that conservation of mass and barycenter can also be defined for solutions so irregular that boundary conditions do not make sense -this is particularly relevant in the case of the PME and the FDE, which are typically solved in spaces of distributions ([Váz07, Chapt. 10]). May then the condition that mass and barycenter be conserved replace boundary conditions altogether?

Imposing conditions on the moments may appear bizarre. However, since certain moment conditions boil down to boundary conditions if solutions are regular enough [START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF]Cor. 4.10]), at a second glance it looks reasonable to investigate well-posedness under such conditions in the case of a bounded domain.

Beginning with [START_REF] Cannon | The solution of the heat equation subject to the specification of energy[END_REF], many authors have studied linear partial differential equations equipped with conditions on the moments complementing those on the boundary values. In [START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF] the present authors have gone on to observe that, in fact, in the case of the linear heat equation one can drop the boundary conditions and obtain well-posedness under a wide class of linear conditions on the moments of order 0 and 1 -and in particular, whenever both of them are assumed to vanish constantly, i.e., (1.2)

µ n (u(t)) = 0 ∀t ≥ 0, n = 0, 1.

This special case had already been discussed by A. Bouziani and his coauthors starting with [START_REF] Bouziani | Problème mixte avec conditions intégrales pour une classe d'équations hyperboliques[END_REF], see [START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF] for more detailed references. A somehow comparable approach has been followed in [START_REF] Vázquez | The Porous Medium Equation: Mathematical Theory[END_REF]§ 9.6], where an analysis based on mere finiteness of mass is performed.

There exist counterexamples showing that, in general, moments of higher order are not conserved under the evolution of (1.1) on R, cf. [Váz07, § 9.6.4]. Hence it is not natural to expect well-posedness upon imposing a condition analogous to (1.2) for any two moments. Our aim in this article is to show that, however, suitable conditions on µ 0 and a further moment µ n suffice to obtain well-posedness of certain modified evolution equations -which can be looked at as PMEs or FDEs with a potential depending on n.

It turns out that the analysis of diffusion equations on an interval under conditions on µ 0 and µ n for general n can be performed closely following some techniques developed in [START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF]. The extension of such techniques to the more general setting of the present article is discussed thoroughly in Section 2. As it is, the well-posedness results presented in [START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF] are just a special case of those that we obtain in Section 3.

However, serious problems seem to arise in the truly nonlinear case, i.e., whenever we discuss (1.1) for p = 2 -this is the topic of Section 4. Both the PME and the FDE with Dirichlet boundary conditions are well-known to be the flow of the gradient associated with a suitable energy functional (also known as the functional's subdifferential in the language of nonlinear semigroup theory, see e.g. [Sho97, Example IV.6.B] and [Váz07, Chapt. 10], and more generally [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert[END_REF] for the abstract theory) with respect to an H -1 -inner product. (Observe that a different, more involved but also mightier approach based on flows on Riemannian manifolds has been introduced by F. Otto in a celebrated article [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF]). In our setting the gradient flow structure is still present, but unlike in the linear case we are not able to show that for initial data smooth enough this evolution equation is just the PME or the FDE with a certain potential. Nevertheless, we obtain well-posedness of a certain n-dependent nonlinear evolution equation that strongly resembles the PME or the FDE, and we are able to show that its long-time behavior depends on p. In particular, we show that for all n the decay of the H -1 -norm of the solutions is polynomial if p ∈ (2, ∞) and exponential if p ∈ (1, 2]. Our analysis is made different from the classical case by the fact that for this new evolution equation integration by parts is not easily applied -this in turn prevents us from applying the classical method based on the weak formulation of the PME or the FDE, which are the backbone of many proofs in [START_REF] Vázquez | The Porous Medium Equation: Mathematical Theory[END_REF].

The functional analytical setting

If we consider (0, 1) as the torus T , then the test function set D(T ) is in fact the set of smooth functions in [0, 1] such that the derivatives at all orders coincide at 0 and 1. In the same manner we will use the Sobolev space H 1 (T ), by which we denote the subspace of those u ∈ H 1 (0, 1) such that u(0) = u(1) (i.e., of those H 1 -functions supported on the torus). We use L 2 (0, 1) as pivot space and denote by H -1 (T ) (resp., D ′ (T )) the dual of H 1 (T ) (resp., D(T )). For p ∈ (1, ∞), p = 2, we define the Sobolev spaces W 1,p (T ) and W -1,p (T ) likewise. In this paper we refrain from considering the case of (1.1) for p ∈ (0, 1], which is known to require a quite different approach, cf. [Váz06, § 2.2.1]. For all n ∈ N 0 we denote by µ n the linear functional defined by

µ n (f ) := 1 0 (1 -x) n f (x) dx,
which is bounded on L 1 (0, 1) (and even on H -1 (T ) for n = 0).

For n ∈ N 0 and ϕ ∈ D(T ) we denote by J n ϕ the function defined by

J n ϕ(x) := 1 x ϕ(y) dy -µ 0 (ϕ)(1 -x) n , x ∈ (0, 1),
and set for all u ∈ H -1 (0, 1)

(2.1)

P n u, ϕ := u, J n ϕ ∀ϕ ∈ D(T ).
which is meaningful since J n ϕ ∈ H 1 0 (0, 1). We obtain the following analogue of [MN11, Lemma. 2.2], where the attention was devoted to P 1 only.

Lemma 2.2. Let p ∈ [1, ∞) and n ∈ N. Then the following assertions hold.

(1) For all u ∈ L p (0, 1), P n u can be written as

(2.2) P n u(x) = Iu(x) -µ n (u), ∀x ∈ (0, 1),
where

Iu(x) := x 0 u(y)dy, ∀x ∈ (0, 1).
In particular, (P n u) ′ = u whenever u ∈ L p (0, 1). (2) Moreover, P n is a bounded linear operator from W -1,p (T ) to L p (0, 1) and from L p (0, 1) to W 1,p (0, 1) as well as from {f ∈ L p (0, 1) : µ 0 (f ) = 0} to W 1,p (T ).

(3) Finally, (P n u) ′ = u in H -1 (0, 1) for all u ∈ H -1 (0, 1).

Proof. Let u ∈ L p (0, 1) be fixed. Denote for a moment Pn u the right-hand side of (2.2) (which is by construction an element of W 1,p (0, 1)), i.e., Pn u(x) := Iu(x)µ n (u),

x ∈ (0, 1).

By integrations by parts for all ϕ ∈ D(T ) one has

1 0 Pn u(x)ϕ(x) dx = 1 0 (Iu(x) -µ n (u))ϕ(x) dx = 1 0 x 0 u(y) dy ϕ(x) dx -µ n (u)µ 0 (ϕ).
Accordingly, for 0 ≤ y ≤ x ≤ 1 by Fubini's Theorem we find

1 0 Pn u(x)ϕ(x) dx = 1 0 1 y ϕ(x) dx u(y) dy -µ n (u)µ 0 (ϕ) = 1 0 1 y ϕ(x) dx -µ 0 (ϕ)(1 -y) n u(y) dy = 1 0 u(x)J n ϕ(x) dx.
This proves that P n u = Pn u, hence (1), and furthermore P n u ∈ W 1,p (0, 1).

In a second step we first easily check that for ϕ ∈ D(T ), J n ϕ is in W 1,q (T ) with 1 p + 1 q = 1 (it is even in W 1,q 0 (0, 1)) and that J n ϕ W 1,q 0 (0,1) ϕ L q (0,1) .

According to (2.1) we then get

| P n u, ϕ | ≤ u W -1,p (T ) J n ϕ W 1,q 0 (0,1)
u W -1,p (T ) ϕ L q (0,1) .

Therefore P n u L p (0,1) u W -1,p (T ) , ∀u ∈ D(T ), which proves that P n is a bounded operator from W -1,p (T ) to L p (0, 1) due to the density of D(T ) into W -1,p (T ). As the boundedness of P n from L p (0, 1) to W 1,p (0, 1) has been observed above, point (2) is proved.

Finally, the assertion (3) follows the definition of P n and the fact that J n (ϕ ′ ) = -ϕ for all ϕ ∈ H 1 0 (0, 1).

In particular,

(P n 1)(x) = x - 1 0 (1 -x) n dx = x - 1 n + 1 .
Observe that, unlike for n = 1, for n ≥ 2 one has in general (2.3)

P n ∈ L {f ∈ H -1 (T ) : µ 0 (f ) = 0}, {f ∈ L p (0, 1) : µ 0 (f ) = 0} , since P n u, 1 = 0.
Remark 2.3. Recall that by [MN11, Lemma 2.3]

(P 1 u|P 1 v) + µ 0 (u)µ 0 (v), u, v ∈ H -1 (T ),
defines an equivalent inner product on H -1 (T ), and in particular

(2.4) Id -1 m u H -1 (T ) ≃ P 1 Id -1 m u L 2 (0,1) , ∀u ∈ H -1 (T ).
In fact, we can say more. Taking into account Remark 2.4.(1) and reasoning just like in the proof of [MN11, Lemma 2.3] one can easily see that

u H -1 (T ) P n u L 2 + |µ 0 (u)|, ∀u ∈ H -1 (T ).
Accordingly, we can even say that

(P n u|P n v) + µ 0 (u)µ 0 (v), u, v ∈ H -1 (T ),
defines an equivalent inner product on H -1 (T ) for all n ∈ N.

Remark 2.4. Let n ∈ N.

(1) Observe that for all n ∈ N

µ n (f ) = n 1 0 (1 -x) n-1 x 0 f (z)dzdx = nµ n-1 (If ) ∀f ∈ L 1 (0, 1).
(2) Let f ∈ L 1 (0, 1). Then by Lemma 2.2.(1) and integrating by parts

µ n-1 (P n f ) = 1 0 (1 -x) n-1 If (x) dx -µ n (f ) 1 0 (1 -x) n-1 dx = (1 -x) n n x 0 f (x)dx x=1 x=0 - 1 0 (1 -x) n n f (x) dx + 1 n µ n (f ) = - 1 n µ n (f ) + 1 n µ n (f ) = 0.
(3) Observe also that

P n δ 1 = 0 in D ′ (T ), since for all φ ∈ D(T ) P n δ 1 , φ = δ 1 , J n φ = J n φ(1) = 0.
(4) It is also important for the following that (2.5)

P n h(0) = -µ n (h) and P n h(1) = µ 0 (h) -µ n (h) ∀h ∈ L 1 (0, 1)
as well as

(2.6) µ 0 (P n (h)) = 1 0 P n h(x) dx = µ 1 (h) -µ n (h) ∀h ∈ L 1 (0, 1) :
both identities are direct consequences of (2.2).

(5) It is a straightforward observation that [MN11, Rem. 2.6] can be generalized as follows: For all p ∈ [1, ∞), all f ∈ W 2,p (0, 1) and all ψ ∈ L p (0, 1), we have

(2.7) Id -1 m (f ′′ + ψ) = f ′′ + ψ + (µ 0 (f ′′ + ψ))δ 1 in W -1,p (T )
, and in particular by (3)

P n (Id -1 m (f ′′ + ψ)) = P n (f ′′ + ψ) in L p (0, 1
). (6) Finally, observe that by Lemma 2.2 and (2.5) P n is bounded from {f ∈ L p (0, 1) : µ 0 (f ) = 0} to W 1,p (T ) and from {f ∈ L p (0, 1) : µ 0 (f ) = µ n (f ) = 0} to W 1,p 0 (0, 1). The crucial point for our investigation is that an integration-by-parts-type formula holds. Recall that we are denoting by Id m the isomorphism between {u ∈ H -1 (T ) : µ 0 (u) = 0} and H -1 (0, 1) and let u ∈ H 1 (0, 1). Then for all u ∈ H 1 (0, 1) and all h ∈ H 1 (T ), by Remark 2.3 one has

Id -1 m (u ′′ )|h H -1 (T ) = (P n Id -1 m (u ′′ )|P n h) L 2 = (u ′ -a|P n h) L 2 , where (2.8) a := u ′ -P n (u ′′ ), 1 .
Hence, (2.6) and a standard integration by parts yield

Id -1 m (u ′′ )|h H -1 (T ) = (u ′ |P n h) L 2 -a(µ 1 (h) -µ n (h)) = -(u|h) L 2 + [uP n h] 1 0 -a(µ 1 (h) -µ n (h)) (2.9) since by Lemma 2.2.(3) (P n h) ′ = h. Remark 2.5. Note that a = u(1) -u(0) -u ′′ , (1 -id) -(1 -id) n because
P n (u ′′ ), 1 = u ′′ , J n 1 -this is the duality between H -1 (0, 1) and H 1 0 (0, 1), with the function J n 1 ∈ H 1 0 (0, 1) given by (J n 1)(x) = (1x) -(1x) n ∀x ∈ (0, 1).

For n = 1, J n 1 ≡ 0 and we get P n (u ′′ ), 1 = 0, but in any case, this is does not really matter since a is multiplied by 0 in (2.9). Indeed, we can still improve the formula in (2.9) for n ≥ 2.

Theorem 2.6. Let u ∈ H 1 (0, 1) and h ∈ L 2 (0, 1). Then

Id -1 m (u ′′ )|h H -1 (T ) = -(u|h) L 2 +     u(1) nu(0) -n(n -1)µ n-2 (u) (1 -n)u(0) -u(1) + n(n -1)µ n-2 (u)     µ 0 (h) µ 1 (h) µ n (h)     C 3 .
Proof. The case n = 1 has already been discussed. For n ≥ 2, our starting point is again (2.9). From the property J n 1 ∈ H 1 0 (0, 1) and an integration by parts, we can write

P n (u ′′ ), 1 = -u ′ , (J n 1) ′ = 1 0 u ′ (x)(1 -n(1 -x) n-1 ) dx = u(1) -u(0) -n(n -1) 1 0 u(x)(1 -x) n-2 ) dx -n[u(x)(1 -x) n-1 ] 1 0 = u(1) -u(0) -n(n -1)µ n-2 (u) + nu(0).
Hence we deduce that for n ≥ 2, the constant in (2.8) is given by

a = n(n -1)µ n-2 (u) -nu(0).

Now, applying (2.5) we find [uP

n h] 1 0 = u(1)P n h(1) -u(0)P n h(0) = u(1)(µ 0 (h) -µ n (h)) + u(0)µ n (h) = u(1)µ 0 (h) + (u(0) -u(1)) µ n (h).
Applying the previous identities to (2.9) we obtain

Id -1 m (u ′′ )|h H -1 (T ) = -(u|h) L 2 + u(1)µ 0 (h) + (u(0) -u(1))µ n (h) -n(n -1)µ n-2 (u) -nu(0) (µ 1 (h) -µ n (h)),
and the claimed formula follows.

If n = 1, this is the formula already obtained in [MN11, Lemma 2.13]. Otherwise, we have obtained a more general identity that allows us to extend the study of diffusion processes under linear conditions on µ 0 and µ 1 (as e.g. in [START_REF] Bouziani | Problème mixte avec conditions intégrales pour une classe d'équations hyperboliques[END_REF][START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF]) to linear conditions on µ 0 and µ n for general n ∈ N.

For each subspace Y of C 2 and each p > 1 we can consider the reflexive Banach space

V (1)
Y,p := f ∈ L p (0, 1) :

µ 0 (f ) µ 1 (f ) ∈ Y . Lemma 2.7. Let k ∈ N.
Then for all m ∈ N k with pairwise distinct entries, the operator

   µ m1 . . . µ m k    : C ∞ [0, 1] → C k is surjective.
Proof. We are going to prove more, namely that the vector valued mapping

   µ 0 . . . µ m k   
is surjective from the space P m k (0, 1) of all polynomials of degree at most m k to C m+1 . For m ∈ N fixed, it is well known that there exists a (m + 1) × (m + 1) invertible matrix M m such that

     1 (1 -id) . . . (1 -id) m      = M m      Q 0 Q 1 . . . Q m     
, where Q k is the Legendre-type polynomial of degree k defined on (0, 1) through the standard Legendre polynomial P k of degree k defined on (-1, 1) by

Q k (x) = P k (2x -1), ∀x ∈ (0, 1). Setting μk (h) := 1 0 h(x)Q k (x) dx, h ∈ L 1 (0, 1), we deduce that (2.10)    µ 0 . . . µ m    = M m    μ0 . . . μm    .
As the Legendre polynomials are pairwise orthogonal, the mapping

   μ0 . . . μm    : P m (0, 1) → C m+1
is trivially surjective. By (2.10) and the invertibility of M m , we conclude that

   µ 0 . . . µ m    : P m (0, 1) → C m+1
is also surjective.

We denote by H Y the Hilbert space

(2.11)

H Y := {f ∈ H -1 (T ) : µ 0 (f ) = 0}, if Y = {0} 2 or Y = {0} × C, H -1 (T ),
otherwise.

It has been shown in [START_REF] Mugnolo | Well-posedness and spectral properties of heat and wave equations with non-local conditions[END_REF] that an equivalent inner product is given in either case by

(2.12) (f |g) HY :=

1 0 P 1 f (x)P 1 g(x)dx + µ 0 (f )µ 0 (g).
Lemma 2.8. The space V

(1)

Y,p is dense in H Y for each Y subspace of C 2 . Proof. By construction, V (1)
Y,p ⊂ H Y and hence the inclusion V

(1) (1) To begin with, observe that it follows from (2.12) that

Y,p ⊂ H Y is clear. The proof
(2.13) (f |g) H {0} 2 = 1 0 P 1 f (x)P 1 g(x)dx, ∀f, g ∈ H {0} 2 . and (2.14) u H -1 (T ) ≃ P 1 u L 2 ∀u ∈ H {0} 2 .
To prove the claimed inclusion, we show that each f ∈ H {0} 2 that is orthogonal to V

(1)

{0} 2 ,p for the inner product (•|•) H {0} 2 is identically zero. In fact, let f ∈ H {0} 2 be orthogonal to V (1) {0} 2 ,p . Then it satisfies (P 1 f |P 1 g) L 2 = 0, ∀g ∈ V (1)
{0} 2 ,p . But according to its definition (2.1) we get equivalently

(2.15) f, J 1 P 1 g = 0, ∀g ∈ V (1) {0} 2 ,p . Now define (2.16) W := {u ∈ H 1 (T ) ∩ W 2,p (0, 1) : u ′ (0) = u ′ (1) = 0}.
This space is dense in H 1 (T ) since for u ∈ H 1 (T ), uu(0) belongs to H 1 0 (0, 1). Hence for all u ∈ H 1 (T ) there exists a sequence of ϕ n ∈ D(0, 1) such that

ϕ n → u -u(0) in H 1 0 (0, 1), and therefore ϕ n + u(0) ∈ W with ϕ n + u(0) → u in H 1 (T ). Now for h ∈ W , we take g := -h ′′ ∈ V (1) {0} 2 ,p . By construction J 1 P 1 g(x) = h(x) -h(1), ∀x ∈ (0, 1).
Plugging this identity in (2.15) yields (2.17)

f, h -h(1) = 0, ∀h ∈ W.
But due to the fact that µ 0 (f ) = 0, we deduce that

f, h = 0, ∀h ∈ W.
As W is dense in H 1 (T ), we conclude that f = 0.

If Y = {0} × C, we see that

H Y = V (1) {0} 2 ,p ⊂ V (1)
{0}×C,p ⊂ H Y , and the assertion follows.

(2) We consider the remaining cases. More precisely, it suffices to study the case of 1-dimensional Y with Y = {0} × C, because once proved this for Y = C 2 and any 1-dimensional subspace Y 0 we have

H Y = V (1) Y0,p ⊂ V (1) C 2 ,p = L p (0, 1) = H Y .
Hence, assume that there exists α ∈ R such that Y is the set of all (z 0 , z 1 ) ∈ C 2 satisfying (2.18)

z 1 = αz 0 . Let f ∈ H -1 (T ) be such that (2.19) (f |g) HY = 0, ∀g ∈ V (1) Y,p . Since V (1) {0} 2 ,p ⊂ V (1) Y,p , one has (f |g) H -1 (T ) = 0, ∀g ∈ V (1)
{0} 2 ,p , and reasoning as in (1) we deduce that (2.17) holds for the space W defined in (2.16). Since W is dense in H 1 (T ), this is equivalent to

f, h -h(1) = 0, ∀h ∈ H 1 (T ), or again (2.20) f = µ 0 (f )δ 1 in H -1 (T ).
Coming back to (2.19) and taking into account Remark 2.3 we get

(2.21) 0 = µ 0 (f ) (P 1 δ 1 |P 1 g) L 2 + µ 0 (δ 1 )µ 0 (g) = µ 0 (f )µ 0 (g), ∀g ∈ V (1)
Y,p . Now, by Lemma 2.7 there exists g ∈ L p (0, 1) such that

µ 0 (g) µ 1 (g) = 1 α ,
where α is as in (2.18). Hence, we can plug such a g ∈ V

(1) Y,p in (2.21) and we find µ 0 (f ) = 0, hence, by (2.20), f = 0. This concludes the proof. Lemma 2.9. Let n ∈ N. Then the vector space

(2.22) V (n) {0} 2 ,p := {u ∈ L p (0, 1) : µ 0 (u) = µ n (u) = 0}, and hence also V (n) {0}×C,p := {u ∈ L p (0, 1) : µ 0 (u) = 0}, are dense in H := {f ∈ H -1 (T ) : µ 0 (f ) = 0}.
Proof. The proof is based on Lemma 2.8, i.e., on the validity of the same assertion in the special case of n = 1. Let w ∈ H: we are looking for a sequence (w k ) k∈N ⊂ Kerµ n here and below in the proof, µ n is seen as a mapping from H ∩ L p (0, 1) into C) that approximates w in H -1 (T ). By Lemma 2.8, we already know that this is possible for n = 1: that is, there exists a sequence (v k ) k∈N ⊂ Kerµ 1 that approximates w in H -1 (T ). Now, for n ≥ 2 observe that the binomial formula yields

(1 -t) n = (1 -t) + p(t), t ∈ (0, 1),
with p ∈ H 1 (T ), and accordingly

µ n (v) = µ 1 (v) + p, v H 1 (T ),H -1 (T ) ∀v ∈ L 1 (0, 1),
and in particular

0 = µ 1 (v k ) = µ n (v k ) -p, v k H 1 (T ),H -1 (T ) ∀k ∈ N. Thus, by continuity, lim k→∞ µ n (v k ) = lim k→∞ p, v k = p, w =: c.
We distinguish the two cases c = 0 and c = 0.

(1) If c = 0, it suffices to take

w k := v k - µ n (v k ) µ n (p) p (observe that µ n (p) = 1 0 |(1 -t)| 2n dt = 0)
, so that w k ∈ Ker µ n for all k ∈ N and moreover lim k→∞ w k = w.

(2) If c = 0, write w = w 1 + αv 0 , with v 0 ∈ Ker µ n such that µ 1 (v 0 ) = p, v 0 = 0, i.e., v 0 ∈ Ker µ n \ Ker µ 1 . Now, set α := p, w p, v 0 and observe that for w 1 := wαv 0 one has p, w 1 = 0, hence w 1 ∈ Ker µ n owing to the first case. As v 0 ∈ Ker µ n we conclude that w ∈ Ker µ n as well. This concludes the proof. Also the following holds.

Lemma 2.10. Let n ∈ N and Y be a subspace of C 2 with Y = {0} 2 and Y = {0} × C. Then the vector space

V (n) Y,p := u ∈ L p (0, 1) : µ 0 (u) µ n (u) ∈ Y is dense in H -1 (T ).
Proof. The proof closely follows that of [MN11, Thm. 4.2]. Like in that proof, we begin by observing that if

Y = {0} 2 and Y = {0} × C, then either Y = C 2 or there exists α ∈ C such that Y is the set of all (z 0 , z 1 ) ∈ C 2 satisfying (2.23) z 1 = αz 0 .
Let us first assume that Y = C 2 and let f ∈ H -1 (T ) be such that

(2.24) (f |g) H -1 (T ) = 0, ∀g ∈ V (n)
Y,p , and in particular

(f |g) H -1 (T ) = 0, ∀g ∈ V (n)
{0} 2 ,p . Just like in the proof of Corollary 2.8 we deduce that (2.17) holds for the space W defined in (2.16). Since W is dense in H 1 (T ), this is equivalent to

f, h -h(1) = 0, ∀h ∈ H 1 (T ), or again (2.25) f = µ 0 (f )δ 1 in H -1 (T ).
Plugging into (2.24) we get

(2.26) 0 = µ 0 (f ) (P 1 δ 1 |P 1 g) L 2 + µ 0 (δ 1 )µ 0 (g) = µ 0 (f )µ 0 (g), ∀g ∈ V (n)
Y,p , because by Remark 2.4.(3) P 1 δ 1 = 0. By Lemma 2.7 there exists g ∈ L p (0, 1) such that

µ 0 (g) µ n (g) = 1 α ,
where α is as in (2.23). Hence, we can plug such a g ∈ V

Y,p in (2.26) and we find µ 0 (f ) = 0, hence, by (2.25), f = 0. This shows that

V (n) Y,p is dense in H -1 (T ) if Y = C 2 . For the case Y = C 2 , however, by definition V (n)
Y,p = L p (0, 1), and the claimed density is clear. Summing up, we have proved the following n-th-moment-analogue of Lemma 2.8.

Corollary 2.11. Let n ∈ N and Y be a subspace of C 2 . The space

V (n) Y,p is dense in H Y for each Y subspace of C 2 .

The linear case

It is instructive to consider the linear case of p = 2 first. Lemma 3.1. Let n ∈ N and Y be a subspace of C 2 . Then the sesquilinear form

a(u, v) := 1 0 u(x)v(x) dx, u, v ∈ V (n) Y,2 ,
is densely defined, symmetric, continuous and coercive in H Y .

Then, well-posedness of first and second order abstract Cauchy problem associated with this form follow directly, by means of the general theory of quadratic forms.

1 (0, 1) → C by γ(f ) := (n -1)(2n -1)f (0) -(n -1) 2 (2n -1)µ n-2 (f ), if n ≥ 2, 0 if n = 1. (3.1) If either Y = {0} 2 or Y = {0} × C, then the operator (A Y , D(A Y )
) associated in H Y with the quadratic form a has domain given by

D(A Y ) = {u ∈ H 1 (0, 1) : µ 0 (u) = µ n (u) = 0} or D(A Y ) = {u ∈ H 1 (0, 1) : µ 0 (u) = 0, u(0) = u(1)}
respectively, and its action is given in both cases by

A Y u = Id -1 m (-u ′′ + γ(u)(1 -id) n-2
). We recall that Id m is the isomorphism introduced in Remark 2.1. The above theorem shows that in particular

A Y u = -u ′′ + γ(u)(1 -id) n-2 in D ′ (0, 1).
Proof. We only consider the case of n ≥ 2, as the case of n = 1 has been proved in [MN11, Thm. 3.1 and 3

.3]. Denote K := {u ∈ H 1 (0, 1) : µ 0 (u) = µ n (u) = 0}. Let us first show the inclusion D(A Y ) ⊂ K. Let f ∈ D(A Y ).
Then there exists g ∈ H -1 (T ) for which µ 0 (g) = 0 (i.e., g ∈ H Y ) and such that

(f |h) L 2 = a(f, h) ! = (g|h) H {0} 2 = 1 0 (P n g)(x)(P n h)(x) dx ∀h ∈ V (n) Y,2 ,
by virtue of Remark 2.3. Now for g ∈ H -1 (T ) it follows from Lemma 2.2.(1) that P n g ∈ L 2 (0, 1), hence (P n (P n g)) ′ = P n g. Integrating by parts we obtain that in particular for all h ∈ V

(n) {0} 2 ,2 1 0 (P n g)(x)(P n h)(x) dx = 1 0 (P n (P n g)) ′ (x)(P n h)(x) dx = - 1 0 (P n (P n g))(x)(P n h) ′ (x) dx = - 1 0 (P n (P n g))(x)h(x) dx. This shows that (f |h) L 2 = -(P n (P n g)|h) L 2 ∀h ∈ V (n) {0} 2 ,2
because of (2.5). Let us now denote by Π the orthogonal projection of L 2 (0, 1) onto the closed subspace of polynomials of one variable spanned by 1 and (1 -id) n . Then we have f = -(I -Π)P n (P n g) + Πf = -P n (P n g) + Π(P n (P n g) + f ).

Accordingly, in either case f ∈ H 1 (0, 1) in view of Lemma 2.2.(2). Moreover,

f ′′ = -g + (ΠP n (P n g))
′′ in D ′ (0, 1).

As Π (P n (P n g) + f ) (x) = α + β(1x) n , for some α, β ∈ R, we have found that

f ′′ = -g + n(n -1)βx n-2 in D ′ (0, 1).

By [MN11, Lemma 2.4] this yields that

A Y f = Id -1 m (-f ′′ + n(n -1)β(1 -x) n-2
). Clearly β depends on f and its dependence will be given below.

Let us finally check that the additional conditions in the definition of

D(A Y ) hold. For all h ∈ V (n) Y,p (f |h) L 2 = a(f, h) ! = (A Y f |h) HY = Id -1 m (-f ′′ + γ(f )(1 -id) n-2 )| h H -1 (T ) = Id -1 m (k ′′ )|h H -1 (T ) where -k(x) := f (x) - 1 n(n -1) γ(f )(1 -x) n )), x ∈ (0, 1),
for some γ(f ) ∈ C to be determined below. Hence in view of Theorem 2.6, and since µ 0 (h) = 0, we find that

(f |h) L 2 = Id -1 m (k ′′ )|h H -1 (T ) = -(k|h) L 2 + nk(0) -n(n -1)µ n-2 (k) (1 -n)k(0) -k(1) + n(n -1)µ n-2 (k) µ 1 (h) µ n (h) C 2 Observe that -(k|h) L 2 = (f |h) L 2 - 1 n(n -1) γ(f )µ n (h)
and furthermore due to Lemma 2.7

(3.2) 0 = -k(0) + (n -1)µ n-2 (k)
in case Y = {0} 2 (and hence µ n (h) = 0); or else

(3.3) nk(0) = n(n -1)µ n-2 (k) 1 n(n-1) γ(f ) = (1 -n)k(0) -k(1) + n(n -1)µ n-2 (k) if Y = {0} × C.
Hence, (3.2) can be re-written as

f (0) - 1 n(n -1) γ(f ) -(n -1)µ n-2 (f ) + 1 n γ(f )µ n-2 ((1 -id) n ) = 0 or rather f (0) -(n -1)µ n-2 (f ) - 1 n(n -1) γ(f ) + 1 n(2n -1) γ(f ) = 0.
Finally, we find

(3.4) f (0) -(n -1)µ n-2 (f ) - 1 (n -1)(2n -1) γ(f ) = 0. If Y = {0} × C, (3.3) can be re-written as 0 = -k(0) + (n -1)µ n-2 (k) n(n -1)γ(f ) = (1 -n)k(0) -k(1) + nk(0),
i.e., (3.4) is satisfied and moreover

1 n(n -1) γ(f ) ! = k(0) -k(1) = -f (0) + 1 n(n -1) γ(f ) + f (1).
Summing up, we see that necessarily

γ(f ) = (n -1)(2n -1)f (0) -(n -1) 2 (2n -1)µ n-2 (f ) if Y = {0} 2 ,
or else

γ(f ) = (n -1)(2n -1)f (0) -(n -1) 2 (2n -1)µ n-2 (f ) f (0) = f (1) if Y = {0} × C.
The converse inclusion can be proven likewise, exploiting our integration-by parts-type formula as in the first part of the proof.

A Y u = Id -1 m (-u ′′ ) for n = 1 whereas we have just proved that A Y u = Id -1 m (-u ′′ + 3u(0)) if n = 2. In general, Id -1 m γ(u)(1 -id) n-2
can be regarded as some sort of potential. In the linear case, this potential can be easily dealt with and, if desired, switched off. We will see in Section 4 that things are different in the nonlinear case.

∂u ∂t (t, x) = ∂ 2 u ∂x 2 (t, x) + ηγ(u)(1 -x) n-2 , t > 0, x ∈ (0, 1),
(where γ(u) is the same term defined in (3.1)) with moment conditions

µ 0 (u(t)) = µ n (u(t)) = 0, t > 0, or µ 0 (u(t)) = 0, u(t, 0) = u(t, 1), t > 0,
and initial condition u(0,

•) = u 0 ∈ {f ∈ H -1 (T ) : µ 0 (f ) = 0}
is well-posed.

We remark explicitly that letting η = 0 we recover well-posedness of the standard heat equation with the above conditions on the moments.

In the proof of this theorem we will need the following two results.

Lemma 3.7. Let p ∈ (1, ∞). If f ∈ W 2,p (0, 1), then

P n (Id -1 m (f ′′ + γ(f )(1 -x) n-2 )) = P n (f ′′ + γ(f )(1 -x) n-2 ) in L 2 (0, 1).
Proof. The claim follows from [MN11, Rem. 2.6] and Remark 2.4.(3).

Lemma 3.8. Let n ≥ 2. Then the following assertions hold.

(1) If Y = {0} 2 , then one has

(3.5) D(A 2 Y ) = {u ∈ H 3 (0, 1) : µ 0 (u) = µ 0 (u ′′ -γ(u)(1 -id) n-2 ) = µ n (u) = µ n (u ′′ -γ(u)(1 -id) n-2 ) = 0}. (2) If Y = {0} × C, then one has (3.6) D(A 2 Y ) = {u ∈ H 3 (0, 1) : µ 0 (u) = µ 0 (u ′′ -γ(u)(1 -id) n-2 ) = 0, u(0) = u(1), u ′′ (0) -γ(u) = u ′′ (1)}. In either case, A Y u = -u ′′ + γ(u)(1 -id) n-2 , ∀u ∈ D(A 2 Y ). Proof.
In either cases, the inclusion "⊃" holds because for u in the right-hand side of (3.5),

-u ′′ + γ(u)(1 -id) n-2 clearly belongs to D(A Y ) and A Y u = -u ′′ + γ(u)(1 -id) n-2 , which also belongs to D(A Y ).
We only prove that "⊂" holds in (1), the corresponding proof in (2) being analogous. Let us take u ∈ D(A 2 Y ). Then u ∈ H 1 (0, 1) and

A Y u = Id -1 m -u ′′ + γ(u)(1 -id) n-2 ∈ H 1 (0, 1
). We first prove that u ′′ ∈ H 1 (0, which clearly implies that u ∈ H 3 (0, 1). Now, set for shortness f := -u ′′ + γ(u)(1 -id) n-2 that clearly belongs to H -1 (0, 1) and consequently

f, v, H -1 (0,1)-H 1 0 (0,1) = Id -1 m f, v H-H 1 (T ) = A Y u, v H-H 1 (T )
∀v ∈ D(0, 1).

Now, because by assumption

A Y u ∈ H 1 (0, 1) we deduce that in fact f, v, H -1 (0,1)-H 1 0 (0,1) = (A Y u|v) L 2 , hence u ′′ = -A Y u + γ(u)(1 -id) n-2 ∈ H 1 (
0, 1) and we conclude that u ∈ H 3 (0, 1), as we wanted to prove.

But now as f belongs to H 1 (0, 1), by [MN11, Lemma 2.4]

Id -1 m f = f -µ 0 (f )δ 1 . Because both f and Id -1
m f and hence also µ 0 (f )δ 1 belong to H 1 (0, 1), we have that necessarily µ 0 (f ) = 0 and therefore by Theorem 3.3 and Lemma 3.7,

A Y u = -u ′′ + γ(u)(1 -id) n-2 .
Accordingly, because u ∈ D(A 2 Y ) and hence µ n (A Y u) = 0, it follows that µ n (u ′′γ(u)(1 -id) n-2 ) = 0. This completes the proof.

Proof of Theorem 3.6. We have seen that -A Y generates an analytic semigroup, hence well-posedness of the corresponding parabolic problem follows. By standard analytic semigroup theory each initial data in H Y is immediately mapped by the semigroup into D(A 2 Y ). Hence, by Lemma 3.7 the claim will follow if we show that D(A 2 Y ) ⊂ H 2 (0, 1). But this is just one of the claims of Lemma 3.8. Remark 3.9. Throughout this section we could have considered some perturbations of the quadratic form a, as we have done in [MN11, § 3 and § 4]. In particular, for any 2 × 2-matrix K the additional term

b(u, v) := K µ 0 (u) µ n (u) µ 0 (v) µ n (v)
, ∀u, v ∈ H 1 (0, 1), may be studied. For the sake of brevity, we avoid to discuss this issue thoroughly: It suffices to observe that for all n ∈ N there exists C n > 0 such that

(3.7) |µ n (g)| 2 ≤ C n g L 2 g H -1 (T ) , ∀g ∈ L 2 (0, 1) :
this can be shown following the proof of [MN11, Lemma 2.12] and taking into account Remark 2.4.(1). Accordingly, the sesquilinear form a + b fits the framework of [START_REF] Crouzeix | Operators with numerical range in a parabola[END_REF], hence (minus) the operator associated with this form in H Y generates a cosine operator function and an analytic semigroup of angle π 2 . 3.2. The general case. We now complete our discussion of the linear heat equation by considering the remaining cases.

D(A Y ) = u ∈ H 1 (0, 1) : µ 0 (u) µ n (u) ∈ Y A Y u = Id -1 m (-u ′′ + γ(u)(1 -id) n-2 ) -c(u)δ 1
, where γ(u) is defined as in (3.1) and c(u) ∈ C is uniquely determined by the condition

(3.8) c(u) + u(1) u(0) -u(1) ∈ Y ⊥
Proof. Again, we only treat the case n ≥ and refer the reader to [MN11, Thm. 4.3] for the case n = 1. We denote

K 1 := u ∈ H 1 (0, 1) : µ 0 (u) µ n (u) ∈ Y
and proceed in a way similar to that in the proof of Theorem 3.3 in order to determine A Y , which by definition is given by

D(A Y ) := {f ∈ V (n) Y,2 : ∃g ∈ H -1 (T ) : a(f, h) = (g|h) H -1 (T ) ∀h ∈ V (n) Y,2 }, A Y f := g.

Let us first check the inclusion D(A

Y ) ⊂ K 1 . Let f ∈ D(A Y ). Then f ∈ V (n) Y,2 and there exists g ∈ H -1 (T ) such that (3.9) (f |h) L 2 = 1 0 (P n g)(x)(P n h)(x) dx + µ 0 (g)µ 0 ( h) ∀h ∈ V (n) Y,2 .
Now, because g ∈ H -1 (T ), by Lemma 2.2, we can consider P n g that belongs to L 2 (0, 1). Therefore by integration by parts and taking into account Lemma 2.2.(1) we obtain that 1 0

(P n g)(x)(P n h)(x) dx = 1 0 (P n (P n g)) ′ (x)(P n h)(x) dx = - 1 0 (P n (P n g))(x)h(x) dx + [P n (P n g)(P n h)] 1 0 ∀h ∈ V (n) Y,2 .
Now, observe that the scalar number µ 0 (g)µ 0 (h) + [P n (P n g)(P n h)] 1 0 ∈ C is a linear combination of µ 0 (h) and µ n (h), hence it can be written in the form

c 0 µ 0 (h) + c n µ n (h) = 1 0 (c 0 + c 1 (1 -x) n )h(x)dx, for some c 0 , c n ∈ C. Letting ρ(x) := c 0 + c 1 (1 -x) n , we obtain that (f |h) L 2 = (-P n (P n g) + ρ|h) L 2 ∀h ∈ V (n) Y,2 .
Therefore, denoting by Π as in the proof of Theorem 3.3 the orthogonal projection of L 2 (0, 1) onto the vector space spanned by 1 and (1 -id) n , we obtain (by restricting the previous identity to all

h ∈ V (n) {0} 2 ,2 ⊂ V (n) Y,2 ) (I -Π)(f + P n (P n g) -ρ) = 0,
or equivalently (3.10) f = (I -Π)(-P n (P n g) + ρ) + Πf = -P n (P n g) + Π(P n (P n g) + f ).

This proves that f belongs to H 1 1) and (differentiating (3.10) twice) that

(3.11) g = -f ′′ + γ(1 -id) n-2 ,
in the distributional sense (i.e. in D ′ (0, 1)) for some γ ∈ C. Hence, by [MN11, Lemma 2.4], there exists c(f

) ∈ C such that A Y f = g = Id -1 m (-f ′′ + γ(1 -id) n-2 ) -cδ 1 , and in fact c(f ) = -µ 0 (g).
It remains to check the condition (3.8). But we first notice that, for all f ∈ D(A Y ), (3.10) leads to

f ′ = -P n g - γ n -1 (1 -id) n-1 ,
for the same γ as in (3.11). By (3.9) we obtain

1 0 f (x) h(x) dx = 1 0 (-f ′ (x) - γ n -1 (1 -x) n-1 )(P n h)(x) dx -c(f )µ 0 (h) = - 1 0 f ′ (x)(P n h)(x) dx - γ n -1 µ n-1 (P n h) -c(f )µ 0 (h) ∀h ∈ V (n) Y,2 .
As µ n-1 (P n h) = 0 by Remark 2.4.(2), we deduce that (3.12)

1 0 f (x)h(x) dx = - 1 0 f ′ (x)(P n h)(x) dx -c(f )µ 0 (h) ∀h ∈ V (n) Y,2 .
Integrating by parts the first term on the right-hand side we obtain

- 1 0 f ′ (x)(P n h)(x) dx = 1 0 f (x) h(x) dx + f (0)(P n h)(0) -f (1)(P n h)(1) = 1 0 f (x) h(x) dx -f (0)µ n ( h) -f (1)(µ 0 (h) -µ n ( h)),
owing to Lemma 2.2.(1) and (2.5), respectively. Plugging this identity in (3.12) we find that

-(f (1) + c(f ))µ 0 (h) + (f (1) -f (0))µ n ( h) = 0 ∀h ∈ V (n) Y,2
. By the surjectivity result from Lemma 2.7, we have shown (3.8): Let us notice that (3.8) determines in a unique way c.

We now prove the converse inclusion. Let then f ∈ K 1 . Then we can take

g := Id -1 m (-f ′′ + γ(1 -id) n-2 ) -cδ 1 , with c ∈ C
fixed by the condition (3.8) (and is equal to -µ 0 (g)) and γ will be fixed later on. Hence by definition of the inner product in H -1 (T ) and the fact that P n δ 1 0 by Remark 2.4.(3), we will have for any

h ∈ V (n) Y,2 (g|h) H -1 (T ) = (P n (Id -1 m (-f ′′ + γ(1 -id) n-2 ))|P n h) L 2 -cµ 0 (h) = -(P n (Id -1 m f ′′ |P n h) L 2 + γ(P n (1 -id) n-2 )|P n h) L 2 -cµ 0 (h).
But simple calculations and an integration by parts yield

(3.13) (P n (1 -id) n-2 )|P n h) L 2 = n (n -1)(2n -1) (µ 1 ( h) -µ n ( h)).
Hence by Theorem 2.6, for all h ∈ V (n)

Y,2 we get

(3.14) (g|h) H -1 (T ) = (f |h) L 2 +   -(f (1) + c) -nf (0) + n(n -1)µ n-2 (f ) + nγ (n-1)(2n-1) -f (0) + f (1) -n(n -1)µ n-2 (f ) + nf (0) - nγ (n-1)(2n-1)     µ 0 (h) µ 1 (h) µ n (h)   .
From this identity we see that f will belong to D(A Y ) if the second entry of the second term in the right-hand side of this identity is zero. This motivates us to choose γ such that

-nf (0) + n(n -1)µ n-2 (f ) + nγ (n -1)(2n -1) = 0,
which is equivalent to (3.1). With this choice we see that (3.14) is equivalent to

(g|h) H -1 (T ) = (f |h) L 2 + -(f (1) + c) -f (0) + f (1) µ 0 (h) µ n (h) = (f |h) L 2 , ∀h ∈ V (n) Y,2 ,
the second identity following from (3.8). This shows that

a(f, h) = (g|h) H -1 (T ) , ∀h ∈ V (n)
Y,2 , and proves that f belongs to D(A Y ).

Corollary 3.11. Under the assumptions of Theorem 3.10, let η ∈ C. Then the operator A Y given by

D(A Y ) := u ∈ H 1 (0, 1) : µ 0 (u) µ n (u) ∈ Y , A Y u := Id -1 m (u ′′ ) + ηγ(u)Id -1 m (1 -id) n-2
, generates on H Y = H -1 (T ) an analytic, uniformly exponentially stable semigroup of angle π 2 that is immediately of trace class.

Proof. Again the assertion follows directly from the observation that u → γ(u)Id -1 m (1 -id) n-2 is a relatively compact perturbation of A Y . Hence a well known perturbation result, cf. [ABHN01, Thm. 3.7.25], allows us to conclude.

As a consequence we obtain the following existence result. Theorem 3.12. Let n ∈ N and η, y ∈ C. Then the heat equation

∂u ∂t (t, x) = ∂ 2 u ∂x 2 (t, x) + ηγ(u(t, •))(1 -x) n-2 , t > 0, x ∈ (0, 1),
(where γ(u) is the same term defined in (3.1)) with moment conditions

µ n (u(t, •)) = yµ 0 (u(t, •)), t > 0, -µ n (u ′′ (t, •)) + γ(u(t, •)) 2n -1 = y(-µ 0 (u ′′ (t, •)) + γ(u(t, •)) n -1 ), t > 0,
and This concludes the proof. Lemma 3.13. Under the assumptions of Theorem 3.10 one has

-µ 0 (u ′′ (t, •)) + γ(u(t, •)) n -1 = (u(t, 1) -u(t, 0))ȳ -u(t, 1) t > 0,
D(A 2 Y ) = u ∈ H 3 (0, 1) : µ n (u) = yµ 0 (u), µ n (u ′′ -γ(u)(1 -id) n-2 ) = yµ 0 (u ′′ -γ(u)(1 -id) n-2 ), µ 0 (u ′′ -γ(u)(1 -id) n-2 ) = (u(1) -u(0))ȳ -u(1) ,
when Y is spanned by (1, y) ⊤ and

A Y u = -u ′′ + γ(u)(1 -id) n-2 , ∀u ∈ D(A 2 Y ).
Proof. The inclusion "⊃" holds because for u in the right-hand side of (3.15), -u ′′ + γ(u)(1 -id) n-2 clearly belongs to D(A Y ) and A Y u = -u ′′ + γ(u)(1 -id) n-2 , which also belongs to D(A Y ).

We now prove the converse inclusion "⊂". Let us fix u ∈ D(A 2 Y ). Then u ∈ H 1 (0, 1) and

A Y u = Id -1 m -u ′′ + γ(u)(1 -id) n-2 -c(u)δ 1 ∈ H 1 (0, 1
). This identity implies that

-u ′′ = A Y u -γ(u)(1 -id) n-2 in D ′ (0, 1),
and the assumption A Y u ∈ H 1 (0, 1) implies that u ∈ H 3 (0, 1). With this regularity at hands, by [MN11, Rem. 2.6] we see that

A Y u = -u ′′ + γ(u)(1 -id) n-2 ) + µ 0 (u ′′ ) -γ(u)µ 0 ((1 -id) n-2 ) -c(u) δ 1 ∈ H 1 (0, 1). Hence necessarily µ 0 (u ′′ ) -γ(u)µ 0 ((1 -id) n-2 ) -c(u) = 0, and A Y u = -u ′′ + γ(u)(1 -id) n-2 . Because u ∈ D(A 2 Y ), -u ′′ + γ(u)(1 -id) n-2 belongs to D(A Y
) and the characterization of D(A Y ) completes the proof.

The porous medium equation

Throughout this section, we consider real valued functions spaces and we let

• p ∈ (1, ∞), • Y be a subspace of R 2 , • n ∈ N. We consider the functional E Y : H Y → R + ∪ {+∞} defined by E Y : f → 1 p 1 0 |f (x)| p dx, if f ∈ V (n) Y,
E ′ Y (f )h = 1 0 |f (x)| p-2 f (x)h(x) dx, f, h ∈ V (n) Y,p .
Thus, E Y is in particular lower semicontinuous as a functional on

V (n) Y,
D(A Y ) := {f ∈ L p (0, 1) : |f | p-2 f ∈ H 1 (0, 1) : µ 0 (f ) = µ n (f ) = 0} or D(A Y ) := {f ∈ L p (0, 1) : |f | p-2 f ∈ H 1 (0, 1) : µ 0 (f ) = 0, f (0) = f (1)}
respectively, whose action is given in both cases by

A Y f := Id -1 m (-(|f | p-2 f ) ′′ + γ(f )(1 -id) n-2 ). (2) If Y = {0} 2 and Y = {0} × R, then the subdifferential ∂E Y of E Y with respect to H Y agrees with the nonlinear operator A Y given by D(A Y ) := f ∈ L p (0, 1) : |f | p-2 f ∈ H 1 (0, 1) and µ 0 (f ) µ n (f ) ∈ Y ,
whose action is given by

A Y f := Id -1 m (-(|f | p-2 f ) ′′ + γ(|f | p-2 f )(1 -id) n-2 ) -c(f )δ 1 . Here c(f ) ∈ R is uniquely determined by the condition (4.2) c(f ) + |f | p-2 (1)f (1) |f | p-2 (0)f (0) -|f | p-2 (1)f (1) ∈ Y ⊥ .
Here, by definition, for a given reflexive Banach space X the subdifferential of a functional E with respect to a Hilbert space H such that X is continuously and densely embedded in H is the single-valued operator given by

D(∂E) := {f ∈ H Y : ∃ g ∈ H Y s.t. E ′ (f )h = (g|h) HY ∀h ∈ V (n) Y,p }, ∂E(f ) := g, cf. [Nit10, Lemma 2.8.9].
Proof. We do not deliver the proof, which can be performed closely following those of Theorems 3.3 and 3.10 if n ≥ 2, or rather the proofs of [MN11, Thm. 3.1, 3.3, 4.3] if n = 1, up to replacing f by |f | p-2 f throughout. The only noteworthy modifications are the following ones: On one hand, one has to replace the orthogonal projection Πψ ∈ L 2 of a vector ψ ∈ L 2 (0, 1) onto the space of polynomials spanned by 1 and (1 -id) n by the vector Πψ ∈ L p ′ (0, 1) of best approximation of ψ ∈ L p ′ (0, 1) in the subspace of R[x] spanned by 1 and (1 -id) n : This makes sense even if Π is not an orthogonal projection, and of course R

[x] = ΠR[x] + (I -Π)R[x]
is dense in L p (0, 1). On the other hand, one must check directly that each f ∈ D(∂E) is of class L p (0, 1) -this can in turn be done observing that because by assumption |f | p-2 f ∈ H 1 (0, 1) ֒→ L p p-1 (0, 1), f p ∈ L 1 (0, 1), i.e., f ∈ L p (0, 1). By virtue of Theorems4.1 and 4.3, the following assertion is a direct consequence of the general theory of nonlinear semigroups associated with subdifferentials, see e.g. [Sho97, Prop. IV.5.2] and [Bré73,

Théo. 3.2]. Proposition 4.4. Let p ∈ (1, ∞). Let T > 0, u 0 ∈ V (n) Y,p and f ∈ L 2 (0, T ; H Y ). Then the abstract Cauchy problem (4.3) du dt (t) = -A Y u(t) + f (t), t > 0, u(0) = u 0 , admits a unique solution u ∈ H 1 (0, T ; H Y ) ∩ L ∞ (0, T ; V (n) Y,p ).
If f ≡ 0, then the the following additional assertions hold:

• the solution of (4.3) is given by a strongly continuous semigroup of nonlinear contractions,

• u(t, •) ∈ D(A Y ) for all t > 0, • u ∈ C([0, ∞); H Y )
is everywhere right differentiable and satisfies (4.3) -and in particular the initial condition -for all t ≥ 0, and finally • t → u(t) p L p is convex and monotonically decreasing on each interval [δ, ∞) for all δ > 0.

Remark 4.5. Observe that A Y is not the operator associated with the (non-signed) PME. Hence, unlike in the setting discussed in Section 3, in the present nonlinear case we are not able to show that the Cauchy problem associated with -A Y takes the form of a common PME (possibly with potential), because the general theory of subdifferentials apparently does not ensure enough regularity of solutions to allow us to drop the term Id -1 m . However, taking into account Remark 2.4.(5) one observes that A Y does indeed act on twice weakly differentiable functions as a second order differential operator with a potential term.

Observe that a possible, weaker but still sufficient approach would consist in restricting ourselves to consider initial values in V (n) Y,p , or some other subspace of L 2 (0, 1) that is left invariant under the semigroup; and then showing that the restriction of the semigroup is strongly continuous, and that its generator is a restriction of A Y to some subspace of H 2 (0, 1). In view of Lemma 3.7, this would do the job. Unfortunately, it seems to be unclear to which extent this kind of procedure is allowed by the general theory of nonlinear semigroups. However, it is for instance known that the operators associated with the PME on bounded domains Ω "generate semigroups of order-preserving contractions in L 1 (Ω)", cf. [Váz07, § 20.1.2].

We have already seen that in the linear case the problem is governed by a uniformly exponentially stable semigroup on H Y . This means that the H Y -norm of each solution tends to 0 as t → ∞, uniformly for all initial data. What about the nonlinear case? We alreay know from Theorem 4.4 that if the inhomogeneous term f ≡ 0, then the L p -norm of the solution is decreasing. Further information can be obtained. Inspired by the classical analysis of the PME, see e.g. [START_REF] Vázquez | The Porous Medium Equation: Mathematical Theory[END_REF]Thm. 11.9], we obtain the following, where for the sake of simplicity we restrict to the case of Y = {0} 2 . Observe that for p = 2 we recover information already yield from the exponential stability of the linear semigroup (cf. Corollary 3.5), whereas for p < 2 (the case of the FDE) we deduce exponential decay -with decay rate depending on the initial data. if n ≥ 2, and a similar but simpler computations holds for n = 1. We finally arrive at (Without loss of generality we can assume that v(t) > 0, for all t > 0, otherwise as v is decaying, v would be zero after a time t 0 > 0). By direct calculations, we have

w ′ = (α -1)(C + v -α v ′ ), ∀t > 0,
and by (4.4), we get that w is decaying. Hence w(t) ≤ w(0), ∀t > 0, or equivalently v(t) ≤ Kt -1 α-1 , ∀t > 0, with K := (C(α -1))

-1

α-1 > 0, which shows the polynomial decay. 2) If α ≤ 1, then again by the decay of v, we have v(t) ≤ v(0) 1-α v(t) α , ∀t > 0, and therefore it follows from (4.4) that (4.5) v ′ (t) ≤ -Kv(t) α , with K := C v(0) 1-α . As before this property implies that the mapping t → v(t)e Kt is decaying and therefore we obtain v(t) ≤ v(0)e -Kt , ∀t > 0, which shows the claimed exponential decay.

  of the converse inclusion is divided in several steps, but follows closely [MN11, Cor. 2.4 and Thm. 4.2]. We first prove the assertion for Y = {0} 2 .

Corollary 3. 2 .

 2 Let n ∈ N and Y be a subspace of C 2 . Then the operator (A Y , D(A Y )) associated with (a, V (n) Y,2 ) is positive definite on H. In particular, -A Y generates a cosine operator function with associated phase space V (n) Y,2 × H Y and hence an exponentially stable, contractive, analytic semigroup (e -tAY ) t≥0 of angle π 2 on H Y . This semigroup is immediately of trace class. 3.1. The case of µ 0 (u) = 0. It remains to determine the operator associated with the form, and hence the Cauchy problems actually solved by the cosine operator functions and the operator semigroup. To begin with, we consider the case where we impose µ 0 (u) = 0, corresponding to setting Y = {0} 2 or Y = {0} × C. Theorem 3.3. Let n ∈ N and define a bounded linear functional γ : H

Remark 3. 4 .

 4 Let us emphasize that taking into account [MN11, Thm. 3.1 and 3.3] and Theorem 3.3 one sees that the domains of A Y coincide for all n ∈ N, in the case of Y = {0} × C. Comparing the special cases of n = 1 and n = 2 one sees that

Corollary 3. 5 .

 5 Let n ∈ N and define the bounded linear functional γ : H 1 (0, 1) → C as in (3.1). Let η ∈ C. Then the operator given byu → Id -1 m (u ′′ ) + ηId -1 m γ(u)(1 -id) n-2 . either with domain {u ∈ H 1 (0, 1) : µ 0 (u) = µ n (u) = 0} or {u ∈ H 1 (0, 1) : µ 0 (u) = 0, u(0) = u(1)}generates on {f ∈ H -1 (T ) : µ 0 (f ) = 0} an analytic, uniformly exponentially stable semigroup of angle π 2 that is immediately of trace class. Proof. The assertion follows directly from the observation thatu → Id -1 m γ(u)(1 -id) n-2 is a relatively compact perturbation of A Y ,and from well known perturbation results, cf. [ABHN01, Thm. 3.7.25]. The following generalizes the well-posedness result [MN11, Thm. 3.7]. Theorem 3.6. Let n ∈ N and η ∈ C. The heat-type equation

Theorem 3. 10 .

 10 Let n ∈ N and Y be a subspace of C 2 , Y = {0} 2 and Y = {0} × C. Then the operator (A Y , D(A Y )) associated in H Y with the quadratic form a is given by

  and initial condition u(0, •) = u 0 ∈ H -1 (T ) is well-posed.Proof. It suffices to apply the previous Corollary with Y spanned by (1, y) ⊤ and Lemma 3.13 below. Note that the condition (3.8) is then equivalent to c(u) = -u(1) -(u(0)u(1))ȳ.

Proposition 4. 6 .

 6 Consider the abstract Cauchy problem (4.3) with u 0 ∈ V (n) Y,p and f = 0. If Y = {0} 2 , then the solution u given by Theorem 4.4 satisfiesu(t) 2 HY ≤ Kt -2 p-2 if p > 2,for some K > 0 only depending on p, whereasu(t) 2 HY ≤ u(0) 2 HY e -Kt if p ≤ 2,for some K > 0 proportional to u(0)

Proof.=

  By the definition of the inner product in H Y and since by assumption µ 0 (u(t)) = µ n (u(t)) = 0, we find thatu(t) (1 -id) n-2 HY = , y)dy(1x) n-1 dx -n (n -1)(2n -1) µ 1 (u(t)) = 1 n µ n (u(t)) -n (n -1)(2n -1) µ 1 (u(t)) = -n (n -1)(2n -1) µ 1 (u(t)),where the second identity follows integrating by parts. Hence for all t ≥ -(u(t)|A Y u(t)) HY =u(t)|Id -1 m (-(|u(t)| p-2 u(t)) ′′ + γ(u(t))(1 -id) n-2 ) HY =u(t)|(|u(t)| p-2 u(t)) L2γ(u(t)) u(t) (1 -id) n-2 HY =u(t) p V (n) Y,p+ (u(t, 0) -(n -1)µ n-2 (u(t))) (n -1)(2n -1) , y)dy(1x) n-1 dx.

  ֒→ H Y , we get 1 2d u(t) 2 HY dt ≤ -C 0 u(t) p HY , for some C 0 > 0. Setting v(t) := u(t) 2 HY , t ≥ 0,we have shown that(4.4) v ′ (t) ≤ -Cv(t) α ,with α := p 2 and C := 2C 0 . Now we distinguish the case α > 1 and α ≤ 1.1) In the first case we consider the function w defined by w(t) := C(α -1)tv(t) 1-α , t ≥ 0.

  Let Y be a subspace of R 2 . Then the functional E Y is proper and convex. Furthermore, E Y is continuously Fréchet differentiable as a functional on V It is clear that E Y is proper and convex. Moreover, it is continuously differentiable -in fact, the Fréchet derivative of E Y is given by

		p ,
	+∞,	otherwise,
	where H Y and V Y,p are the spaces introduced in (2.11) and (2.22). It turns out that the subdifferential (n)
	of E Y has particularly good properties.
	Theorem 4.1. (n) Y,p and lower semicontinuous as a
	functional on H Y .
	Proof. (4.1)

  p , and lower semicontinuity as a functional on H Y follows from [Sho97, Lemma IV.5.2]. If either Y = {0} 2 or Y = {0} × R, then the subdifferential ∂E Y of E Y with respect to H Y agrees with the nonlinear operator A Y given by

	Remark 4.2. An additional term of the form Φ(	µ 0 (f ) µ n (f )	), where Φ : Y → R is a convex continuously
	differentiable function, may be easily dealt with by means of simple perturbation results for nonlinear
	forms, see e.g. [MP11].
	Theorem 4.3. Let γ(•) be defined by (3.1). Then the following assertions hold.
	(1)


Serge Nicaise

Cristina Pignotti

Well-posedness and stability results for nonlinear abstract evolution equations with time delays

We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the C 0 -semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a general exponential decay estimate for small time delays if the nonlinear term is globally Lipschitz and an exponential decay estimate for solutions starting from small data when the nonlinearity is only locally Lipschitz and the linear part is a negative selfadjoint operator. In the latter case we do not need any restriction on the size of the time delays. In both cases, concrete examples are presented that illustrate our abstract results.

Introduction

Let H be a fixed Hilbert space with inner product (•, •) H and norm • H , and consider an operator A from H into itself that generates a C 0 -semigroup (S(t)) t≥0 that is exponentially stable, i.e., there exist two positive constants M and ω such that S(t) L(H) ≤ Me -ωt , ∀ t ≥ 0, (1.1) where, as usual, L(H) denotes the space of bounded linear operators from H into itself.

We consider the evolution equation U t (t) = AU(t) + I i=1 F i (U(t), U(tτ i )) in (0, +∞), U(tτ ) = U 0 (t), ∀t ∈ (0, τ ], (1.2) where I is a positive natural number and τ i > 0, i = 1, • • • , I, are time delays. Without loss of generality we can suppose that the delays are different from each other and that

τ i < τ = τ 1 , ∀i = 2, • • • , I.
The nonlinear terms F i : H × H → H satisfy some Lipschitz conditions, while the initial datum U 0 satisfies U 0 ∈ C([0, τ ]; H). We denote by U 0 the initial datum at t = 0, namely

U 0 = U 0 (τ ) ∈ H . (1.3)
Time delay effects frequently appear in many practical applications and physical models. On the other hand, it is well-known (cfr. [START_REF] Bátkai | Semigroups for delay equations[END_REF][START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF][START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF]) that they can induce some instability. Then, we are interested in giving an exponential stability result for such a problem under a suitable smallness condition on the delay τ or on the initial data. In [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF] we have studied the case of linear dependency on the delayed state U(tτ ), i.e., we have considered the model U t (t) = AU(t) + G(U(t)) + kBU(tτ ) in (0, +∞), U(0) = U 0 , BU(tτ ) = g(t), ∀t ∈ (0, τ ), (1.4) where B is a fixed bounded operator from H into itself, G : H → H satisfies some Lipschitz conditions, the initial datum U 0 belongs to H and g ∈ C([0, τ ]; H). For some concrete examples, mainly for G ≡ 0, it was known that the above problem, under certain smallness conditions on the delay feedback kB, is exponentially stable, the proof being from time to time quite technical because some observability inequalities or perturbation methods are used. For instance the case of wave equation with interior feedback delay and dissipative boundary condition was considered in [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF] by constructing suitable Lyapunov functionals while the case of a locally damped wave equation with distributed delay has been analyzed in [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF] by introducing an auxiliary model easier to deal with and using a perturbative argument. The last approach has then been extended to a general class of second order evolution equations in [START_REF] Nicaise | Stabilization of second-order evolution equations with time delay[END_REF]. The case of a Timoshenko system with delay has been studied in [START_REF] Said-Houari | Stability result of the Timoshenko system with delay and boundary feedback[END_REF] by using appropriate Lyapunov functionals. Also wave type equations with viscoelastic dissipative damping and time delay feedback have been considered (see e.g. [START_REF] Guesmia | Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay[END_REF][START_REF] Alabau-Boussouira | Exponential stability of the wave equation with memory and time delay[END_REF]). Moreover we quote the book [START_REF] Bátkai | Semigroups for delay equations[END_REF] for several examples, also in the parabolic case.

Therefore, in [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF] our main goal was to furnish a general stability result based on a direct and more simple proof obtained by using the so-called Duhamel's formula (or variation of parameters formula). Indeed we proved an exponential stability result under a suitable condition between the constant k and the constants M, ω, τ, the nonlinear term G and the norm of the bounded delay operator B . Now, we extend the results in [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF] by analyzing the more general model (1.2). In particular, while in [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF] we have considered only linear dependency on the delayed state U(tτ ) (see (1.4)), we now consider a nonlinear dependency on the state and also on the delayed states. We first treat the case when F i , i = 1, . . . , I, are globally Lipschitz continuous and give an exponential stability result under a smallness assumption on the time delays. The proof, when adapted to model (1.4), is easier with respect to the one that we gave in [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF] for the globally Lipschitz case. However, the result from [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF] allowed a bit larger size on the time delay (see Remark 2.5) in order to have an exponential stability estimate for problem (1.4).

Then, we assume that the nonlinearity is only locally Lipschitz continuous (with a stronger topology) and we prove global existence and an exponential decay estimates for small initial data, independently of the size of the time delays. In this last case, we need to restrict ourselves to the case of a negative self-adjoint operator A. Nevertheless, this last case has various applications in population dynamics (see section 4).

The paper is organized as follow. In section 2 we deal with the globally Lipschitz case; we give a well-posedness result and an exponential stability estimate for small delays. Some concrete examples are also presented. In section 3 we treat the case where the reaction term is only locally Lipschitz. We prove the local well-posedness and then, for small initial conditions, a global existence result and an exponential decay estimate. Finally, in section 4 we give some illustrative applications of the abstract results of section 3 to models in population dynamics. In particular we consider model for single species and competition models for two species.

The case of small delays with F i globally Lipschitz

In this section we assume that the nonlinear terms are globally Lipschitz and study existence and asymptotic behavior of solutions, generalizing Proposition 2.1 and Theorem 2.2 of [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF].

Well-posedness and stability estimate

Let the functions F i , i = 1, . . . , I, be globally Lipschitz continuous, namely for every i = 1, . . . , I,

∃γ i > 0 such that F i (U 1 , U 2 ) -F i (U * 1 , U * 2 ) H ≤ γ i ( U 1 -U * 1 H + U 2 -U * 2 H ), ∀ (U 1 , U 2 ), (U * 1 , U * 2 ) ∈ H × H . (2.
1) The following well-posedness result holds. Proposition 2.1 Assume that the functions F i satisfy (2.1), for all i = 1, . . . , I. For any initial datum U 0 ∈ C([0, τ ]; H), there exists a unique (mild

) solution U ∈ C([0, +∞), H) of problem (1.2). Moreover, U(t) = S(t)U 0 + t 0 S(t -s) I i=1 F i (U(s), U(s -τ i )) ds.
(2.2)

Proof. Like in [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF], we use an iterative argument. Namely in the interval (0, τ min ), where τ min = min i=1,...,I τ i , problem (1.2) can be seen as a standard evolution problem

U t (t) = AU(t) + g 1 (U(t)) in (0, τ min ) U(0) = U 0 , (2.3)
where

g 1 (U(t)) = I i=1 F i (U(t), U(t -τ i ))
. Note that the terms U(tτ i ) can be regarded as known data for t ∈ [0, τ min ) . This problem has a unique solution U ∈ C([0, τ min ], H) (see Th. 1.2, Ch. 6 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]) satisfying

U(t) = S(t)U 0 + t 0 S(t -s)g 1 (U(s)) ds.
This yields U(t), for t ∈ [0, τ min ]. Therefore on (τ min , 2τ min ), problem (1.2) can be seen as the evolution problem

U t (t) = AU(t) + g 2 (U(t)) in (τ min , 2τ min ) U(τ min ) = U(τ min -), (2.4)
where g 2 (U(t)) = I i=1 F i (U(t), U(tτ i )) , since the terms U(tτ i ) can be regarded as data being known from the first step. Hence, this problem has a unique solution U ∈ C([τ min , 2τ min ], H) given by

U(t) = S(t -τ min )U(τ min -) + t τ min S(t -s)g 2 (U(s)) ds, ∀t ∈ [τ min , 2τ min ].
By iterating this procedure, we obtain a global solution U satisfying (2.2). Now we will prove the following exponential stability result.

Theorem 2.2 Assume that the functions F i satisfy (2.1) and

F i (0, 0) = 0, (2.5)
for all i = 1, . . . , I. With M, ω from (1.1), we assume that (see (2.1))

γ = I i=1 γ i < ω 2M . ( 2
U(t) H ≤ Me -(ω-M γ(1+e ωτ ))t ( U 0 H + α) , t ≥ 0 . (2.17)
Then, exponential decay is ensured under the condition

ω > Mγ(1 + e ωτ ) . (2.18)
Under the assumption (2.6), inequality (2.18) is satisfied if and only if τ satisfies the smallness condition (2.7). Then, the statement is proved.

We can restate the previous theorem for the particular case of model (1.4):

Theorem 2.3 Assume that the nonlinear term G satisfies the Lipschitz condition [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF][START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF][START_REF] Nicaise | Stabilization of second-order evolution equations with time delay[END_REF] and subsections 2.2-2.3 below) and Timoshenko models (cfr. [START_REF] Said-Houari | Stability result of the Timoshenko system with delay and boundary feedback[END_REF]) eventually with the addition of a nonlinear term depending on the not delayed state and on a finite number of delayed states. Also, it includes recent stability results for problems with viscoelastic damping and time delay (cfr. [START_REF] Guesmia | Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay[END_REF][START_REF] Alabau-Boussouira | Exponential stability of the wave equation with memory and time delay[END_REF]).

∃γ > 0 such that G(U 1 ) -G(U * 1 ) H ≤ γ U 1 -U * 1 H , ∀ U 1 , U * 1 ∈ H . ( 2
Remark 2.5 Note that the condition (2.20) is a bit less general with respect to the one obtained in [START_REF] Nicaise | Exponential stability of abstract evolution equations with time delay[END_REF]. Indeed, here we need

k B H e ωτ + γ < ω M ,
instead of the condition k B H e ωτ + γ < e ωτ -1 Mτ assumed there. On the other hand, the present approach allows to extend the class of the problems for which the stability result holds.

We now present a couple of examples, among the many, illustrating our previous abstract result.

The damped wave equation

Let Ω be a bounded domain in IR n with a boundary Γ of class C 2 . Let f j : IR → IR be globally Lipschitz continuous functions, j = 1, 2, satisfying f 1 (0) = f 2 (0) = 0. Let us consider the following semilinear damped wave equation:

u tt (x, t) -∆u(x, t) + a(x)u t (x, t) = f 1 (u(x, t)) + f 2 (u(x, t -τ )) in Ω × (0, +∞), u(x, t) = 0 in Ω × (0, +∞), u(x, t -τ ) = u 0 (x, t), u t (x, t -τ ) = u 1 (x, t) in Ω × (0, τ ] ,
where τ > 0 is the time delay and the damping coefficient

a ∈ L ∞ (Ω) satisfies a(x) ≥ a 0 > 0, a.e. x ∈ ω , (2.22)
for some nonempty open subset ω of Ω satisfying some control geometric properties (see e.g. [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]). The initial datum (u

0 , u 1 ) is taken in C([0, τ ], H 1 0 (Ω) × L 2 (Ω)). Setting U = (u, u t ) T , this problem can be rewritten in the form (1.2) with H = H 1 0 (Ω) × L 2 (Ω), A = 0 1 ∆ -a and F 1 (U(t), U(t -τ )) = (0, f 1 (u(t)) + f 2 (u(t -τ ))) T .
It is well-known that A generates a strongly continuous semigroup which is exponentially stable (see e.g. [START_REF] Zuazua | Exponential decay for the semi-linear wave equation with locally distributed damping[END_REF][START_REF] Komornik | Exact controllability and stabilization, the multiplier method[END_REF]), thus the assumptions on f 1 , f 2 ensure that Proposition 2.1 and Theorem 2.2 apply to this model giving a well-posedness result and an exponential decay estimate of the energy for small values of the time delay τ.

The wave equation with memory

Let Ω ⊂ IR n be an open bounded set with a smooth boundary and let f j , j = 1, 2, as in the previous example. Let us consider the following problem:

u tt (x, t) -∆u(x, t) + ∞ 0 µ(s)∆u(x, t -s)ds = f 1 (u(x, t)) + f 2 (u(x, t -τ )) in Ω × (0, +∞), (2.23) u(x, t) = 0 on ∂Ω × (0, +∞), (2.24) u(x, t) = u 0 (x, t) in Ω × (-∞, 0], (2.25)
where the initial datum u 0 belongs to a suitable space, the constant τ > 0 is the time delay and the memory kernel

µ : [0, +∞) → [0, +∞) is a locally absolutely continuous function satisfying i) µ(0) = µ 0 > 0; ii) +∞ 0 µ(t)dt = μ < 1; iii) µ ′ (t) ≤ -αµ(t), for some α > 0.
As in [START_REF] Dafermos | Asymptotic stability in viscoelasticity[END_REF], we denote η t (x, s) := u(x, t)u(x, ts).

(2.26)

Then we can restate (2.23)-(2.25) as

u tt (x, t) = (1 -μ)∆u(x, t) + ∞ 0 µ(s)∆η t (x, s)ds = f 1 (u(x, t)) + f 2 (u(x, t -τ )) in Ω × (0, +∞) (2.27) η t t (x, s) = -η t s (x, s) + u t (x, t) in Ω × (0, +∞) × (0, +∞), (2.28) u(x, t) = 0 on ∂Ω × (0, +∞) (2.29) η t (x, s) = 0 in ∂Ω × (0, +∞), t ≥ 0, (2.30) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, (2.31) η 0 (x, s) = η 0 (x, s) in Ω × (0, +∞), (2.32) where u 0 (x) = u 0 (x, 0), x ∈ Ω, u 1 (x) = ∂u 0 ∂t (x, t)| t=0 , x ∈ Ω, η 0 (x, s) = u 0 (x, 0) -u 0 (x, -s), x ∈ Ω, s ∈ (0, +∞). (2.33) Let us denote U := (u, u t , η t ) T . Then we can rewrite problem (2.27)-(2.32) in the abstract form U t (t) = AU(t) + F 1 (U(t), U(t -τ )), U(0) = (u 0 , u 1 , η 0 ) T , (2.34)
where the operator A is defined by

A   u v w   :=   v (1 -μ)∆u + ∞ 0 µ(s)∆w(s)ds -w s + v   , (2.35)
with domain

D(A) := (u, v, η) T ∈ H 1 0 (Ω) × H 1 0 (Ω) × L 2 µ ((0, +∞); H 1 0 (Ω)) : (1 -μ)u + ∞ 0 µ(s)η(s)ds ∈ H 2 (Ω) ∩ H 1 0 (Ω), η s ∈ L 2 µ ((0, +∞); H 1 0 (Ω)) , (2.36)
where L 2 µ ((0, ∞); H 1 0 (Ω)) is the Hilbert space of H 1 0 -valued functions on (0, +∞), endowed with the inner product

ϕ, ψ L 2 µ ((0,∞);H 1 0 (Ω)) = Ω ∞ 0 µ(s)∇ϕ(x, s)∇ψ(x, s)ds dx.
Denote by H the Hilbert space

H = H 1 0 (Ω) × L 2 (Ω) × L 2 µ ((0, ∞); H 1 0 (Ω)), equipped with the inner product   u v w   ,   ũ ṽw   H := (1 -μ) Ω ∇u∇ũdx + Ω vṽdx + Ω ∞ 0 µ(s)∇w∇ wdsdx.
(2.37) It is well-known (see e.g. [START_REF] Giorgi | Global attractors for a semilinear hyperbolic equation in viscoelasticity[END_REF]) that the operator A generates an exponentially stable semigroup. Thus, Proposition 2.1 and Theorem 2.2 guarantee well-posedness and exponential stability, for small delays, also for the model (2.23)-(2.25).

The case of small data with general nonlinearities

Here we consider a more general class of nonlinearities but assume that A is a negative selfadjoint operator in H. In this case, A generates an analytic semi-group (see [14, Example IX.1.25]) and existence results for problem (1.2) can be obtained for nonlinear terms satisfying the next hypothesis (3.2). More precisely, we recall that V = D((-A)

1 2
) is a Hilbert space with the norm

U 2 V = ((-A) 1 2 U, (-A) 1 2 U), ∀ U ∈ V.
Furthermore if λ 1 is the smallest eigenvalue of -A, we have

λ 1 U 2 H ≤ U 2 V , ∀ U ∈ V. (3.1)
Then we assume that there exist a positive real number β < 1 2 , a constant C 0 and two continuous functions h 1 , h 2 from [0, ∞) 4 to [0, ∞) such that, for all i = 1, . . . , I,

F i (U 1 , V 1 ) -F i (U 2 , V 2 ) H ≤ C 0 U 1 -U 2 H (3.2) +h 1 ( U 1 V , U 2 V , V 1 V , V 2 V ) U 1 -U 2 V +h 2 ( U 1 V , U 2 V , V 1 V , V 2 V ) V 1 -V 2 D((-A) β ) , for all (U 1 , U 2 ), (V 1 , V 2 ) ∈ V × V.
The following local existence result holds (compare with Theorem 1 of [START_REF] Oliva | Reaction-diffusion equations with nonlinear boundary delay[END_REF]).

Proposition 3.1 Assume that the functions F i satisfy (3.2). Then for any initial datum

U 0 ∈ C([0, τ ]; V) ∩ C 0,θ ([0, τ ], D((-A) β )), with β < 1 2 from the assumption (3.
2) and θ = min{β, 1 2 -β}, there exist a time T ∞ ∈ (0, +∞] and a unique solution

U ∈ C([0, T ∞ ), V) ∩ C 1 ((0, T ∞ ), H) of problem (1.2).
Proof. Like in the proof of Proposition 2.1, we look at the problem (1.2) in the time interval [0, τ min ) where, as before, τ min = min i=1,...,I τ i . Then the model can be rewritten as

U t (t) = AU(t) + g(t, U(t)) in (0, τ min ), U(0) = U 0 , (3.3)
where g(t, U(t)) = I i=1 F i (U(t), U(t-τ i )). Recall that the terms U(t-τ i ) can be regarded as known data for t ∈ [0, τ min ) . Then owing to (3.2) the nonlinear part satisfies

g(t 1 , U 1 ) -g(t 2 , U 2 ) H = I i=1 [F i (U 1 , U 0 (t 1 -τ i + τ )) -F i (U 2 , U 0 (t 2 -τ i + τ ))] H ≤ I i=1 F i (U 1 , U 0 (t 1 -τ i + τ )) -F i (U 2 , U 0 (t 2 -τ i + τ )) H ≤ IC 0 U 1 -U 2 H + I i=1 h 1 ( U 1 V , U 2 V , U 0 (t 1 -τ i + τ ) V , U 0 (t 2 -τ i + τ ) V ) U 1 -U 2 V + I i=1 h 2 ( U 1 V , U 2 V , U 0 (t 1 -τ i + τ ) V , U 0 (t 2 -τ i + τ ) V )• • U 0 (t 1 -τ i + τ ) -U 0 (t 2 -τ i + τ ) D(-A β ) .
(3.4) From (3.4), using (3.1), easily follows

g(t 1 , U 1 ) -g(t 2 , U 2 ) H ≤ L(R)(|t 1 -t 2 | θ + U 1 -U 2 V ) , for all (U 1 , U 2 ) ∈ V × V with U 1 V , U 2 V ≤ R and for all t 1 , t 2 ∈ [0, τ min ]
. Therefore, we can apply Theorem 6.3.1 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] and deduce the existence of a unique local solution U ∈ C([0, t), V)∩C 1 ((0, t), H) defined in a time interval [0, t) with t ≤ τ min . If t = τ min , and U(τ min ) V < +∞, then one can extend the solution U for times t > τ min , by considering on the time interval (τ min , 2τ min ) the problem

U t (t) = AU(t) + g 1 (t, U(t)) in (τ min , 2τ min ) U(0) = U(τ - min ), (3.5)
where g 1 (t, U(t)) = I i=1 F i (U(t), U(tτ i )). Note that, since we know the solution U(t) for t ∈ [0, τ min ] from the first step, the terms U(tτ i ) can be regarded as known data for t ∈ [τ min , 2τ min ) .

Observe also that

U ∈ C 0,θ ([0, τ min ], D(-A) β )). (3.6)
Indeed from the proof of Theorem 6.3.1 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], we see that

U(t) = (-A) -1 2 y(t),
with y ∈ C([0, τ min ], H) given by

y(t) = S(t)(-A) 1 2 U 0 + t 0 (-A) 1 2 S(t -s)g(s, (-A) -1 2 y(s)) ds, ∀t ∈ [0, τ min ].
Hence for any t ∈ [0, τ min ) and h > 0 such that t + h ≤ τ min , we have

y(t + h) -y(t) = (S(h) -I)S(t)(-A) 1 2 U 0 + R 2 (t, h) + R 3 (t, h),
where

R 2 (t, h) = t 0 (S(h) -I)(-A) 1 2 S(t -s)g(s, (-A) -1 2 y(s)) ds R 3 (t, h) = t+h t (-A) 1 2 S(t + h -s)g(s, (-A) -1 2 y(s)) ds.
Accordingly, we have

(-A) β (U(t + h) -U(t)) = (-A) β-1 2 (y(t + h) -y(t)) = (S(h) -I)(-A) β S(t)U 0 + (-A) β-1 2 (R 2 (t, h) + R 3 (t, h)),
and since (-A) β-1 2 is a bounded operator from H into itself (see [25, Lemma 2.6.3]), one gets

(-A) β (U(t + h) -U(t)) H (S(h) -I)(-A) β S(t)U 0 H + R 2 (t, h) H + R 3 (t, h) H .
(3.7) The estimates (6.3.15) and (6.3.16) of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] give

R 2 (t, h) H + R 3 (t, h) H h β h θ , (3.8)
hence to get the Hölder continuity (3.6), it remains to show that

(S(h) -I)(-A) β S(t)U 0 H h θ . (3.9)
But we first notice that the estimates (2.6.26) of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] yields

(S(h) -I)(-A) β S(t)U 0 H h 1 2 -β (-A) 1 2 S(t)U 0 H ,
and since (-A)

1 2 S(t) = S(t)(-A) 1 2
and the semigroup is of contractions, we get

(S(h) -I)(-A) β S(t)U 0 H h 1 2 -β (-A) 1 2 U 0 H h 1 2 -β U 0 V ,
which proves (3.9). Then, as before, one can apply Theorem 6.3.1 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] extending the previously found solution. One can eventually iterate this procedure by obtaining a solution

U ∈ C([0, T ∞ ), V) ∩ C 1 ((0, T ∞ ), H) of problem (1.2), satisfying lim t→T - ∞ U(t) V = +∞ if T ∞ < +∞ .
We now give an exponential stability result for small initial data. Note that we do not require here any restriction on the size of the time delays. For that purpose, we need the additional assumption on our nonlinear functions, namely we suppose that there exist a positive constant C 1 and a continuous function Then there exist K 0 > 0 small enough and γ 0 < 1 (depending on K 0 ) such that for all K ∈ (0, K 0 ] and

h 3 from [0, ∞) I+1 to [0, ∞) satisfying h 3 (0) = 0 and such that | I i=1 (W, F i (U, V i )) H | ≤ W H (C 1 U H + h 3 ( U V , V 1 V , • • • , V I V ) U V ) , (3.10) for all W ∈ H, U, V i ∈ V, i = 1, • • • , I.
U 0 ∈ C([0, τ ]; V) satisfying U 0 (t) V < γ 0 K, ∀ t ∈ [-τ, 0], (3.12)
problem (1.2) has a global solution U that satisfies the exponential decay estimate

U(t) H ≤ Me -ωt ∀t ≥ 0 , (3.13)
for a positive constant M depending on U 0 and a suitable positive constant ω.

Proof. By Proposition 3.1, there exists

T ∞ > 0 such that problem (1.2) has a unique solution U ∈ C([0, T ∞ ), V) ∩ C 1 ((0, T ∞ ), H).
As in [START_REF] Liu | Asymptotic behavior of solutions of time-delayed Burgers' equation[END_REF], for K ∈ (0, K 0 ] with K 0 fixed later on, we look at

T 0 = sup{δ ∈ (0, ∞) : U(t) V ≤ K, ∀t ∈ (0, δ)}. (3.14)
Our assumption (3.12) clearly guarantees that T 0 > 0. We will now show that T 0 = +∞ by a contradiction argument. Indeed if we assume that T 0 is finite, then by its definition, we will have

U(t) V < K, ∀t ∈ [-τ, T 0 ), (3.15)
and

U(T 0 ) V = K. ( 3
F i (U(t), U(t -τ i ))) H = -2 U(t) 2 V + 2 I i=1 (U(t), F i (U(t), U(t -τ i ))) H .
Hence using the assumption (3.10), we get

d dt U(t) 2 H ≤ -2 U(t) 2 V + 2C 1 U(t) 2 H + 2h 3 ( U(t) V , U(t -τ 1 ) V , • • • , U(t -τ I ) V ) U(t) H U(t) V .
Therefore by (3.15), we deduce that

d dt U(t) 2 H ≤ -2 U(t) 2 V + 2C 1 U(t) 2 H + 2C 4 (K) U(t) H U(t) V ,
where

C 4 (K) = max 0≤y,z i ≤K h 3 (y, z 1 , • • • , z I
) is a constant that depends on K and is a non-decreasing function of K. Using (3.1), we arrive at

d dt U(t) 2 H ≤ 2 -1 + C 1 λ 1 + C 4 (K) √ λ 1 U(t) 2 V .
From our assumption (3.11), we can choose K 0 small enough such that

-ω = -1 + C 1 λ 1 + C 4 (K 0 )
is negative. Hence for all K ∈ (0, K 0 ], the previous estimate implies that

d dt U(t) 2 H ≤ -2ω U(t) 2 V ≤ -2ωλ 1 U(t) 2 H , ∀ t ∈ [0, T 0 ). (3.17)
This estimate obviously implies that

U(t) 2 H ≤ e -2ωλ 1 t U(0) 2 H , ∀ t ∈ [0, T 0 ). (3.18)
But the first estimate of (3.17) means that

d dt U(t) 2 H + 2ω U(t) 2 V ≤ 0,
and if we multiply this estimate by e ω ′ t for some ω ′ ∈ (0, ω] fixed later on, we find that

d dt (e ω ′ t U(t) 2 H ) + 2ω ′ e ω ′ t U(t) 2 V ≤ ω ′ e ω ′ t U(t) 2 H ,
and using (3.18), we get

d dt (e ω ′ t U(t) 2 H ) + 2ω ′ e ω ′ t U(t) 2 V ≤ ωe (ω ′ -2λ 1 ω)t U(0) 2 H . (3.19)
Hence we fix ω ′ ∈ (0, ω] such that ω ′ < 2λ 1 ω (possible since λ 1 and ω are positive), and then integrate (3.19) between 0 and T 0 to find

e ω ′ T 0 U(T 0 ) 2 H + 2ω ′ T 0 0 e ω ′ t U(t) 2 V dt ≤ 1 + ω 2λ 1 ω -ω ′ (1 -e (ω ′ -2λ 1 ω)T 0 ) U(0) 2 H ≤ 1 + ω 2λ 1 ω -ω ′ U(0) 2 H .
This clearly implies that there exists a positive constant C(ω, λ 1 ) depending only on ω and λ 1 such that

T 0 0 e ω ′ t U(t) 2 V dt ≤ C(ω, λ 1 ) U(0) 2 V . (3.20)
We now estimate d dt U(t) 2 V . First we notice that

d dt U(t) 2 V = 2((-A) 1 2 U, (-A) 1 2 U t ) H = -2(AU, U t ) H = -2(U t (t) - I i=1 F i (U(t), U(t -τ i )), U t ) H = -2 U t (t) 2 H + 2 I i=1 (F i (U(t), U(t -τ i )), U t ) H .
By the assumption (3.10), we get

d dt U(t) 2 V ≤ -2 U t (t) 2 H + 2 U t (t) H (C 1 U(t) H + C 4 (K) U(t) V ).
Young's inequality and (3.1) lead to

d dt U(t) 2 V ≤ ( C 1 √ λ 1 + C 4 (K)) 2 U(t) 2 V . (3.21)
Now we proceed as in the proof of Lemma 2.1 of [START_REF] Liu | Asymptotic behavior of solutions of time-delayed Burgers' equation[END_REF], namely we multiply this estimate by e ω ′ t to get

d dt (e ω ′ t U(t) 2 V ) ≤ (ω ′ + ( C 1 √ λ 1 + C 4 (K)) 2 )e ω ′ t U(t) 2 V .
Integrating this estimate between 0 and t ∈ (0, T 0 ], one obtains

e ω ′ t U(t) 2 V ≤ U(0) 2 V + (ω ′ + ( C 1 √ λ 1 + C 4 (K)) 2 ) t 0 e ω ′ s U(s) 2 V ds,
and therefore owing to (3.20), one finally finds

U(t) 2 V ≤ 1 + (ω ′ + ( C 1 √ λ 1 + C 4 (K)) 2 )C(ω, λ 1 ) U(0) 2 V e -ω ′ t , ∀t ∈ (0, T 0 ]. (3.22)
Therefore if we define γ 0 > 0 by

γ -2 0 = 4 1 + (ω ′ + ( C 1 √ λ 1 + C 4 (K 0 )) 2 )C(ω, λ 1 ) ,
we clearly see that γ 0 < 1. Furthermore for any U 0 satisfying (3.12), and reminding that 0 < K ≤ K 0 , we have

1 + (ω ′ + ( C 1 √ λ 1 + C 4 (K)) 2 )C(ω, λ 1 ) U(0) 2 V ≤ (1 + (ω ′ + ( C 1 √ λ 1 + C 4 (K 0 )) 2 )C(ω, λ 1 ) γ 2 0 K 2 = 1 4 K 2 .
Consequently, the estimate (3.22) guarantees that

U(t) V ≤ K 2 e -ω ′ t 2 , ∀t ∈ (0, T 0 ]. (3.23)
In particular, on gets

U(T 0 ) V ≤ K 2 ,
that clearly contradicts (3.16). This means that T 0 is infinite and we conclude by the estimate (3.23).

Examples with general nonlinearities

In the whole section, Ω denotes an arbitrary bounded domain of R d , d ≥ 1, with a Lipschitz boundary Γ.

Delay-diffusion equations

We consider the semilinear diffusion equation with time delay

u t (t) -∆u(t) = f (u(t), u(t -τ )) in Ω × (0, +∞), (4.1) Bu(x, t) = 0 on Γ × (0, +∞), (4.2) u(x, t)) = u 0 (x, t) in Ω × [-τ, 0], (4.3)
where the constant τ > 0 is the time delay and the initial datum u 0 belongs to the space C([0, τ ]; L 2 (Ω)). The operator B is in the form

Bu = α∂ n u + α ′ u,
with either α = 0 and α ′ = 1 corresponding to the case of Dirichlet boundary conditions or α = 1 and α ′ ≥ 0 (with α ′ ∈ L ∞ (∂Ω)) corresponding to the case of Neumann-Robin boundary conditions. Analogous problems have been considered by Friesecke in [START_REF] Friesecke | Convergence to equilibrium for delay-diffusion equations with small delay[END_REF], where an exponential stability result is obtained for small time delay, under some growth conditions on the locally Lipschitz function f , by using a Lyapunov functional approach. We may also quote the paper of Pao [START_REF] Pao | Dynamics of nonlinear parabolic systems with time delays[END_REF] where a coupled system of parabolic semilinear equations with delays is studied. Using the method of upper and lower solutions to investigate existence and asymptotic behavior, sufficient conditions for stability and instability are given, under some monotonicity properties on the reaction term f , independent of the time delays. In the same spirit, in [START_REF] Freedman | Global asymptotics in some quasimonotone reaction-diffusion systems with delays[END_REF] the global asymptotic behavior of some quasimonotone reaction-diffusion system with delays is analyzed. A trichotomy of the global dynamics is established via linearization.

For further uses, in the case of Neumann conditions (α = 1, α ′ = 0), we fix a positive real parameter ε (that may depend on f ), otherwise we set ε = 0. Now we assume that the nonlinearity f : IR 2 → IR satisfies the following assumption: there exist a non-negative constant α 0 (that may depend on ε) and two polynomials P 1 and P 2 (of one real variable) of degree n 1 and n 2 in the form

P i (X) = n i j=1 α i,j X j ,
with non negative real numbers α i,j such that

|f (x 1 , y 1 ) -f (x 2 , y 2 ) + ε(x 1 -x 2 )| ≤ α 0 |x 1 -x 2 | + P 1 (|x 1 | + |y 1 | + |x 2 | + |y 2 |)|x 1 -x 2 | +P 2 (|x 1 | + |y 1 | + |x 2 | + |y 2 |)|y 1 -y 2 |, ∀(x 1 , y 1 ), (x 2 , y 2 ) ∈ IR 2 . (4.4)
In particular this assumption means that f is only locally Lipschitz.

Under this assumption, let us now show that problem (4.1)-( 4.3) enters in the framework of section 3. Indeed in such a situation, we take H = L 2 (Ω) and define A as follows: Note that by Lemmas 1.5.3.7 and 1.5.3.9 of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], for any u ∈ {u ∈ H 1 (Ω) : ∆u ∈ L 2 (Ω)}, Bu has a meaning (as element of H -1 2 (Γ) if α > 0). It is not difficult to show that -A is a positive selfadjoint operator in H since it is the Friedrichs extension of the triple (H, V, a) where V = H 1 0 (Ω) in case of Dirichlet boundary conditions otherwise V = H 1 (Ω), and the sesquilinear form a is defined by

D(A) := {u ∈ H 1 (Ω) : ∆u ∈ L 2 (Ω)
a(u, v) = Ω (∇u • ∇v + εu • v) dx + Γ α ′ u • v dσ(x),
that is symmetric, continuous and strongly coercive on V. Before going on, let us notice that in the case of Neumann boundary conditions, the smallest eigenvalue λ 1 of -A is ε, otherwise it does not depend on ε and corresponds to the smallest eigenvalue of the Dirichlet problem or to the Robin eigenvalue problem.

By introducing the function

F 1 (x, y) = f (x, y) + εx,
we see that problem (4.1)-( 4.3) can be written as (1.2) with I = 1, U(t) = u(•, t) and U 0 (t) = u 0 (•, t). Consequently the next local existence result follows from Proposition 3.1.

Proposition 4.1 Assume that (4.4) holds and that d satisfies

d ≤ 2 1 + 1 n 1 and d < 2 1 + 1 n 2 . (4.5)
Then there exists β ∈ (0, 1 2 ) such that for any initial datum

u 0 ∈ C([0, τ ]; V) ∩ C 0,θ ([0, τ ], D((-A) β )), with θ = min{β, 1 2 -β}, there exist a time T ∞ ∈ (0, +∞] and a unique solution u ∈ C([0, T ∞ ), V) ∩ C 1 ((0, T ∞ ), H) of problem (4.1) -(4.3).
Proof. It suffices to check that F 1 defined above satisfies (3.2). For that purpose, we see that (4.4) implies that (for shortness we write F instead of F 1 )

|F (x 1 , y 1 ) -F (x 2 , y 2 )| ≤ α 0 |x 1 -x 2 | + P 1 (|x 1 | + |y 1 | + |x 2 | + |y 2 |)|x 1 -x 2 | +P 2 (|x 1 | + |y 1 | + |x 2 | + |y 2 |)|y 1 -y 2 |, ∀(x 1 , y 1 ), (x 2 , y 2 ) ∈ IR 2 .
and consequently for any u i , v i ∈ V,

F (u 1 , v 1 ) -F (u 2 , v 2 ) H ≤ α 0 u 1 -u 2 H + P 1 (|u 1 | + |v 1 | + |u 2 | + |v 2 |)|u 1 -u 2 | H + P 2 (|u 1 | + |v 1 | + |u 2 | + |v 2 |)|v 1 -v 2 | H .
Hence by the form of P i and Hölder's inequality, we arrive at

F (u 1 , v 1 ) -F (u 2 , v 2 ) H ≤ α 0 u 1 -u 2 H + C n 1 j=1 (|u 1 | + |v 1 | + |u 2 | + |v 2 |) j L q 1 (Ω) u 1 -u 2 | L p 1 (Ω) + C n 2 j=1 (|u 1 | + |v 1 | + |u 2 | + |v 2 |) j L q 2 (Ω) v 1 -v 2 L p 2 (Ω) ,
for some C > 0 and p i , q i > 2 chosen below and such that

1 p i + 1 q i = 1 2 , i = 1, 2. Now as D((-A) 1 2
) ֒→ H 1 (Ω) 1 and D((-A) 0 ) = L 2 (Ω), by interpolation we get that

D((-A) β ) ֒→ H 2β (Ω), ∀ β ∈ [0, 1 2 ].
Then we notice that the Sobolev embedding theorem guarantees that H 1 (Ω) ֒→ L p 1 (Ω) and H 2β (Ω) ֒→ L p 2 (Ω) as soon as

1 - d 2 ≥ - d p 1 and 2β - d 2 ≥ - d p 2 .
Since β can be taken as close to 1 2 as we want, we get equivalently

1 - d 2 ≥ - d p 1 and 1 - d 2 > - d p 2 . (4.6)
In a second step we now need to estimate terms like

(|u 1 | + |v 1 | + |u 2 | + |v 2 |) j L q i (Ω)
for all 1 ≤ j ≤ n i . But clearly we have

(|u 1 | + |v 1 | + |u 2 | + |v 2 |) j L q i (Ω) ≤ C 2 ( u 1 j L jq i (Ω) + v 1 j L jq i (Ω) + u 2 j L jq i (Ω) + v 2 j L jq i (Ω) ),
for some C 2 > 0 (that depends on j). Therefore our last assumptions are

1 - d 2 ≥ - d jq i , ∀ 1 ≤ j ≤ n i ,
or equivalently (since p i and q i are conjugates)

1 - d 2 ≥ d j ( 1 p i - 1 2 ), ∀ 1 ≤ j ≤ n i . (4.7)
1 by X ֒→ Y , we mean that X is continuously embedded into Y This condition and (4.6) guarantee that

F (u 1 , v 1 ) -F (u 2 , v 2 ) H ≤ α 0 u 1 -u 2 H + C n 1 j=1 ( u 1 j V + v 1 j V + u 2 j V ) u 1 -u 2 | V + C n 2 j=1 ( u 1 j V + v 1 j V + u 2 j V ) v 1 -v 2 D((-A) β ) ,
for some C > 0 and some β ∈ (0, 1 2 ), which yields (3.2) where C 0 = α 0 and h i are polynomials with positive coefficients.

Finally it is an easy exercise to check that conditions (4.5) are equivalent to the existence of p 1 > 2 and p 2 > 2 satisfying (4.6) and (4.7).

Concerning global existence and exponential decay, owing to Theorem 3.2, we can state the next result. are satisfied. With α 0 from the assumption (4.4), we assume that

α 0 < ε, (4.9)
in the case of Neumann boundary conditions, and

α 0 < λ 1 , (4.10)
otherwise. Then there exist K 0 > 0 small enough and γ 0 < 1 (depending on K 0 ) such that for all K ∈ (0, K 0 ] and u 0 ∈ C([0, τ ]; V) satisfying

u 0 (t) V < γ 0 K, ∀ t ∈ [-τ, 0], (4.11)
problem (4.1) -(4.3) has a global solution u that satisfies the exponential decay estimate

u(t) H ≤ Me -ωt ∀ t ≥ 0 , (4.12)
for a positive constant M depending on u 0 and a suitable positive constant ω. in the other cases.

2. The modified Hutchinson equation. In that case, f is given by (see [START_REF] Memory | Stable and unstable manifolds for partial functional-differential equations[END_REF][START_REF] Friesecke | Convergence to equilibrium for delay-diffusion equations with small delay[END_REF])

f (x, y) =
f (x, y) = -x 2 y
considered in [START_REF] Inoue | Some properties of solutions for semilinear heat equations with time lag[END_REF][START_REF] Lightbourne | Global existence for a delay differential equation[END_REF][START_REF] Friesecke | Convergence to equilibrium for delay-diffusion equations with small delay[END_REF] is also covered by our setting in the case of Dirichlet or Robin boundary conditions, since (4.4) holds with α 0 = 0 and n 1 = n 2 = 2. Therefore local existence and exponentiel decay for sufficiently small initial data hold for any d ≤ 2.

Remark 4.3 If f is globally Lipschitz, i.e., P 1 = 0 and P 2 is a positive constant, we can alternatively use the theory from section 2 and obtain exponential decay for small delays.

Predator-prey or two species competition systems with delays

Here we consider the semilinear diffusion system with time delays

u 1,t (x, t) = d 1 ∆u 1 (x, t) + u 1 (x, t) a 1 + a 11 u 1 (x, t) + 2 j=1 a ′ 1j u j (x, t -τ 1j ) in Ω × (0, +∞), (4.15) u 2,t (x, t) = d 2 ∆u 2 (x, t) + u 2 (a 2 + a 22 u 2 (x, t) + 2 j=1 a ′ 2j u j (x, t -τ 2j )
in Ω × (0, +∞), (4.16)

B 1 u 1 (x, t) = B 2 u 2 (x, t) = 0 on Γ × (0, +∞), (4.17) (u 1 (x, t), u 2 (x, t)) = (u 0,1 (x, t), u 0,2 (x, t)) in Ω × [-τ, 0], (4.18)
where the constants τ ij > 0 are the time delays, τ = max τ ij and the initial datum (u 0,1 , u 0,2 ) ⊤ belongs to C([0, τ ]; L 2 (Ω) 2 ). The operator B i are in the form

B i u = α i ∂ n u i + β i u i ,
with either α i = 0 and β i = 1 corresponding to the case of Dirichlet boundary conditions or α i = 1 and β i ≥ 0 (with β i ∈ L ∞ (∂Ω)) corresponding to the case of Neumann-Robin boundary conditions. Here d i are positive constants, while a i , a ij and a ′ ij are simply functions in L ∞ (Ω).

Systems for competitive species are studied in [START_REF] Pao | Dynamics of nonlinear parabolic systems with time delays[END_REF] under some monotonicity properties on the nonlinear functions. Conditions for stability or instability of solutions are given independently of the time delays. Two species prey-predator models and competition system with delays are also analyzed in [START_REF] Ruan | Persistence and extinction in two species reactiondiffusion systems with delays[END_REF]. By using the infinite-dimensional dissipative system theory and comparison arguments, persistence criteria and also global extinction criteria are established.

System (4.15)-(4.18) enters in the framework of section 3. Indeed in such a situation, we take H = L 2 (Ω) 2 and define A as follows:

D(A) := {u = (u 1 , u 2 ) ⊤ ∈ H 1 (Ω) 2 : ∆u i ∈ L 2 (Ω) and satisfying B i u i = 0 on Γ, i = 1, 2}, and Au = (d 1 ∆u 1 -ε 1 u 1 , d 2 ∆u 2 -ε 2 u 2 ) ⊤ , ∀ u ∈ D(A),
where ε i > 0 if Neumann condition is imposed on u i (fixed later on), and ε i = 0 otherwise. As before, -A is a positive selfadjoint operator in H with

V = D((-A) 1 2 ) = H 1 0 (Ω) 2 ,
if Dirichlet boundary conditions are imposed on u 1 and u 2 ,

V = D((-A) 1 2 ) = H 1 0 (Ω) × H 1 (Ω)
if Dirichlet boundary condition is imposed on u 1 and Neumann or Robin type on u 2 ,

V = D((-A) 1 2 ) = H 1 (Ω) × H 1 0 (Ω)
if Dirichlet boundary condition is imposed on u 2 and Neumann or Robin type on u 1 , and finally V = D((-A)

1 2 ) = H 1 (Ω) 2 , otherwise.
We now distinguish four different cases: Case 1: when Neumann boundary conditions are imposed on both u 1 and u 2 , then we take ε 1 = ε 2 = ε and the smallest eigenvalue λ 1 of -A is ε. Case 2: when Neumann boundary condition is imposed on u 2 and Dirichlet or Robin type on u 1 , then the smallest eigenvalue λ 1 of -A is equal to min{µ 1 , ε 2 }, where µ 1 > 0 corresponds to the smallest eigenvalue of -d 1 ∆ with Dirichlet or Robin boundary conditions.

Case 3: when Neumann boundary condition is imposed u 1 and Dirichlet or Robin type on u 2 , then the smallest eigenvalue λ 1 of -A is equal to min{µ 2 , ε 1 }, where µ 2 > 0 corresponds to the smallest eigenvalue of -d 2 ∆ with Dirichlet or Robin boundary conditions. Case 4: when Dirichlet or Robin boundary conditions are imposed on both u 1 and u 2 , then the smallest eigenvalue λ 1 of -A is equal to min{µ 1 , µ 2 }.

Without loss of generality we can suppose that τ = max{τ 11 , τ 12 } and therefore we set

F 1 (t, u, v) = ((a 1 + ε 1 )u 1 + a 11 u 2 1 + a ′ 11 u 1 v 1 , 0) ⊤ , if τ 11 > τ 12 , and F 1 (t, u, v) = ((a 1 + ε 1 )u 1 + a 11 u 2 1 + a ′ 12 u 1 v 2 , 0) ⊤ , if τ 11 < τ 12 .
Then in the first case, we set τ 2 = τ 21 , τ 3 = τ 12 , τ 4 = τ 22 and for all u

= (u 1 , u 2 ), v = (v 1 , v 2 ) ∈ C 2 : F 2 (t, u, v) = (0, (a 2 + ε 2 )u 2 + a 22 u 2 2 + a ′ 21 u 2 v 1 ) ⊤ , F 3 (t, u, v) = (a ′ 12 u 1 v 2 , 0) ⊤ , F 4 (t, u, v) = (0, a ′ 22 u 2 v 2 ) ⊤ .
The second case is similar by simply changing τ 3 and F 3 accordingly. With these notations, we see that problem (4.15)-(4.18) can be written as (1.2) with I = 2, U(t) = (u 1 (•, t), u 2 (•, t)) ⊤ and U 0 (t) = (u 0,1 (•, t), u 0,2 (•, t)) ⊤ . Consequently the next local existence result follows from Proposition 3.1.

Proposition 4.4 Assume that d ≤ 3, then there exists β ∈ (0, 1 2 ) such that for any initial datum (u 0,1 , u 0,2 ) ⊤ ∈ C([0, τ ]; V) ∩ C 0,θ ([0, τ ], D((-A) β )), with θ = min{β, 1 2 -β}, there exist a time T ∞ ∈ (0, +∞] and a unique solution (u 1 , u 2 ) ⊤ ∈ C([0, T ∞ ), V)∩C 1 ((0, T ∞ ), H) of problem (4.15) -(4.18).

Proof. It is a direct consequence of Proposition 3.1 since it is easy to check that F i satifies (3.2) with n 1 = n 2 = 1 and C 0 = max i=1,2 sup Ω |a i + ε i |.

As before global existence and exponential decay follow from Theorem 3. in case 4. Then there exist K 0 > 0 small enough and γ 0 < 1 (depending on K 0 ) such that for all K ∈ (0, K 0 ] and (u 0,1 , u 0,2 ) ⊤ ∈ C([0, τ ]; V) satisfying (u 0,1 , u 0,2 ) ⊤ (t) V < γ 0 K, ∀ t ∈ [-τ, 0], ( These conditions hold if and only if there exists δ > 0 small enough such that -µ 1 + δ ≤ a 1 ≤ µ 1δ andε 2 + δ ≤ a 1 < ε 2δ in Ω, -µ 1 + δ ≤ a 2 + ε 2 ≤ µ 1δ andε 2 + δ ≤ a 2 + ε 2 ≤ ε 2δ in Ω.

By re-arranging those conditions, we find -µ 1 + δ ≤ a 1 ≤ µ 1δ and a 2 ≤ -δ in Ω, (4.25)

max{a 1 + δ, -a 1 + δ, δ -a 2 2 , -a 2 -µ 1 + δ} ≤ ε 2 ≤ µ 1 -δ -a 2 in Ω. ( 4.26)
A necessary condition to find such a ε 2 is that

max{a 1 + δ, -a 1 + δ, δ -a 2 2 , -a 2 -µ 1 + δ} ≤ µ 1 -δ -a 2 in Ω,
which is indeed true if δ is small enough and if (4.25) holds. Therefore we fix

ε 2 = inf Ω (µ 1 -δ -a 2 ) = µ 1 -sup Ω a 2 -δ,
with δ > 0 small enough so that this quantity is positive. We then easily check that (4.26) holds under the assumption (4.25) and the condition 3δ + 2 sup

Ω a 2 -inf Ω a 2 ≤ 2µ 1 .
Hence this condition and (4.25) hold for δ small enough if and only if (4.20) is valid.

Remark 4.6 Clearly, our approach can be used to study a general system of N-competing species with delays like the system (1.3) of [START_REF] Pao | Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays[END_REF] since the nonlinear terms appearing in this system are similar to the ones from system (4.15)-(4.18). We then let the details to the reader.

Theorem 3 . 2 λ 1 - 1

 3211 Assume that (3.2) and (3.10) are satisfied. With C 1 from the assumption (3.10), we assume that C 1 < 0. (3.11)

  and satisfying Bu = 0 on Γ}, and Au = ∆uεu, ∀ u ∈ D(A).

Theorem 4 . 2

 42 Assume that conditions (4.4), (4.5) as well asf (0, y) = 0, ∀ y ∈ IR,(4.8)

Proof.

  It suffices to check that (3.10) and (3.11) hold. The first condition directly follows from Cauchy-Schwarz's inequality and the assumptions (4.8) and (4.4) where C 1 = α 0 . Then the second condition (3.11) simply becomes either (4.9) in the Neumann case or (4.10) in the two other cases. Our general setting covers a very large number of concrete examples. Let us mention the following cases: 1. Diffusive logistic equations with delay. In that case, f is given by f (x, y) = axbx 2 + cxy, with a, b, c ∈ L ∞ (Ω). This example covers the Hutchinson equation by taking a = α ∈ IR, c = -α and b = 0. In such a case, the condition (4.4) holds with α 0 = sup Ω |a + ε|, and n 1 = n 2 = 1. Hence local existence (i.e. Theorem 4.1) holds for any d ≤ 3, while exponentiel decay for sufficiently small initial data (i.e. Theorem 4.2) holds under the additional assumption that sup Ω a < 0, (4.13) in the case of Neumann boundary conditions (by chosing ε > 0 large enough), and sup Ω |a| < λ 1 , (4.14)

  αx(1 + βy + γy 2 + δy 3 ), with α, β, γ, δ ∈ IR. In such a case, the condition (4.4) holds with α 0 = |α + ε|, and n 1 = n 2 = 3. Hence local existence (i.e. Theorem 4.1) holds for any d ≤ 2, while exponentiel decay for sufficiently small initial data (i.e. Theorem 4.2) holds under the additional assumption α < 0 in the case of Neumann boundary conditions, and -λ 1 < α < λ 1 in the other cases. 2. A cubic nonlinearity. The case where f is given by

  2.

	Theorem 4.5 Assume that d ≤ 3 and that
		sup Ω	|a i | < λ 1 , i = 1, 2,		(4.19)
	in case 1, that
	sup Ω	|a 1 | < µ 1 , sup Ω	a 2 < 0, 2 sup Ω	a 2 -inf Ω	a 2 < 2µ 1 ,	(4.20)
	in case 2, that
	sup Ω	|a 2 | < µ 2 , sup Ω	a 1 < 0, 2 sup Ω	a 1 -inf Ω	a 1 < 2µ 2 ,	(4.21)
	in case 3, and that
		sup

Ω a 1 < 0, sup Ω a 2 < 0,

(4.22)

  ) problem (4.1) -(4.3) has a global solution (u 1 , u 2 ) ⊤ that satisfies the exponential decay estimate (u 1 (•, t), u 2 (•, t)) ⊤ H ≤ Me -ωt ∀t ≥ 0 , (4.24) for a positive constant M depending on u 0 and a suitable positive constant ω. Proof. It suffices to check that (3.10) and (3.11) hold. The first condition directly follows from Cauchy-Schwarz's inequality where C 1 = C 0 = max i=1,2 sup Ω |a i +ε i |. In the first and fourth cases, the second condition (3.11) simply becomes either (4.19) or (4.22). Hence it remains to check that in cases 2 and 3, one can find ε 1 and ε 2 appropriately. By symmetry, it suffices to look at case 2. In such a situation, (3.11) becomes max{sup

Ω |a 1 |, sup Ω |a 2 + ε 2 |} < min{µ 1 , ε 2 }, that is clearly equivalent to sup Ω |a 1 | < µ 1 and sup Ω |a 1 | < ε 2 , sup Ω |a 2 + ε 2 | < µ 1 and sup Ω |a 2 + ε 2 | < ε 2 .