ATTRACTORS AND ENTROPY BOUNDS FOR A NONLINEAR RDES WITH DISTRIBUTED DELAY IN UNBOUNDED DOMAINS

. A nonlinear reaction-diﬀusion problem with a general, both spatially and delay distributed reaction term is considered in an unbounded do- main R N . The existence of a unique weak solution is proved. A locally compact attractor together with entropy bound is also established.

The problem is posed in the unbounded spatial domain x ∈ Ω = R N . The equation can be seen as an abstract prototype of a nonlinear reaction diffusion system, which combines three nontrivial mathematical features: (i) nonlinear diffusion term − div a(∇u), (ii) temporally and spatially distributed delay terms and (iii) the setting of unbounded domains. We will begin by discussing the difficulties related to these three issues, together with a selection of recent references.
Let us start with the last point (iii). It can be said that the dynamics in unbounded domains has attracted a growing attention of the PDE community during the last decade. The problem obviously has an inherent non-compactness or even non-separability. This calls for a careful rethinking of the proper choice for the functional setting, so that the results on the global attractor and its finite-dimensionality, which are typical in bounded domain setting, can find a proper generalized expression. A natural choice seems to be some space of uniformly locally integrable functions, see [4], [19], [2], [1]. In such a setting, the existence of locally compact attractor admitting natural entropy estimates is the expected result; see also [10] and [3].
Concerning the point (ii), we would remark that presence of temporally and spatially non-local reaction lower order terms arise naturally in describing both living and non-living nature. We can mention the birth-death dynamics of maturing population or the spread of infection on the one hand, and the phenomena of yield or creep occurring in viscoelastic materials, or nonlocal interactions in phase transitions, on the other hand. The available mathematical techniques and results depend essentially on the complexity of non-local terms. In the case of linear delay of convolution type, linear techniques (theory of C 0 -semigroups, linear stability results) can be used [7,11,5].
For a more general non-linear problems, perturbation and topological methods provide sufficient conditions for the existence of robust nontrivial structures like travelling waves, see [6], [17]. In the case of a bounded spatial domain, the existence of a global compact attractor was shown for the equation (1) in [13] with a linear diffusion a(∇u) = ∇u; cf. also [12]. The existence of a global attractor for a similar linear equation in unbounded domain with b(y, u(t+θ, y)) instead of b(u(t+θ, y)) was proved in [9]. However, the authors in [9] study the equation in classical Sobolev spaces with ξ ≡ 1 and certain restrictions on d and r have to be met to obtain the existence of a compact attractor. Another similar linear equation with fixed delay and N = 1 was studied in [18] in the setting of bounded uniformly continuous functions. The existence of generalized attractors for delayed systems in unbounded domains was recently established in [16] and [15].
In the following we analyze the equation (1) in locally uniform spaces L 2 b in the spirit of [19]. The main advantage of this setting, as compared to standard Lebesgue or weighted Lebesgue spaces, is the possibility to capture arbitrary spatial complexity of the dynamics, including (spatially) periodic patterns. The spatial uniformity of L 2 b spaces, however, makes them similar to L ∞ spaces and thus not a good choice as a target spaces for the underlying dynamical system. For example, one cannot in general expect that the solution will be continuous with values in L 2 b . Several auxiliary weaker spaces are thus necessary to be introduced in the course of the analysis. Here in particular, following [8], we employ a sort of parabolic version of locally uniform spaces L 2 b,1 (0; T ; L 2 ) (see Section 2 below for definitions). The smoothing property of the dynamics can be easily proved in this parabolic setting, very much in the spirit of the so-called method of -trajectories. This leads to the existence and entropy estimates of the global (locally uniform) attractor A. No higher order regularity estimates and in particular, no restrictions on d or r other than r, d > 0 are needed. This low-cost (in terms of regularity) approach also enables us to work with a more general assumptions on the diffusion term (i): a general non-linear elliptic diffusion is possible, further generalizing the results common in the existing literature, where most often a linear dissipation (i.e. the Laplace operator) is considered.
The paper is organized as follows: the locally uniform spaces, corresponding duals and also their parabolic variants are briefly reviewed in Section 2. Existence and uniqueness of the weak solution are proven in Section 3. Locally compact attractor and its entropy estimates are established in Sections 4 and 5.
Definition 2.1. Letx ∈ R N and ε > 0. The weighted Lebesgue space L 2 x,ε (Ω) is defined by the norm Similarly the spaces W 1,2 x,ε (Ω) and W −1,2 x,ε (Ω) are defined by the norms where the last supremum is taken over v ∈ W 1,2 x,ε (Ω) with unit norm. We use the notation Similarly we can define the weighted Lebesgue spaces for a general p ∈ [1, ∞) and the same actually holds for all the spaces defined in the rest of this section.
Definition 2.2. The space of locally uniform L 2 functions is defined by Here B(x 0 , r) stands for an r-ball centered in x 0 . Let x k , k ∈ N, enumerate the points with half-integer coordinates, i.e. (Z/2) N , and let C k = C(x k ), k ∈ N, be the unit cubes, centered in x k . Then clearly the space L 2 b (Ω) has an equivalent norm Definition 2.3. Let µ ≥ 0. An admissible weight function of growth rate µ is a measurable bounded function φ : R N → (0, ∞) satisfying the inequalities for some C ≥ 1 and every x, y ∈ R N .
A typical example of an admissible weight function is the exponential φ(x) = e −q|x−x| withx ∈ R N and q ∈ [0, 1]. Trivially, φ(x) ≡ 1 is and admissible weight function of growth rate µ = 0. In fact we can define the locally uniform space in a more general manner and arrive at similar relations between weighted Lebesgue spaces and (weighted) locally uniform spaces. For more information see e.g. [2], Section 4. Definition 2.4. Let φ be an admissible weight function. We define the space of weighted locally uniform L 2 functions L 2 b,φ (Ω) by For φ ≡ 1, we simply write L 2 b (Ω). Similarly as in the non-weighted case one may observe that the space L 2 b,φ (Ω) has an equivalent norm Theorem 2.5 ([8], Theorem 2.1). Let φ be an admissible weight function with growth rate strictly smaller than ε > 0. The space L 2 b,φ (Ω) admits an equivalent norm Following the notation of [8], we define the L 2 b,φ seminorms corresponding to a subdomain O ⊆ Ω. For O ⊆ Ω we define We will need to use the so-called parabolic locally uniform spaces introduced in [8].
Definition 2.6. Let φ be an admissible weight function and ε > 0. We define the parabolic locally uniform spaces by their respective norms Once again, the symbol φ is dropped if φ ≡ 1.
A simple variant of Theorem 2.4 from [8] implies that for φ of growth rate µ strictly smaller than ε > 0, the parabolic locally uniform spaces admit equivalent norms where the first supremum in the last equivalence is taken over the functions v ∈ L 2 b,φ (−r, ; W 1,2 (Ω)) with unit norm. The parabolic uniformly bounded spaces and the Bochner spaces constructed over locally uniform spaces are related in the following way: × Ω). Recall that for ε > 0, a metric space M and a precompact set K ⊆ M , the Kolmogorov ε-entropy is defined by where N ε (K, M ) is the smallest number of balls of radius ε that cover the set K in M .
Let R > 0 and θ ∈ (0, 1). Then there exists c 0 > 0 such that where B R (x 0 ; X) denotes a ball in the space X with radius R centered at x 0 . The constant c 0 depends on c 1 , , θ and µ, C in (2), but does not depend on χ, r and the particular form of the weight function φ.
Observe that a ball in R N satisfies (7) with c 1 independent of the radius R ≥ 1.
We conclude this section with four auxiliary lemmata. The proofs are elementary and therefore omitted. Lemma 2.11 is the standard L p -estimate for the convolution.
(Ω) be bounded. Then for every δ > 0 there exists R > 0 such that Then the estimate 3. Well-posedness. We impose the following assumptions on the nonlinearities: Let a : R N → R N be a continuous function satisfying for some κ > 0, γ ≥ 1.
Next we choose 0 < ε < 1 small enough, namely We remark that due to the unbounded domain setting, the absolute term du is indespensable for dissipation as well as is some smallness condition on ε, which corresponds to boundary condition at infinity. This particular choice of ε does not affect the well-posedness of the problem, in fact the well-posedness can be proved for arbitrary 0 < ε < 1 by the same argument as below.
Finally, concerning the form of the distributed delay, we impose the following conditions on the function ξ : the following holds: Note that the condition (ii) implies that F (u t ) ∈ L 2 x,ε (Ω) uniformly for every x,ε (Ω)), and u satisfies the variational formulation (18) for every ψ ∈ D((0, T ) × Ω) and the initial conditions hold true. For arbitraryx ∈ Ω we may use a standard density argument and arrive to the duality with respect to L 2 x,ε (Ω): for any ψ ∈ L 2 (0, T ; W 1,2 x,ε (Ω)) ∩ W 1,2 (0, T ; L 2 x,ε (Ω)). Indeed, one can replace ψ in (18) by ψχ n e −ε|x−x| , where χ n is some sequence of cut-off functions such that χ n → 1, ∇χ n → 0 and |χ n | + |∇χ n | ≤ c a.e. It is clear that (20) in turn implies (18). Proof. The proof is a variant of the original proof for the linear case in a bounded domain (see [13], Theorem 1). We need to handle the limit of nonlinear diffusion term (cf. [8], Theorem 3.2); otherwise, standard techniques for unbounded domains are used ( [19]).
We approximate the problem (18) by a sequence of problems solvable on bounded domains and then pass to the limit. Let Ω n = B n (0) ⊆ R N and let ψ n ∈ C ∞ (Ω, [0, 1]) satisfy ψ n ≡ 1 onΩ n−1 , supp ψ n ⊆ Ω n , and define u 0,n = u 0 ψ n and ϕ n (θ) = ϕ(θ)ψ n for θ ∈ [−r, 0]. Using Theorem 2.5 and the Lebesgue's dominated convergence theorem we immediately obtain for everyx ∈ Ω. Next we define the operator and the approximate problem A nonlinear variant of Theorem 1 from [13] implies that the equation (22) with the initial equation (23) has a solution x,ε (Ω)). Let us extend u n by zero outside of Ω n (note that then u n ∈ L ∞ (0, T ; (Ω)) and thus ξ(θ, u n (t), u t n ) makes good sense) and test (22) by u n (t, x)e −ε|x−x| to get 1 2 Observe that the integration in the previous equation is over Ω instead of Ω n . This is possible since u n ≡ 0 outside of Ω n and b(0) = 0, a(0) = 0. Using the boundedness of the functions b and f , from (ii) it follows that The previous estimate, (25), (9), (10) and Young's inequality immediately give d dt u n (t) 2 x,ε + σ ∇u n (t) 2 x,ε + u n (t) 2 x,ε ≤ C 1 u n (t) 2 x,ε + C 2 for some σ > 0. Gronwall's inequality applied to x,ε (Ω)) and using a standard argument we finally have (24). Observe that if we take the supremum of (26) overx ∈ Ω, using (3) and (5) we obtain Following a similar argument we can show x,ε (Ω)) and therefore we immediately have u ∈ C([0, T ], L 2 x,ε (Ω)). Also note that since the norms L 2 x,ε (Ω) are equivalent for differentx ∈ Ω, the function u is independent ofx. Next we show u n → u in L 2 (0, T ; L 2 x,ε (Ω)), The first step is to establish the convergence for m ∈ N fixed. We proceed similarly as in the proof of Theorem 3.2 from [8]. The weak convergence (24) implies Then we choose n > m and test the equation (22) with v ∈ L 2 (0, T ; W 1,2 0 (Ω m )) extended by zero outside of Ω m to get The desired convergence (29) then follows from (30), (31) and the Aubin-Lions lemma.
Clearly the equality (39) holds also a.e. with respect to the standard Lebesgue measure in (0, T ) × Ω and therefore α is independent of the choice ofx. Therefore we may use (39) to substitute in (36), which finishes the proof of existence. The proof of the uniqueness is analogous to the proof for the bounded domain. Let u and v be two solutions with the respective initial conditions (u 0 , ϕ), (v 0 , ψ) ∈ H and denote w(t) = u(t) − v(t). Test the equations for u and v by w. Subtracting these and using a similar argument as in the derivation of the inequalities (32) and (33) we obtain for some σ > 0. The Gronwall's lemma applied to the function The estimate can be rewritten in the form which gives the uniqueness of the solutions.
where u(t) is the solution from Theorem 3.1.
As is usual in locally uniform spaces, we cannot generally expect the solution u(t) to be continuous in the space L 2 b (Ω). Continuity in L 2 b (Ω) can be achieved for more regular initial data; cf. [18]. In the following we will compensate for the lack of additional regularity by working in weighted Lebesgue spaces and by using the method of -trajectories.

Corollary 1. The operator
x,ε (Ω)) is Lipschitz continuous on H uniformly with respect to t ∈ [0, T ] for everyx ∈ Ω and 0 < ε < 1. The solution operator S(t) paired with H as its phase space form a dynamical system.
Proof. The continuity in time follows from the continuity of the solution and the locally uniform Lipschitz continuity from (41). The semigroup property is obvious by uniqueness. The joint continuity w.r. t. (t, χ), where χ ∈ H, follows by time continuity of solutions and locally uniform continuity w.r.t. χ.

Locally compact attractor.
Theorem 4.1. Let the assumptions of Theorem 3.1 hold and let φ be an admissible weight function with growth rate smaller than ε. Then for t ≥ r we have the estimate where σ, C 1 , C 2 > 0 are dependent of the data of the equation and independent of the initial data u 0 , ϕ.
Proof. The proof follows the proof of Theorem 3.2 in [8]. Let u be the solution of (18), (19). Then using the Cauchy-Schwartz and Young's inequalities, (9) and the initial choice of ε in (15) we get for some C 1 , σ > 0. The Gronwall's lemma implies x,ε e −σt + C 3 and integrating (43) from t − r to t leads to Finally we multiply the previous estimate by φ(x) and take the supremum over x ∈ Ω. From the definition of parabolic locally uniform spaces we obtain (42).

Corollary 2. The solution operator
x,ε (Ω)) admits a positively invariant bounded absorbing set W ⊆ H. Moreover, W absorbs not only bounded subsets of H, but also the sets of the form Proof. Using the notation of Theorem 4.1 (cf. (42)), letW = B(0, C 2 + 1) ⊆ H. The setW is clearly absorbing. Then we find t 0 > 0 such that S(t)W ⊆W for every t ≥ t 0 and set The fact that the set W absorbs even the sets of the form B × L 2 b (−r, 0; L 2 (Ω)) for B ⊆ L 2 b (Ω) bounded follows immediately from the form of the estimate (42).
Since all the solutions from Theorem 3.1 are continuous for t ≥ 0 and we are interested in the asymptotic dynamics, from now on we may consider only S(t) : x,ε (Ω)); u is a (weak) solution from Theorem 3.1} is equipped with L 2 (−r, 0; L 2 x,ε (Ω)) topology for fixedx ∈ Ω. The choice of particularx does not play any role in the analysis as the solution u(t) satisfies (27). Corollary 2 implies that the dynamical system (X, S(t)) admits a bounded absorbing set B ⊆ L 2 (−r, 0; L 2 x,ε (Ω)) ∩ L 2 b (−r, 0; L 2 (Ω)). Moreover, the continuity of the solutions allows us to assume x,ε (Ω)). Clearly all the long time dynamics will take place in the absorbing set B. Now we may define the space of short trajectories similarly as in [12].
Definition 4.2. The space of short trajectories is given by x,ε (Ω)); χ is a solution of (1) in [0, ] with χ| [−r,0] ∈ B}, together with the L 2 (−r, ; L 2 x,ε (Ω)) topology. The evolution operator L(t) : X → X is defined by where u is the solution from Theorem 3.1 satisfying u| (−r,0) = χ. Finally, the operator e : X → X is defined by We note that X in general is not complete with respect to its metric; we will see however in Lemma 4.6 that L(t) is asymptotically compact on X .
Theorem 4.3. The operator L(t) : X → X is Lipschitz continuous uniformly with respect to t ∈ [0, T ]. Moreover, the pair (X , L(t)) forms a dynamical system.
Proof. Let χ 1 , χ 2 ∈ X and let u 1 and u 2 be the respective solutions from Theorem 3.1. Define w(t) = u 1 (t) − u 2 (t). We start from the estimate (40), use the condition (10) and integrate over τ ∈ [s, t], where s ∈ (0, ), t ∈ ( , + T ), to obtain Then we integrate over s ∈ (0, ) and get Again we apply the Gronwall's lemma to the function x,ε (Ω) dτ and obtain the estimate which gives sup It remains to prove that (X , L(t)) is a dynamical system. The continuity of the operator L(t) in time follows from the continuity of solutions and the uniform continuity of L(t) in X follows from the previous part of the proof.
Definition 4.4. We define the space W as the set X with the norm x,ε (Ω) ds.
Proof. Let χ n ∈ X be a bounded sequence and t n → ∞. We aim to show where χ ∈ X , up to a subsequence. From Theorem 4.1 and Lemma 4.5 we see that L(t n )χ n is bounded in the norms L 2 (−r, ; W 1,2 (B)), W 1,2 (−r, ; W −1,2 (B)), where B ⊆ Ω is an arbitrary compact set. The Aubin-Lions lemma implies that and therefore L(t n )χ n → χ in L 2 loc ((−r, ) × Ω). Since the sequence is also bounded in L ∞ (−r, ; L 2 b (Ω)), Lemma 2.10 immediately gives us the strong convergence (55). Theorem 4.1 and Lemma 4.5 also imply x,ε (Ω)), which together with the strong convergence (55), (10) and the Lipschitz continuity of F (34) justifies taking a limit in the equation in a similar manner as in the proof of existence (see Theorem 3.1) and thus χ ∈ X .
Theorem 4.7. The dynamical system (X, S(t)) has a locally compact attractor, more precisely a L 2 b (−r, 0; L 2 (Ω)), L 2 loc ((−r, 0) × Ω) -attractor. Proof. First we observe that the dynamical system (X , L(t)) has a global attractor A . By Theorem 4.1 it has a bounded absorbing set; the asymptotic compactness was proved in Lemma 4.6 and we apply a standard result (see e.g. [14,Theorem 23.12]). Define A = e(A ).
(56) It remains to check that A is the desired attractor.
Observe that S(t)e(χ) = e(L(t)χ), therefore A is invariant under S(t). The compactness in L 2 loc ((−r, 0) × Ω) follows from the compactness of A , the continuity of e : X → X and Lemma 2.10. To show that A attracts bounded sets of X, we observe that x,ε (Ω)); χ is a solution from Theorem 3.1 with the initial condition (ϕ(0), ϕ) for ϕ ∈ B}.
By Theorem 4.1, the set B is bounded in X for B bounded in X. Then we have the estimate dist X (S(t + r + )B, A) = dist X (S(t)S(r + )B, A) = C 1 dist X (S(t)e(B ), A) where the last estimate uses the Lipschitz continuity of e, cf. (54).

Entropy estimates.
We estimate the entropy of the attractor constructed in Theorem 4.7 using the general method presented in [19], that has been adapted to the setting of -trajectories and parabolic uniform spaces in [8]. Actually, the rest of the proof follows the latter article quite closely. We need some preliminary results. First we formulate the Lipschitz continuity of the operators L(t), e and the smoothing property in the context of parabolic uniformly bounded spaces.  (8), is Lipschitz. The Lipschitz constants only depend on C and µ in (2) and not on the particular form of the weight function ψ.
Proof. Multiply (45) by ψ(x) and take supremum overx ∈ Ω to obtain The first assertion then follows from the equivalence of the norms (4).
The remaining assertions can be proved in a similar manner from a variant of (54), the equivalence of norms (4)-(6) and from Lemma 4.5.
We are now ready to prove the entropy estimate.