Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions

In this work we consider a family of nonautonomous partial differential inclusions governed by \begin{document}$ p $\end{document} -laplacian operators with variable exponents and large diffusion and driven by a forcing nonlinear term of Heaviside type. We prove first that this problem generates a sequence of multivalued nonautonomous dynamical systems possessing a pullback attractor. The main result of this paper states that the solutions of the family of partial differential inclusions converge to the solutions of a limit ordinary differential inclusion for large diffusion and when the exponents go to \begin{document}$ 2 $\end{document} . After that we prove the upper semicontinuity of the pullback attractors.


1.
Introduction. It is well known that the solutions of reaction-diffusion systems with suitable boundary conditions and large diffusion are close to the solutions of a suitable ordinary differential system (see e.g. the classical papers [17,26,27,28,24,14,15,13] and, more recently, [1,44,11,37,38,12]). Such property is very important, as in particular it allows us to extrapolate information about the dynamics inside the global attractor for the reaction-diffusion system once we know the dynamics of the ordinary differential equation. Due to the complexity of studying the structure of attractors in infinite dimensions, this method is very helpful.
In the last years some authors have considered this problem for parabolic equations generated by a p-laplacian operator with variable exponent (see [34,40,23,36]), which appear in models of electrorheological fluids and image processing (see e.g. [20,25] and the references therein).
As far as we know, little is done so far in this direction concerning parabolic differential inclusions with large diffusion. For differential parabolic inclusions with multivalued right-hand side of Lipschitz type the convergence of solutions and global attractors to the corresponding ones of a limit ordinary differential inclusion was proved in [35].
In particular, we are interested in a type of differential inclusions generated by a nonlinear function having a discontinuity, which can be expressed as a differential inclusion making use of a Heaviside function. Such inclusions have been used for modelling processes of combustion in porous media [22], the conduction of electrical impulses in nerve axons (see [42,43]) or the surface temperature on Earth (see [7,19]) among others.
It is quite interesting, challenging and very difficult to study the structure of the global attractors for this type of inclusions. Nevertheless, some progress has been done already in the papers [2,4,5].
In this paper we study a parabolic differential inclusion governed by an elliptic p-laplacian operator with variable exponent, driven by a nonautonomous Heaviside forcing term and with homogeneous Neumann boundary conditions. As in the aforementioned papers, when the diffusion becomes large the solutions of the inclusion tend to be constant in space, and in this way we are able to prove that they converge to the solutions of the scalar ordinary differential inclusion considered in [10] when the exponents of the p-laplacian operator go to 2. Moreover, we prove that this problem generates a nonautonomous multivalued dynamical system possessing a pullback attractor and that the sequence of attractors behaves upper semicontinuously with respect to the pullback attractor of the limit inclusion. We observe that the structure of the pullback attractor of this ordinary inclusion is detaily described in [10]. We can expect that the dynamics in the attractor of the partial differential inclusion is similar to some extent to the dynamics inside the attractor of the limit problem. This question will be addressed in a future work.

2.
Setting of the problem. Throughout the paper for any two sets A, B in a Banach space X we denote the Hausdorff semidistance from A to B by dist (A, B), and the Hausdorff distance between A and B by dist H (A, B) .
Let Ω ⊂ R n , n ≥ 1, be a smooth bounded domain with boundary ∂Ω and H := L 2 (Ω), with norm · H and scalar product (·, ·). The aim of this work is to study the asymptotic behavior of the solutions as s → ∞ for the multivalued initial value problem with homogeneous Neumann boundary conditions, where λ > 0, D s ∈ [1, ∞), p s (·) ∈ C(Ω), p − s := min x∈Ω p s (x) ≥ 2, and there exists a constant p 0 ≥ 2 such that p + s := max x∈Ω p s (x) ≤ p 0 , for all s ∈ N. We assume that p s (·) → 2 in L ∞ (Ω) and D s → ∞ as s → ∞.
Let b : R → R + be a continuous function satisfying and consider the Heaviside function H given by The function F in problem (1) is defined as F : We denote by f : R × R → P (R) the multivalued function given by f (t, u) = b(t)H(u). Then f has nonempty, closed, bounded and convex values, and for all t ∈ R the map f (t, ·) : R → P (R) is upper semicontinuous. Moreover, for any and |f (t, u)| + := sup Following the same arguments as in the proof of Lemma 6.28 in [29] we obtain: iv) For all y ∈ H and for a.a. t ∈ R + , Now we recall the properties of elliptic operators in variable spaces in order to establish existence of solutions for problem (1).
In the sequel for p s (·) as defined above we shall use the family of spaces Y s = W 1,ps(·) (Ω). In this case the modular ρ will be denoted by ρ s .
Consider the operator A s defined in Y s in such a way that to each u ∈ Y s we associate the element A s u : Y s → R of Y * s given by The authors in [39] (see also [23,30]) proved that the operator A s : Y s → Y * s , with domain Y s , is maximal monotone, A s (Y s ) = Y * s , the realization operator of A s at H = L 2 (Ω), denoted by A s H , is maximal monotone in H and A s H is the subdifferential ∂ϕ ps(·) of the convex, proper and lower semicontinuous map ϕ ps(·) : Moreover, as our operator is of subdifferential type, the level sets M s R = {u ∈ H : u H ≤ R, ϕ ps(·) (u) ≤ R} are compact in H for all R > 0 and cl H (D(ϕ ps(·) )) = H.
Hence, since F satisfies the conditions given in Lemma 2.1, we can obtain the existence of global strong solutions for problem (1) using the results in Chapter 6 of [29].
We put D s (u τ , t, τ ) := {u(t) : u(·) is a strong solution of (1) with u(τ ) = u τ }, By Theorem 6.11, Lemma 6.16 and 6.17 in [29] we obtain the existence of a global strong solution for problem (1). Let us denote by B r the ball of radius r centered at 0. Theorem 2.3. If F is as in (2), then the multivalued problem (1) has a strong solution for every u τ ∈ H. Moreover, for arbitrary r ≥ 0, T > τ, t ∈ [τ, T ] the set D s (B r , t, τ ) := ∪ uτ ∈Br D s (u 0 , t, τ ) is connected in H and for arbitrary > 0 the set ; H) such that f (t) ∈ F (t, u(t)), for a.a. t ∈ (τ, T ), and for any η ∈ D(∂ϕ ps(·) ), v ∈ −∂ϕ ps(·) (η) we have that It is well known that if the selection f satisfies f ∈ L 2 ([τ, T ]; H), then u (·) is a strong solution of (4) if and only if it is an integral solution (see for example the proof of Lemma 6.16 [29]). For our problem (1) due to property iv) in Lemma 2.1 the fact that f ∈ L 2 ([τ, T ]; H) is true for arbitrary strong or integral solutions. Therefore, the sets of strong and integral solutions coincide.
The following lemma establishes that the translation and concatenation of strong solutions are also strong solutions.
If u (·) is a strong solution of (1) on [τ, s] and v (·) is a strong solution of (1) on [s, T ] such that v (s) = u (s) , then the function is a strong solution.
Proof. The proof of these results for integral solutions is quite similar to the ones in Lemmas 6.31 and 6.32 of [29]. Taking into account that in our case strong solutions and integral solutions are the same, both statements follow.
3. Existence and properties of pullback attractors. In this section we will define a multivalued nonautonomous dynamical systems from the solutions of the family of problems (1). We will start by obtaining some estimates of the solutions which are uniform on s, proving after that the existence of a sequence of pullback attractors and the compactness of its union with respect to the parameter s and the time t.
It is worth noticing that by concatenation every strong solution can be extended to a globally defined one, that is, which exists for any t ≥ τ . Denote by R s (u τ , τ ) the set of all globally defined strong solutions with initial condition u τ at time τ .
First, we will prove the existence of an absorbing ball which does not depend on s. We start with an auxiliary lemma.
, for any u ∈ L ps(·) and s ∈ N.
Proof. Let q s (x) be such that 1 Hence, By the Hölder inequality [20] we have that We estimate now 1 L qs(·) by a constant K (Ω) independent of s. We assume that 1 L qs (·) > 1, as otherwise we are done. From
In a similar way we can obtain this inequality for the other cases, so for γ = is satisfied for almost every t > τ. Therefore, Gronwall's lemma gives Putting R 0 = 1 + δ γ , we obtain that for any bounded set B there is a constant T (B) such that

JACSON SIMSEN, MARIZA STEFANELLO SIMSEN AND JOSÉ VALERO
The second statement follows for R(B) = sup y∈B y 2 H + δ γ . This completes the proof. Now, we will establish the existence of a compact absorbing set which does not depend on s.
Taking into account (9), we obtain that the ball K 0 of radius R = K(Ω)R in the space H 1 (Ω) defined by is absorbing for any s. Since the embedding H 1 (Ω) ⊂ H is compact, we obtain that K 0 is a compact set of H.
If we replace in the previous arguments the constants R 0 , T (B) by the constant R(B) from Theorem 3.2 and 0, respectively, then we obtain (16)- (17).
Let us prove now the last statement. As B is bounded in H, there exists T (B) such that (22) holds for any u ∈ R s (u τ , τ ), u τ ∈ B. Integrating (19) over the interval (τ, t) with t − τ ≤ T (B) and using that ϕ ps(·) (u τ s ) ≤ D, for any u τ s ∈ B and s ∈ N, we obtain ϕ ps(·) (u(t)) ≤ ϕ ps(·) (u(τ )) + Arguing as in the proof of the first statement we obtain the existence of a constant Hence, using (22) we have where κ = κ(D, B) = max{κ 1 , R}.
Remark 1. (15) implies that the set K 0 is both forward and pullback attracting.
The multivalued evolution process U is called strict if U (t, r, x) = U (t, τ, U (τ, r, x)), for all x ∈ X, r ≤ τ ≤ t. Definition 3.5. Let U be a multivalued evolution process on X and t ∈ R. The set D(t) ⊂ X attracts (pullback) the nonempty bounded subset B of X at time t if: lim The set D(t) is said to be (pullback) attracting at time t if (23) is satisfied for any nonempty bounded subset B ⊂ X.
For a nonempty and bounded subset B ⊂ X and t ∈ R, let us put γ r (t, B) = τ ≤r U (t, τ, B) and ω(t, B) = r≤t γr(t, B). The set ω(t, B) is called the pullback ω-limit set of B at time t with respect to the multivalued evolution process U . Then ω(t, B) is nonempty, compact and the minimal closed set attracting B at time t.
Definition 3.7. A family of sets {A(t) : t ∈ R} of X is called a pullback attractor for the multivalued evolution process U if: (1) A(t) is pullback attracting at time t for all t ∈ R; (2) it is semi-invariant (or negatively invariant), that is, U (t, r, A(r)), for any (t, r) ∈ R d ; (3) it is minimal, that is, for any closed attracting set Y at time t, we have A(t) ⊂ Y.
It is called strictly invariant if, moreover, A(t) = U (t, r, A(r)), for any (t, r) ∈ R d .
Theorem 3.8 (Theorem 18 in [9]). Let us suppose that for all (t, r) ∈ R d the map x → U (t, r, x) ∈ P (X) is closed. If, moreover, for any t ∈ R there exists a nonempty compact set D(t) which is attracting, then the set A = {A(t)} t∈R , with where B(X) = {B ∈ P (X) : B is bounded}, is the pullback attractor of U . Moreover, the sets A(t) are compact.
We come back now to our problem (1). Using Lemma 2.5 it is easy to check that the map U s : is a strict multivalued evolution process. We observe that as every strong solution can be extended to a globally defined one, the set D s (u τ , t, τ ) defined in (5) is equal to U s (t, τ, u τ ). Theorem 3.9. Let t ∈ R and let B be a nonempty and bounded subset of H. Then the ω-limit set ω s (t, B) corresponding to the multivalued evolution process associated with problem (1) is nonempty, compact and the minimal closed set attracting B at time t.
Proof. Note that by Theorem 3.3 the constant family K(t) ≡ K 0 of compact sets of H pullback attracts bounded sets of H at time t. Hence, by Theorem 3.6, ω s (t, B) is nonempty, compact and the minimal closed set attracting B at time t. Proposition 1. Let ξ n → ξ in H and u n (·) ∈ R s (ξ n , τ ). Then there exists a subsequence u n k (·) and u ∈ R s (ξ, τ ) such that u n k → u in C([τ, T ]; H) for any T > τ .
By Lemma 1 in [31] we have that v n converges in C([τ, T ]; H) to the solution u of ∂u ∂t (t) + ∂ϕ ps(·) (u(t)) = f (t) a.e. on [τ, T ], Now, using (24) and (26) we have So we conclude that u n → u in C([τ, T ]; H). Now, to finish the proof, it follows by Theorem 3.3 in [18] that f (t) ∈ F (t, u(t)) a.e. on [τ, T ], so u (·) is a strong solution of problem (1) on [τ, T ]. By a diagonal argument we obtain the convergence for any T > τ.
Theorem 3.10. The multivalued evolution process associated with problem (1) has a pullback strictly invariant attractor A s = {A s (t) : t ∈ R}. Moreover, the sets A s (t) are compact and A s (t) ⊂ K 0 for every t ∈ R.
Proof. By Theorem 3.3 the constant family K(t) ≡ K 0 of compact sets of H pullback attracts bounded sets of H at time t, and by Corollary 2 for all (T, τ ) ∈ R d the map H ξ → U (T, τ, ξ) ∈ P (H) is closed. Hence, the existence and compactness of the pullback attractor follows from Theorem 3.8. Since the pullback attractor by definition is the minimal closed pullback attracting family, we obtain that A s (t) ⊂ K 0 for every t ∈ R. Finally, the invariance follows from Lemma 2.5 in [10].
As a consequence of A s (t) ⊂ K 0 , where K 0 is independent of s and t, we obtain the following important property.
We will finish this section by giving a characterization of the pullback attractor as the union of all bounded complete trajectories.
Let us denote W τ = C([τ, ∞); H) and let R s = {R s (τ )} τ ∈R , where R s (τ ) = ∪ uτ ∈H R s (u τ , τ ). Then it follows from Theorem 2.3, Lemma 2.5 and Proposition 1 that for any s the set of solutions R s satisfies the following axiomatic properties: (H1) For any τ ∈ R and u τ ∈ H there exists u ∈ R s (τ ) such that u (τ ) = u τ . (H2) u r = u | [τ +r,+∞) ∈ R s (τ + r) for any r ≥ 0, u ∈ R s (τ ) (translation property). (H3) Let u, v ∈ R s be such that u ∈ R s (τ ), v ∈ R s (r) and u(p) = v(p) for some p ≥ r ≥ τ . Then the function z defined by belongs to R s (τ ) (concatenation property). (H4) For any sequence u n ∈ R s (τ ) such that u n (τ ) → u τ in H, there exists a subsequence u n k and u ∈ R s (τ ) such that A complete trajectory is called bounded if the set ∪ t∈R γ(t) is bounded in H.
It is well known [10, Corollaries 2.10 and 2.12] (see also [41]) that if either (H1)− (H3) or (H1) − (H2), (H4) hold and the multivalued process U s generated by R s posseses a bounded pullback attractor A s = {A s (t) : t ∈ R}, which means that ∪ t∈R A s (t) is bounded, then A s can be characterized by the union of all bounded complete trajectories of R s . Therefore, as Corollary 3 implies that ∪ t∈R A s (t) is bounded, we obtain the following result for the pullback attractor of problem (1).
Theorem 3.11. The pullback attractor of problem (1) is charaterized by A s (t) = {γ (t) : γ is a bounded complete trajectory of R s }.

4.
The limit problem. The following differential inclusion was considered in [10]: where λ > 0 and the maps H(u), b(t) are the same as for problem (1). When D s → ∞ and p s (·) → 2, that is, when the diffusion becomes large and the variable exponent tends to 2, it is natural to expect that the solutions of problem (1) converge to the solutions of problem (28) in some sense. This will be proved rigorously in the following section. Moreover, in the last section we will prove that the pullback attractors of the family of problems (1) are upper semicontinuous on s at ∞ with respect to the pullback attractor of problem (28). We observe that in [10] a precise characterization of the pullback attractor for problem (28) was given. Therefore, if the structure of the pullback attractor were robust, the attractors of problem (1) would inherit it for large diffusion.
We sumarize briefly the results proved in [10]. The function u : [τ, T ] → R is called a solution of (28) if u ∈ C ([τ, T ], R), du dt ∈ L ∞ (τ, T ; R) and there exists h ∈ L ∞ (τ, T ; , R) such that h(t) ∈ H(u(t)), for a.a. t ∈ (τ, T ) , and It is easy to see that this definition is equivalent to the one of strong solution given in Definition 2.2 if we replace the operator ∂ϕ ps(·) (u) by the linear operator λu and the space H by R.
The solutions of problem (28) generate a strict multivalued process denoted by U ∞ : R d ×R →P (R), which possesses a strictly invariant pullback attractor A ∞ = {A ∞ (t)} t∈R . Moreover, this attractor is bounded in the sense that the union of all sections ∪ t∈R A ∞ (t) is bounded and it is characterized by the set of all bounded complete trajectories, that is, We recall that γ : R → R is a complete trajectory if u = γ | [τ,∞) is a solution of (28) for any τ ∈ R. A complete trajectory is bounded if ∪ t∈R γ(t) is bounded.
More precisely, the pullback attractor is characterized by three nonautonomus equilibria (one of which is 0) and the heteroclinic connections which go from 0 to the two non-zero equilibria. We describe this structure in more detail.
There exists a maximal bounded complete trajectory ξ M (·), which means that for any bounded complete trajectory ψ (·) we have It is defined by This trajectory is the unique bounded complete positive trajectory, which means that, if ψ (·) is a bounded complete trajectory such that ψ (t) > 0 (ψ (t) < 0) for all t ∈ R, then ψ (t) ≡ ξ M (t) (ψ (t) ≡ −ξ M (t)). Moreover, every solution with positive (negative) initial data approaches to the complete solution ξ M (t) (−ξ M (t)) asymptotically as t → +∞, that is, if u τ > 0, v τ < 0, then Theorem 6.1. The family of global attractors {A s ; s ∈ N} associated with problem (1) is upper semicontinuous on s at infinity, in the topology of H, i.e., for each τ ∈ R, lim s→+∞ dist(A s (τ ), A ∞ (τ )) = 0, where A ∞ is the pullback attractor of the limit problem (28).
Proof. Let τ ∈ R and {v j } j be an arbitrary sequence with v j ∈ A j (τ ), ∀ j ∈ N. By Corollary 3, there exists a subsequence, that we still denote in the same way, such that v j → v 0 in H as j → +∞. By [16,Lemma 1.2] it is enough to prove that v 0 ∈ A ∞ (τ ). Using the characterization of the pullback attractor given in (30), what we have to do is to construct a complete bounded trajectory through v 0 . From Lemma 5.3 v 0 is constant, so we can assume that v 0 ∈ R.
By Theorem 3.11 there exist bounded complete trajectories γ j (·) of problem (1) with s = s j , γ j (τ ) = v j such that γ j (τ ) = v j → v 0 in H as j → +∞. Denote u 0 j := γ j | [τ,+∞) ∈ R j (τ ). By Theorem 5.2 there exists a solution g 0 of the limit problem (28) with g 0 (τ ) = v 0 and a subsequence of {u 0 j }, that we still denote the same, such that u 0 j (t) → g 0 (t) in H as j → +∞, for any t ≥ τ. Now we consider u 1 j := γ j | [τ −1,∞) ∈ R j (τ − 1). Again by Corollary 3 up to a subsequence u 1 j (τ − 1) → v −1 in H as j → +∞. From Lemma 5.3 we can consider that v −1 ∈ R. Using Theorem 5.2 we obtain that there exists a solution g 1 of the limit problem (28) with g 1 (τ − 1) = v −1 and a subsequence of {u 1 j } (relabeled the same) such that u 1 j (t) → g 1 (t) in H as j → +∞ for all t ≥ τ − 1. Now note that g 1 (t) = g 0 (t) for each t ≥ τ. Indeed, Proceeding in this way inductively, we find for each r = 0, 1, 2, · · · , a solution g r of the limit problem (28) with g r (τ − r) = v −r such that g r+1 (t) = g r (t) for t ≥ τ − r. Given t ∈ R, we define g(t) as the common value of g r (t) for t ≥ τ − r. Then we have that g is a complete trajectory with g(τ ) = g 0 (τ ) = v 0 . Note that for each r = 0, 1, 2, · · · we have that g r (t) = lim j→+∞ u r j (t) and u r j (t) ∈ A j (t), for any j ∈ N and t ≥ τ − r. As A j (t) ⊂ s∈N A s (t), for all j ∈ N, we obtain that there exists a constant C > 0 such that g r (t) H ≤ C, ∀ t ≥ τ − r and r = 0, 1, 2, · · · .
Hence, we have that ∪ t∈R g(t) is bounded in H. Then, there exists a constantC > 0 such that |g(t)| = 1 |Ω| 1/2 g(t) H ≤C, ∀ t ∈ R. We conclude that g : R → R is a complete bounded trajectory through v 0 .
If we consider the particular case where p s (x) ≡ 2, then it is obvious that any solution of the limit problem (28) is a solution of (1). This implies readily that every complete trajectory of (28) is a complete trajectory of (1). Thus, A ∞ (τ ) ⊂ A s (τ ) for all τ ∈ R, s ∈ N.
Hence, the family of global attractors {A s ; s ∈ N} associated with problem (1) is lower semicontinuous on s at infinity. Therefore, we obtain the continuity of the pulback attractors with respect to the Hausdorff distance. 7. The autonomous case. If the function b (·) ≡ b does not depend on t, that is, it is a constant, then problems (1) and (28) are autonomous. As a particular case of the previous results we obtain the upper semicontinuity of global attractors as s → ∞.
We notice that in this autonomous setting, as the solution does not depend on the initial moment of time τ , the operators U s , G s satisfy the following relationship: U s (t, τ, ξ) = U s (t − τ, 0, ξ) = G s (t − τ, ξ).
Hence, the pullback attractor A s = {A s (t) : t ∈ R} given in Theorem 3.10 does not depend on time, that is, A s (t) ≡ A s . It follows also that A s is compact and dist(G s (t, B), A s ) → 0 as t → +∞, for any bounded set B, A s ⊂G s (t, A s ) for any t ≥ 0, which means that A s is a compact global attractor for the semiflow G s . Moreover, it follows from [32,Remark 8] that it is invariant, that is, A s =G(t, A s ) for any t ≥ 0.
In a similar way, for problem (28) we define a strict multivalued semiflow G ∞ possessing the compact invariant global attractor A ∞ , which satisfies that A ∞ (t) ≡ A ∞ , where A ∞ = {A ∞ (t)} t∈R is its pullback attractor.
Problem (28) has three fixed points given by z + 1 = b λ , z − 1 = − b λ , z 0 = 0. The global attractor is characterized by these three equilibria and the heteroclinic connections which go from 0 to either z + 1 or z − 1 . As a particular case of Theorem 6.1 and Corollary 4 we obtain the upper semicontinuity (respectively, continuity) of global attractors for large diffusion.