Global attractors for nonlinear viscoelastic equations with memory

We study the asymptotic properties of the semigroup S(t) arising from a nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain written in the past history framework of Dafermos. We establish the existence of the global attractor of optimal regularity for S(t) for a wide class of nonlinearities as well as within the most general condition on the memory kernel.

The model is subject to the initial conditions (the dependence on x is omitted) where u 0 , v 0 : Ω → R and η 0 : Ω × R + → R are prescribed functions.The external force h is time-independent, while the locally Lipschitz nonlinearity f , with f (0) = 0, fulfills the critical growth restriction along with the dissipation conditions 1 for some ν > 0 and m f ≥ 0.Here λ 1 > 0 denotes the first eigenvalue of the Dirichlet operator −∆ and Finally, the convolution (or memory) kernel µ is a nonnegative, nonincreasing, piecewise absolutely continuous function on R + of finite total mass ∞ 0 µ(s) ds = κ ≥ 0 complying with the further assumption for some Θ > 0. In particular, µ is allowed to exhibit (even infinitely many) jumps, and can be unbounded about the origin.At the same time, µ can be identically zero, yielding the equation Remark 1.1.Assuming the past history of u to be known, from the second equation of (1.1) together with (1.3) one deduces the formal equality (see [10]) Accordingly, the first equation becomes This provides a generalization, accounting for memory effects in the material, of equations of the form arising in the description of the vibrations of thin rods whose density ̺ depends on the velocity ∂ t u (see e.g.[20]).
1 Conditions (1.5)-(1.6)follow for instance by requiring f ∈ C 1 (R) with lim inf 1.2.Earlier contributions.The Volterra version of (1.1) with f = h ≡ 0 µ(s)∆u(t − s) ds = 0, corresponding to the choice of the initial datum η 0 = u 0 , has been considered by several authors, also with different kind of damping terms, in concern with the decay pattern of solutions (see [3,14,15,18,19,21,22,23,24,25,28]).On the contrary, the asymptotic analysis of the whole system (1.1) has been tackled only in the recent work [1], where the authors prove the existence of the global attractor (without any additional regularity) within the following set of hypotheses: • The nonlinearity f has at most polynomial growth 3 and • The nonnegative kernel µ ∈ C 1 (R + ) ∩ L 1 (R + ) fulfills for some δ > 0 and every s ∈ R + the relation In addition, the exponential decay of solutions is obtained when h ≡ 0 (meaning that the global attractor A = {0} is exponential as well).
Remark 1.2.Actually, analogously to what done in all the other papers on the Volterra case, the (exponential) decay rate turns out to depend on the size of the initial data.As we will see in the next Section 4, a simple argument allows to get rid of such a dependence.
The restriction on ρ is also motivated by the fact that the well-posedness result for (1.1), hence the existence of the semigroup, was available only for ρ ∈ (1, 2] (besides for the much simpler case ρ = 0).On the other hand, after [7] now we know that (1.1) generates a strongly continuous semigroup in our more general assumptions, and in particular, for all ρ ∈ [0, 4].This, of course, opens a new scenario which is worth to be investigated.1.3.The result.In this work, we prove that the strongly continuous semigroup S(t) generated by system (1.1) is dissipative (i.e.possesses bounded absorbing sets) for all ρ ∈ [0, 4].In particular, the exponential decay of solutions occurs whenever m f = 0 and h ≡ 0. Besides, we establish the following theorem.4).Then S(t) possesses the global attractor of optimal regularity.With respect to the earlier literature, Theorem 1.3 improves the picture in several directions: • The attractor is bounded in a more regular (in fact, the best possible) space.
• The nonlinearity f is allowed to reach the critical polynomial order 5, under the very general dissipation conditions (1.5)-(1.6),which include for instance terms of the form f (u) = u 5 + au 4 + bu 3 + cu 2 + du + e, not covered by (1.9).
• Condition (1.7) on the memory kernel µ is the most general possible one (among the class of nonincreasing summable kernels), since its failure prevents the uniform decay of solutions to systems with memory, no matter how the equations involved are (see [5]).As shown in [11], the condition can be equivalently stated as for some C ≥ 1, δ > 0, every σ ≥ 0 and almost every s > 0. Observe that the latter inequality with C = 1 boils down to (1.10) (actually, for a.e.s ∈ R + ).At the same time, when C > 1 a much wider class of memory kernels is admissible (see [5] for more comments).• The parameter ρ belongs to the interval [0, 4).Nonetheless, the existence of the global attractor in the case ρ = 4, which is critical for the Sobolev embedding, seems to be out of reach at the moment.
Plan of the paper.After introducing the functional setting (Section 2), we dwell on the existence of the solution semigroup (Section 3), whose dissipative features are discussed in Section 4. Our main results on the existence of the global attractor of optimal regularity are presented in Section 5.The remaining three Sections 6-8 are devoted to the proofs.The final Appendix contains some technical lemmas.

Functional Setting
We denote by A = −∆ the Dirichlet operator on L 2 (Ω) with domain H 2 (Ω) ∩ H 1 0 (Ω).For r ∈ R, we define the scale of compactly nested Hilbert spaces . The index r is omitted whenever zero.In particular, (Ω), and we have the generalized Poincaré inequalities Next, we introduce the history spaces We will also consider the infinitesimal generator T of the right-translation semigroup on M defined as the prime standing for weak derivative.The following inequality holds (see e.g.[12]) Finally, we introduce the extended history spaces Notation.Throughout the paper, c ≥ 0 and Q(•) will stand for a generic constant and a generic increasing positive function, respectively.We will use, often without explicit mention, the usual Sobolev embeddings, as well as the Young, Hölder and Poincaré inequalities.

The Gradient System
Rewriting (1.1)-(1.3) in the form the following result is proved in [7].
Besides, given any initial data z = (u 0 , v 0 , η 0 ) ∈ H and denoting the corresponding solution by (u(t), ∂ t u(t), η t ) = S(t)z, we have the explicit representation formula Moreover, defining the energy at time t of the solution S(t)z as M , we have (see [7]) Proposition 3.2.The uniform estimates Finally, we show the existence of a gradient system structure.We first recall the definition.
for all t > 0, then z is a stationary point for S(t).If there exists a Lyapunov functional, then S(t) is called a gradient system.Proposition 3.4.S(t) is a gradient system on H.
In light of (1.4) and (2.2), it is readily seen that This proves (i).For sufficiently regular initial data z, testing system (3.1) with (∂ t u, η) in H × M and recalling (2.3), we get so establishing (ii).To prove (iii), we note that if L(S(t)z) is constant, it follows that In particular, the second equation of (3.1) reduces to ∂ t η = T η, and a multiplication by η gives thus, in light of (1.11) which forces the equality η 0 = 0.In conclusion, meaning that z is a stationary point.

Dissipativity
The dissipativity of S(t) follows from the existence of a bounded absorbing set.This is a straightforward consequence of the next result.
Theorem 4.1.There exists ω > 0 such that In light of the theorem, every ball B of H centered at zero with radius strictly greater than √ 2R 0 is a (bounded) absorbing set for S(t).Recall that B is called an absorbing set if for every bounded set B ⊂ H there exists a time t B ≥ 0 such that If R 0 = 0 the exponential decay of the energy occurs.
As a first step, we prove the result in a weaker form, allowing the (exponential) decay rate to depend on R. Lemma 4.4.For every R ≥ 0 there exists a constant δ = δ R > 0 such that be the Lyapunov functional of Proposition 3.4.Then, we introduce the further functionals Arguing as in the proof of Lemma A.1 in Appendix, and from (2.1) we obtain and for δ > 0 to be properly chosen later, we consider the functional Collecting (3.5) and (4.1)-(4.2),we end up with We estimate the right-hand side above in the following way.For δ > 0 sufficiently small, standard computations entail Moreover, by Proposition 3.2 and the embedding Therefore, we arrive at Actually, since by (1.7) and an application of the Gronwall lemma leads to We now prove that Therefore, on account of (3.4) and ( 4.3), we readily get hence, possibly by further reducing δ in dependence of R, we obtain The claim follows from (4.4) and (4.5).
Proof of Theorem 4.1.Let z H ≤ R for some R ≥ 0.Then, we infer from Lemma 4.4 the existence of t R ≥ 0 such that and a further application of Lemma 4.4 yields , where ω = δ 1+R 0 .At the same time, again by Lemma 4.4, Collecting the two inequalities we are done.

Main Results
Theorem 5.1.The semigroup S(t) possesses the global attractor A.
By definition, the global attractor of S(t) is the unique compact set A ⊂ H which is at the same time fully invariant and attracting for the semigroup (see e.g.[2,13,16,27]).Namely, (i) S(t)A = A for every t ≥ 0; and (ii) for every bounded set B ⊂ H where dist H denotes the standard Hausdorff semidistance in H.We also recall that, for an arbitrarily fixed τ ∈ R, the global attractor can be given the form (see [16]) where a complete bounded trajectory (cbt) of the semigroup is a function According to [4,13], the existence of a Lyapunov function (Proposition 3.4) ensures that A coincides with the unstable manifold of the set S of equilibria of S(t), which is compact, nonempty and made of all vectors z ⋆ = (u ⋆ , 0, 0) with u ⋆ solution to the elliptic equation Moreover, the following result holds.
In particular, lim On the other hand, S might as well be a continuum (e.g. if F is a double-well potential, see [16]).In such a case, the convergence of a given trajectory to a single equilibrium cannot be predicted, and is false in general.Nonetheless, if f is real analytic, there is a well-known tool which can be used in order to guarantee the convergence of trajectories to equilibria: the Lojasiewicz-Simon inequality (see e.g.[17]).
Coming to the regularity of the attractor, we have Observe that, as S ⊂ A, if h ∈ H without any further assumption we cannot have more than H 2 -regularity for the first component.Thus the inclusion A ⊂ H 1 is optimal.Proposition 5.5.For every cbt ζ = (u, ∂ t u, η) the formal equality (1.8) holds true for every t ∈ R. In particular, it follows that η t ∈ Dom(T ) for all t.
A direct consequence of the proposition is the next corollary, whose proof is left to the reader.
) and defining η = η t (s) for all real t by the formula (1.8), the vector ζ = (u, ∂ t u, η) is a cbt if and only if u solves the equation The proofs of the results stated above will be carried out in the subsequent sections.

Existence of the Global Attractor
In what follows, let ρ ∈ [0, 4).Besides, let B be a given bounded absorbing set, whose existence is guaranteed by Theorem 4.1.The main result of the section is Proposition 6.1.For any t ≥ 0, there exists a compact set K(t) ⊂ H such that dist H (S(t)B, K(t)) ≤ ce −ωt for some c ≥ 0 and ω > 0 depending only on B. Proposition 6.1 tells that S(t) is asymptotically compact.Hence, invoking a general result of the theory of dynamical systems (see e.g.[2,6,13,16,27]), S(t) possesses the global attractor A. This establishes the proof of Theorem 5.1.
In order to prove Proposition 6.1, we need a suitable decomposition of f .Lemma 6.2.The nonlinearity f admits the decomposition for some f 0 , f 1 with the following properties: and fulfills the critical growth restriction • f 0 fulfills for every s ∈ R the bounds Choosing then any smooth function ̺ : R → [0, 1] satisfying ̺ ′ (s)s ≥ 0 and In light of (6.1)-( 6.2), it is immediate to check that f 0 (s)s ≥ 0, implying in turn F 0 (s) ≥ 0. We are left to prove the estimate f 0 (s)s ≥ F 0 (s).We limit ourselves to discuss the case s > 0, being the other one analogous.If s < k, then f 0 (s) = F 0 (s) = 0 by the very definition of ̺.If s ≥ k, using again (6.1)-( 6.2) we infer that Hence Exploiting (1.5)-(1.6)and (6.2), we get This concludes the proof.
Defining now the following result holds.
Lemma 6.3.For any t ≥ 0, there exists a closed bounded set B σ (t) ⊂ H σ such that for some constants c ≥ 0 and ω > 0 depending only on B.
Proof.We write f = f 0 + f 1 as in Lemma 6.2.For an arbitrarily fixed z ∈ B, let be the solutions at time t > 0 to the problems (6.3) In what follows, the generic constant c ≥ 0 is independent of the choice of z ∈ B.
Concerning system (6.3),since the forcing term is null and f 0 (v)v ≥ F 0 (v) ≥ 0, an application of Lemma 4.4 yields the exponential decay (6.5) v(t), ∂ t v(t), ξt H ≤ c z H e −ωt , for some c ≥ 0 and ω > 0, depending only on B. Furthermore, a multiplication of the first equation of (6.3) by ∂ tt v gives By the growth assumption on f 0 , Moreover, and Thus, we infer from (6.5) that Concerning system (6.4),introducing the energy H σ , we want to prove the estimate (6.7) Êσ (t) ≤ e ct .
To this aim, we multiply the first equation of (6.4) by A σ ∂ t ŵ, and the second one by ψ in M σ , so to get Thus, the Sobolev embeddings Therefore, in light of Proposition 3.2, (6.5) and (6.6), we have

Collecting the above inequalities we arrive at
Recalling that Êσ (0) = 0, by the Gronwall lemma we obtain the sought inequality (6.7).This finishes the proof.Lemma 6.3 is not quite enough to conclude, since the embedding H σ ⊂ H is not compact (see [26]).Hence, a further argument is needed.
Proof of Proposition 6.1.In the previous notation, for any z ∈ B and any fixed t ≥ 0 we set Ξ t = z∈B ψt .
Exploiting the representation formula for ψt and taking into account that ∂ t ŵ ∈ L ∞ (0, ∞; H 1 ), we learn that Ξ t ⊂ Dom(T ), and by elementary computations we obtain Besides, by (6.7), sup we infer from Lemma A.2 that Ξ t is precompact in M. At this point, exploiting (6.7) again, let B(t) be the closed ball of H 1+σ × H 1+σ centered at zero of a suitable radius Q(t) such that sup Finally, define K(t) = B(t) × Ξ t , the bar standing for the closure in M. Then K(t) is compact in H and fulfills the claim.Remark 6.4.Actually, relying on the gradient system structure of S(t) provided by Proposition 3.4, one could prove the existence of the global attractor without passing through the existence of a bounded absorbing set, which is then recovered as a byproduct (see e.g.[8,13]).The disadvantage of this scheme is that it does not provide any estimate of the entering time into the absorbing set.Proof.The global attractor A, being fully invariant, is contained in every closed attracting set.Hence, to prove the lemma it is enough to exhibit a (closed) ball B σ ⊂ H σ which attracts the bounded absorbing set B. Indeed, by applying Lemma A.3 with r = σ, we will show that dist H (S(t)B, B σ ) ≤ ce −κt , for some κ > 0. To this end, let z ∈ B be fixed, and let y ∈ B and x ∈ H σ be any pair satisfying y + x = z.We define the operators where (v(t), ∂ t v(t), ξ t ) and (w(t), ∂ t w(t), ψ t ) solve systems (6.3) and (6.4) without the hats, with initial data Condition (i) of Lemma A.3 holds by construction, while (ii) follows by the exponential decay (6.5), which now reads Arguing as in the proof of (6.6) we also get In order to prove (iii), we set which holds for all δ > 0 small enough.Here, γ is defined by We estimate the nonlinearity g as follows: we write and, with analogous computations as in (6.8), we obtain Exploiting the decay estimates (6.5) and (7.1) together with (6.7), we arrive at for some Q(•) independent of x.Besides, arguing exactly as in the proof of Lemma 6.3, where we used Proposition 3.2, (7.1) and (7.2).By analogous computations, we draw for some Q(•) independent of x.We finally end up with the differential inequality In light of (7.3), we infer from the Gronwall lemma that

This proves (iii).
A further regularization for ∂ tt u will be needed.
Lemma 7.2.For initial data z ∈ A we have for some c > 0 depending only on A.
Proof.A multiplication of the first equation of (3.1) by A σ ∂ tt u gives In order to estimate the terms in the right-hand side, we exploit the bound (u, ∂ t u, η) H σ ≤ c.

Optimal Regularity of the Attractor
In this section we prove the optimal regularity of the attractor.The key ingredient is the following lemma.
Proof.Knowing that A is fully invariant and bounded in H r , we split the solution S(t)z with z ∈ A into the sum S(t)z = L(t)z + K(t)z, where L(t)z = (v(t), ∂ t v(t), ξ t ) and K(t)z = (w(t), ∂ t w(t), ψ t ) solve the systems for all δ > 0 sufficiently small.We now exploit the embeddings2 H 1+r ⊂ L Besides, due to (1.4), we have for every s ≤ τ .Letting now k = τ , we obtain (1.8) for all s ≤ τ , and from the arbitrariness of τ > 0 the claim follows.Fixing ε ∈ 0, 1 2Θ such that 1 − 2εΘ 2 κ ≥ 1 2 , inequality (A.3) holds for every δ > 0 small.A.2. Two lemmas.We finally recall two results needed in the investigation.The first is a compactness lemma in the space M proved in [26] (see Lemma 5.5 therein), while the second one is Theorem 3.1 from [9], written here in a suitable form for our scopes.for some Q(•) independent of x.Then, B is exponentially attracted by a closed ball B r of H r centered at zero; namely, there exist (strictly) positive constants c, κ such that dist H (S(t)B, B r ) ≤ ce −κt .

Theorem 5 . 4 .
The global attractor A of S(t) is bounded in H 1 .Theorem 5.1 and Theorem 5.4 subsume the main Theorem 1.3 stated in the introduction.

7. Further Regularity Proposition 7 . 1 .
The global attractor A is bounded in H σ .

Lemma A. 2 .
Let Ξ be a subset of Dom(T ), and let r > 0. If sup η∈Ξ η M r + T η M < ∞ and the map s → sup η∈Ξ µ(s) η(s) application of Lemma A.1 together with the Gronwall lemma to the first system shows that the linear semigroup L(t) decays exponentially in H, i.e.
Proof of Proposition 5.5.Let ζ(t) = (u(t), ∂ t u(t), η t ) be a cbt, that is, a solution lying on A. Fixed an arbitrary k > 0, let us consider the solution at time τ > 0 with initial data Proof.We multiply (A.1) by (∂ t u, η) in H r × M r , so obtainingd dt E r + ∂ t u 2 1+r = T A M + γ, ∂ t u r ≤ γ, ∂ t u r ..2) is easily seen to hold for every ε ≤ 1 2Θ and δ ≤ 1 4 .Taking the time derivative of Ψ r we get M r ≤ γ, ∂ t u r + δ γ, u r .

2 1
belongs to L 1 (R + ), then Ξ is precompact in M.Lemma A.3.Let B ⊂ H be a bounded absorbing set for S(t), and let r > 0. For every z ∈ B, assume there exist two operators V z (t) and U z (t) acting on H and H r , respectively, with the following properties:(i) given any y ∈ B and any x ∈ H r satisfying the relation y + x = z, S(t)z = V z (t)y + U z (t)x;(ii) there exists a positive function d 1 vanishing at infinity such that, for any y ∈ B,sup z∈B V z (t)y H ≤ d 1 (t)y H ; (iii) there exists a positive function d 2 vanishing at infinity such that, for any x ∈ H r , sup z∈B U z (t)x H r ≤ d 2 (t) x H r + Q(t),