Regular global attractors for wave equations with degenerate memory

We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory, J. Differential Equations (2016). Under some additional assumptions, we show the existence of a compact absorbing set. This result provides further regularity for the global attractor and shows that it consists of regular solutions.


Introduction
An elastic body perturbed from equilibrium may undergo a restoring force subject to both frictional and viscoelastic dissipation mechanisms. The problem under consideration is the wave equation with degenerate viscoelastic dissipation in the unknown u = u(x, t) This problem was recently treated, to the extent of global well-posedness and global attractors, in [4]. The novelty here being the degenerate nature of the viscoelasticity. Similar problems have yielded several important results as well. We mention some other works concerning semilinear wave equations with memory. On the asymptotic behavior of solutions (in the sense of global attractors) see [9,10,13,27,28,29], and on rates of decay of solutions one can also see [24,30,31].
To the problem under consideration here, the well-posedness was carried out under the guise of semigroup methods. Here, local mild solutions and regular (or "strong" solutions) are obtained using the fact that the underlying operator is the infinitesimal generator of a strongly continuous semigroup of contractions on the Hilbertian phase space H, and the other condition naturally being that the nonlinear term defines a locally Lipschitz continuous functional also on H.
The main result concerning the asymptotic behavior of (1.1)- (1.3) in [4] consists in demonstrating the existence of a finite dimensional global attractor for the semidynamical system (H, S(t)). For this, the authors of [4] rely on [7,Proposition 7.9.4 and Theorem 7.9.6]. That is, the problem is of the asymptotically smooth gradient system class where the set of stationary points is bounded. The socalled quasi-stability of the dynamical system (H, S(t)) involves finding a suitable (relatively) compact seminorm on H (i.e., the approach is similar to finding a global attractor via an α-contraction method). Instead of characterizing the global attractor as the omega-limit set of some bounded absorbing set B in H, i.e. A = ω(B), the global attractor in this work is characterized with properties from the gradient system so that the global attractor is described by the union of unstable manifolds connecting the set of stationary points N , i.e. A = M u (N ). Unlike the methods used to prove the existence of a global attractor by virtue of the former characterization, in the latter no (explicit) bounded absorbing set B nor any (explicit) uniform bound on solutions is used to prove the existence of the global attractor. Finally, it seems that an explicit bound in terms of some of the parameters of the problem (Lipschitz constant, etc.) can be given to the fractal dimension of the global attractor (indeed, see [6,Theorem 3.4.5]). These results are obtained without assuming the two damping terms satisfy a geometric control condition (cf. e.g. [23]).
To treat the memory term, we define a past history variable using the relative displacement history, for all x ∈ Ω ⊂ R 3 and s, t ∈ R + , η t (x, s) := u(x, t) − u(x, t − s).
(1. 4) In order for this formulation to make sense, we also need to prescribe the past history of u(x, t), t < 0. Observe, from (1.4) we readily find the useful identity where k 0 := ∞ 0 g(s)ds assumed to be sufficiently small below (see (2.4)). Thus, equations (1.1)-(1.3) have an equivalent form in the unknowns u = u(x, t) and η t = η t (x, s), for all x ∈ Ω and s, t ∈ R + , with boundary conditions, for all (x, t) ∈ Γ × R + , u(x, t) = 0 and η t (x, s) = 0, (1.7) and the following initial conditions at t = 0, In this article, we aim to provide a regularity result to the global attractors found in [4] for the problem (1.1)-(1.3).

Preliminaries
This section contains a summary of the assumptions and main results of [4]. A word about notation: we will often drop the dependence on x and even t or s from the unknowns u(x, t) and η t (x, s) writing only u and η t instead. The norm in the space L p (Ω) is denoted · p except in the common occurrence when p = 2 where we simply write the L 2 (Ω) norm as · . The L 2 (Ω) product is simply denoted (·, ·). Other Sobolev norms are denoted by occurrence; in particular, since we are working with the homogeneous Dirichlet boundary conditions (1.7), in H 1 0 (Ω), we will use the equivalent norm u H 1 0 (Ω) = ∇u , and in particular, where λ 1 > 0 denotes the first eigenvalue of the Dirichlet-Laplacian. With D(−∆) = H 2 (Ω) ∩ H 1 0 (Ω), we are able to define, for any s ≥ 0, Given a subset B of a Banach space X, denote by B X the quantity sup x∈B x X . Finally, in many calculations C denotes a generic positive constant which may or may not depend on several of the parameters involved in the formulation of the problem, and Q(·) will denote a generic positive nondecreasing function.
Concerning the model problem, we make the following assumptions.
(H1): Let a ∈ C 1 (Ω) be such that meas{x ∈ Γ : a(x) > 0} > 0, and : is a Hilbert space endowed with the product (Two examples are given in [4].) Above ψ |Γ = 0 is meant in the sense of trace which is well-defined when V 1 a ֒→ W 1,1 (Ω). In addition, we also assume the continuous embeddings hold H 1 0 (Ω) ֒→ V 1 a ֒→ L 2 (Ω), and also that Au := div(a(x)∇u) is a self-adjoint non-positive operator. (H2): Assume b ∈ L ∞ (Ω) is a non-negative function and c 0 is a constant satisfying, for all x ∈ Ω, We also impose on g the smallness condition Remark 2.1. Assumption (H1) allows us to set the space for the past history function η t . Indeed, define which is Hilbert with the product Remark 2.2. It should be noted that in [4], the assumption (H2) allows one to view the role of the frictional damping coefficient b as an arbitrarily small complementary damping in the following sense: if where Now we make our final assumptions.
(H4): Let f ∈ C 2 (Ω) and assume there exists C f > 0 such that, for all s ∈ R, (Hence, the nonlinear term is allowed to attain critical growth.) We also assume that Remark 2.4. The two conditions (2.7) and (2.8) are used in [17] which treats the asymptotic behavior of a phase-field equation with memory. The assumption (2.7) implies there is a constant C > 0 such that for all r, s ∈ R |f (r) − f (s)| ≤ C|r − s|(1 + |r| 2 + |s| 2 ).
Concerning the new regularity results described in section 3, we additionally assume the following assumptions hold along with (H1)-(H4).
(H1r): Suppose a ∈ C 1 (Ω) is such that is a Hilbert space endowed with the product Also, assume the continuous embedding holds (2.13) Remark 2.6. The last assumption (2.13) appears in [5,14,15,16,26]. Such a bound is commonly utilized to obtain the precompactness property for the semiflow associated with evolution equations where the use of fractional powers of the Laplace operator present a difficulty, if they are even well-defined.
The finite energy phase-spaces we study problem P in involve the following Hilbert spaces. First, endowed with the norm whose square is given by, for U = (u, v, η) ∈ H 0 , Later we also require a ds < ∞ and H 1 := H 2 (Ω) × H 1 (Ω) × M 1 , with the norm whose square is given by, for U = (u, v, η) ∈ H 1 , (2.14) for some constant C > 0. So that we may write problem P in an operator formulation, we also define the following spaces, to which we observe that there holds D(L) ⊂ H 1 . On these spaces we defined the associated operators T η := −η s , for η ∈ D(T ), and LU := 15) holds as an ODE in M 0 subject to the initial condition Concerning the IVP (2.15)-(2.16), we have the following proposition (cf. [27]).
Proposition 2.7. The operator T with domain D(T ) the generator of the right-translation semigroup. Moreover, η t can be explicitly represented by (2.17) Next we define the nonlinear functional by Problem P can now be written as the abstract Cauchy problem on H 0 , Later, when we are concerned with the regularity properties of problem P, we will also be interested in a more regular subspace of H 0 (this is discussed further below).
Concerning the spaces V 1 a and V 2 a from above, it is important to note that although the injection V 1 a ←֓ V 2 a is compact, it does not follow that the injection M 0 ←֓ M 1 is. Indeed, see [27] for a counterexample. Moreover, this means the embedding H 1 ֒→ H 0 is not compact. Such compactness between the "natural phase spaces" is essential to obtaining further regularity for the global attractors and even for the construction of finite dimensional exponential attractors. To alleviate this issue we follow [20,27] (also see [11,18]) and define the so-called tail function of η ∈ M 0 by, for all τ ≥ 0, With this we set, The space T 1 is Banach with the norm whose square is defined by Hence, let us now also define the space 20) and the desired compact embedding K 1 ֒→ H 0 holds. Again, each space is equipped with the corresponding graph norm whose square is defined by, for all U = (u, v, η) ∈ K 1 , Concerning the IVP (2.15)-(2.16), we will also call upon the following (cf. [11, Lemmas 3.6]).
We now report some results from [4] who only need to assume (H1)-(H4) hold. The following result is from [4, Theorem 2.1]. The proof follows by relying on classical semigroup theory; namely, the operator L is the infinitesimal generator of a C 0 -semigroup of contractions e Lt in H 0 (cf. [4, Lemma 3.1]) and the local Lipschitz continuity of F : H 0 → H 0 . Theorem 2.9. Given h ∈ L 2 (Ω) and U 0 = (u 0 , u 1 , η 0 ) ∈ H 0 , problem P possesses a unique global mild solution satisfying the regularity , the solution is regular and satisfies are any two mild solutions to problem P corresponding to the initial data Z 1 0 , Z 2 0 ∈ H 0 , respectively, where Z 1 0 H 0 ≤ R and Z 2 0 H 0 ≤ R for some R > 0, then for any T > 0 and for all t ∈ [0, T ], for some positive nondecreasing function Q(·).
The next result depends on [4, Lemma 3.3]. For this we define the "energy functional" which is used to extend local solutions to global ones, as well as demonstrate the gradient structure of problem P.
The energy E(t) is non-increasing along any solution U (t) = (u(t), u t (t), η t ). In addition, there exists δ 0 , C f h > 0, independent of U , such that for all t ≥ 0, The following is [4, Theorem 2.2].
Theorem 2.11. Let h ∈ L 2 (Ω) and U 0 = (u 0 , u 1 , η 0 ) ∈ H 0 . The dynamical system (H 0 , S(t)) generated by the mild solutions of Problem P is gradient and possesses a global attractor A which has finite (fractal) dimension and coincides with the unstable manifold M n (N ) of stationary solutions of problem P.
The final two results here will be useful in the next section. Each result follows from the existence of a (bounded) attractor in H 0 . The first result provides a uniform bound on the mild solutions of problem P and some extremely important dissipation integrals, and the second provides the existence of an absorbing set in a natural way.
Consequently, there also holds Proof. The first result is a consequence of the existence of a global/universal attractor.
Corollary 2.13. The semigroup of solution operators S(t) admits a bounded absorbing set B in H 0 ; that is, for any subset B ⊂ H 0 , there exists t B ≥ 0 (depending on B) such that for all t ≥ t B , S(t)B ⊂ B.
Proof. The proof follows directly from the fact that the attractor A is bounded in H 0 ; e.g., a ball in H 0 of radius A H 0 + 1 is an absorbing set in H 0 .
Remark 2.14. Unfortunately we do not know the rate of convergence of any bounded subset in H 0 to the global attractor A. Moreover, there are several applications in the literature (not containing equations with degeneracies in crucial diffusion or damping terms) in which the rate of convergence of any bonded subset B of H 0 is exponential in the sense that there is a constant ̟ > 0 such that for any nonempty bounded subset B ⊂ H 0 and for all t ≥ 0 there holds, Here, given two subsets U and V of a Banach space X, the Hausdorff semidistance between them is

Regularity
The aim of this section, and indeed the aim of this article, is to show the existence of a smooth compact subset of H 0 containing the global attractor A. This is achieved by finding a suitable subset C of K 1 ֒→ H 0 ; hence, C is compact in H 0 . To this end we decompose the semigroup of solution operators by showing it splits into uniformly decaying to zero and uniformly compact parts. With this we obtain asymptotic compactness for the associated semigroup of solution operators. The procedure requires some technical lemmas and a suitable Grönwall type inequality; the presentation follows [14,16]. The argument developed here will also be relied on to establish the existence of a compact attracting set. As a reminder to the reader, throughout this section (and the next) we assume the hypotheses (H1r), (H3r) and (H4r) hold in addition to (H1)-(H4).
The main result in this section is the following.
Theorem 3.1. There exists a closed and bounded subset C ⊂ K 1 and a conatant ω > 0 such that for every nonempty bounded subset B ⊂ H 0 and for all t ≥ 0, there holds (3.1) Consequently, the global attractor A (cf. Theorem 2.11) is bounded in K 1 and trajectories on A are regular solutions of the form The proof first requires several lemmas. where v + w = u and ξ + ζ = η satisfy, respectively, problem V and problem W which are given by at Ω × {0}.
(3.5) We now define the operators K(t)U 0 := (w(t), w t (t), ζ t ) and Z(t)U 0 := (v(t), v t (t), ξ t ) using the associated global mild solutions to problem V and problem W (the existence of such solutions follows in a similar manor to the semigroup methods used to establish the well-posedness for problem P; cf. Theorem 2.9 and the regularity described in (2.21)).
The first of the subsequent lemmas shows that the operators K(t) are bounded bounded on H 0 .
The following lemma provides an estimate that will be extremely important later in this section.
Lemma 3.2. For each U 0 = (u 0 , u 1 , η 0 ) ∈ H 0 there exists a unique global weak solution to problem W. Moreover, for each R > 0 and for all U 0 ∈ H 0 with U 0 H 0 ≤ R, there holds, for all t ≥ 0, for some nonnegative increasing function Q(·). There also holds In addition, for every ε > 0 there exists a function Q ε (·) ∼ ε −1 such that for every 0 ≤ s ≤ t, R > 0 and (3.10) Proof. As we have already stated above, the existence of global mild solutions satisfying (3.6) follows by arguing as in the proof of Theorem 2.9. The bound (3.7) essentially follows from the existence of a global attractor for problem P (cf. Corollary 2.12). The dissipation property (3.8) follows by arguing exactly as in the proof of Corollary 2.12 keeping in mind both u (1) and u (b) make sense, and that we are able to utilize the bound (2.28) for either one.
To show (3.10), we now add in the bound u t 2 + δ η 2 M 0 + 2 w t 2 ≤ C(R) into (3.11), and this time estimate the right-hand side with C(R) + w t 2 to obtain Integrating (3.13) over (t, t + 1) and applying (2.30) for problem W yields (3.10).
Lemma 3.3. For each U 0 = (u 0 , u 1 , η 0 ) ∈ H 0 there exists a unique global weak solution to problem V. Moreover, for each R > 0 and for all U 0 ∈ H 0 with U 0 H 0 ≤ R, there exists ω 1 > 0 such that, for all t ≥ 0, for some positive nondecreasing function Q(·). Thus, the operators Z(t) are uniformly decaying to zero in H 0 .
Proof. As we have already stated above, the existence of global mild solutions satisfying (3.14) follows by arguing as in the proof of Theorem 2.9. It suffices to show (3.15).
Then multiply the result in L 2 (Ω) by v t + εv, where ε > 0 will be chosen below. When we include the basic identity to the result and use (3.4) 2 , we find that there holds, for almost all t ≥ 0, We now consider the functional defined by We now will show that, given U (t) = (u(t), u t (t), η t ), W (t) = (w(t), w t (t), ζ t ) ∈ H 0 are uniformly bounded with respect to t ≥ 0 by some R > 0, there are constants C 1 , C 2 > 0, independent of t, in which for all To this end we begin by estimating the following product with (2.1), 19) and (3.20) Concerning the terms in the functional V that involve the nonlinear term ψ, using (3.3), (2.7), (2.8) and the embedding H 1 (Ω) ֒→ L 6 (Ω), and also (2.26), there holds where the constant 0 < C ε ∼ ε −1 . From assumption (2.13) and (3.3) Hence, for β = β(ε) sufficiently large, the combination of (3.21) and (3.22) produces, So for a sufficiently small ε > 0 fixed (which also fixes the choice of β), there is m 0 > 0 in which, for all t ≥ 0, we have that Now by the (local) Lipschitz continuity of f , the embedding H 1 0 (Ω) ֒→ L 2 (Ω), the uniform bounds on u and w, and the Poincaré inequality (2.1), it is easy to check that with (2.9) there holds (3.25) Also, using (3.3), (2.7), (2.8) and the bound (2.26), there also holds (3.26) Thus, with (3.25), (3.26) and referring to some of the above estimates, the right-hand side of (3.18) also follows.
Moving forward, we now work on (3.16). In light of the estimates and (here the constants C(R) > 0 also depend on β > 0) we see that with (3.27), (3.28), as well as (2.6), (2.3) and (3.22), the differential identity (3.16) becomes where we also added 3ε v t 2 to both sides (observe, 3ε v t 2 ≤ 3εV). We now seek a suitable control on the product For sufficiently large β > 0, we may omit the positive terms 2 b(x)v t 2 + (2ε(β − ϑ) − 1 β ) v 2 from the left-hand side of (3.29) so that it becomes, with (3.30), For any ε > 0 sufficiently small so that we can find a constant m 1 > 0, thanks to (3.18), such that (3.31) can be written as the following differential inequality, to hold for almost all t ≥ 0, (3.32) Here we recall Proposition A.1 and Lemma 3.2. Applying these to (3.32) yields, for all t ≥ 0, for some positive nondecreasing function Q(·). By virtue of (3.18) and the initial conditions provided in (3.4), Therefore (3.33) shows that the operators Z(t) are uniformly decaying to zero. The proof is finished. <<<<< The remaining lemmas will show that the operators K(t) are asymptotically compact on H 0 . In order to establish this, we prove that the operators K(t) are uniformly bounded in K 1 ֒→ H 0 .
Due to the nature of the proof of the following lemma, we also need to assign the past history for the term w t . Indeed, from below we need to consider the initial condition for some positive nondecreasing function Q(·).
Proof. For all x ∈ Ω and t, s ∈ R + , set H(x, t) := w t (x, t) and X t := ζ t t (s). Differentiating problem W with respect to t yields the system at Ω × {0}.
(3.36) Multiply equation (3.36) 1 by H t + εH for some ε > 0 to be chosen below. To this result we apply the identities and (here we rely on (3.36 so that together we find H). (3.37) Next we recall (2.3) and find where the last inequality follows from (2.4). For all ε > 0 and t ≥ 0, define the functional Thanks to (2.6) and since ψ ′ > 0, there is a constant C > 0, sufficiently small, so that (3.41) At this point we can write (3.37)-(3.39) with (3.40) as H). (3.42) Next, let us rely on the uniform bounds (2.26) and (3.15) to estimate the products on the right-hand side Thus, combining (3.42)-(3.45) yields Since 4ε H t 2 ≤ 4εI, adding this to (3.46) makes the differential inequality (we also omit 2ε b(x)H t 2 ) We now find that for any ε > 0 small so that to which we now apply Proposition A.2 and the bounds (3.9) and (3.10) to conclude that, for all t ≥ 0, there holds (3.49) Moreover, with (3.40) and the initial conditions in (3.36) we find that there is a constant C > 0 (with ε > 0 now fixed) in which This establishes (3.35) and completes the proof.
We derive the immediate consequence of (3.5) and (3.35). Before we continue, we derive a further estimate for ζ t .
Lemma 3. 6. Under the assumptions of Lemma 3.4, there holds for all t ≥ 0, Proof. Formally multiplying (3.5) 2 in L 2 g (R + ; L 2 (Ω)) by −∆ζ t (s) and estimating the result yields the differential inequality (3.52) Hence, applying the bound (3.35) to (3.52), we find the differential inequality which holds for almost all Applying a straight-forward Grönwall inequality and the initial conditions in (3.5) produces the desired bound (3.51). This concludes the proof.
Lemma 3.7. Under the assumptions of Lemma 3.4, the following holds for all t > 0, (3.53) for some positive nondecreasing function Q(·). Furthermore, the operators K(t) are uniformly compact in H 0 .
Proof. The proof consists of several parts. In the first part, we derive further bounds for some higher order terms. We begin by rewriting/expanding (3.5) as (3.57) We now report six identities that will be used below: and Next we multiply (3.57) in L 2 (Ω) by (−∆)w t + (−∆)w to obtain, in light of (3.59)-(3.63), the differential identity We now seek a constant m 2 > 0 sufficiently small so that we can write the above differential identity in the following form (3.66) The important lower bound holds for some constants C 1 , C 2 (R) > 0, and essentially follows from some basic estimates, the bounds (2.26), (3.15), (3.35), (3.51), the Poincaré inequality (2.1) and with the assumptions on the functions a and b. Indeed, we estimate, for all ε > 0, (3.72) Applying (3.68)-(3.72) to (3.66) gives us the lower bound for all ε > 0, For any fixed 0 < ε < min{1, ℓ 0 /(2 + k 0 )}, we obtain (3.67).
We now prove the main theorem.
Proof of Theorem 3.1. Define the subset C of K 1 by where Q(R) > 0 is the function from Lemma 3.7, and R > 0 is such that U 0 H 0 ≤ R. Let now U 0 = (u 0 , u 1 , η 0 ) ∈ B (the bounded absorbing set of Corollary 2.13 endowed with the topology of H 0 ). Then, for all t ≥ 0 and for all U 0 ∈ B, S(t)U 0 = Z(t)U 0 + K(t)U 0 , where Z(t) is uniformly and exponentially decaying to zero by Lemma 3.3, and, by Lemma 3.7, K(t) is uniformly bounded in K 1 . In particular, there holds dist H 0 (S(t)B, C) ≤ Q(R)e −ωt .
The proof is finished.