Regular global attractors for wave equations with degenerate memory Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 59-82

DOI: 10.15826/umj.2019.1.007

REGULAR GLOBAL ATTRACTORS FOR WAVE EQUATIONS

WITH DEGENERATE MEMORY

Joseph L. Shomberg

Department of Mathematics and Computer Science, Providence College, Providence, RI 02918, USA [email protected]

Abstract: We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cav-alcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory, J. Differential Equations (2016). Under certain extra assumptions (namely on the nonlinear term), we show the existence of a compact attracting set which provides further regularity for the global attractor and show that it consists of regular solutions.

Keywords: Degenerate viscoelasticity, Relative displacement history, Nonlinear wave equation, Critical exponent, Regular global attractor.

1. 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)

f <x

utt - Au +/ g(s)div[a(x)Vu(t - s)]ds + b(x)ut + f (u) = h(x) in Q x R+, (1.1) 0

defined on a bounded domain Q in R3 with smooth (at least class C2) boundary r. Here, g(s) is an temporal interaction kernel which transmits memory effects to produce the viscoelastic dissipation mechanisms, the function b(x) is the spatially dependent frictional damping coefficient, the nonlinear term f (u) communicates displacement dependent density in the material, and the function h(x) represents a spatially dependent external forcing mechanism. The equation is subject to Dirichlet boundary conditions

u(x, t) = 0 onr x R+, (1.2)

and the initial conditions

u(x, 0) = u0(x) and ut(x, 0) = u1(x) at Q x {0}. (1.3)

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, 27-29], 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 (sometimes referred to as "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 notions of mild solution and regular solution are described below after equation (2.15).

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 associated semidynamical system (H,S(t)). (Throughout, S(t) denotes the semigroup of solution operators generated by problem (1.1)-(1.3).) 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 so-called 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 a-contraction method). Instead of characterizing the global attractor as the omega-limit set of some bounded absorbing set B in H, i.e. A = w(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 = Mu(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 € Q C R3 and s, t € R+,

nf (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

/ g(s)div[a(x)Vu(t — s)]ds = — / g(s)div[a(x)Vn4(s)]ds + k0 div[a(x)Vu(t)],

J 0 Jo

where k0 := /0° g(s)ds assumed to be sufficiently small below (see (2.3)). Thus, equations (1.1)-(1.3) have an equivalent form in the unknowns u = u(x,t) and rf = rf(x, s), for all x € Q and s, t € R+,

f <x

utt — div[(1 — k0 a(x))Vu] — / g(s)div[a(x)Vn* (s)]ds + b(x)ut + f (u) = h(x), (1.5)

0

nt = —ns + ut,

with boundary conditions, for all (x,t) € r x R+,

u(x,t) = 0 and nt(x,s)=0, (1.6) and the following initial conditions at t = 0,

u(x, 0) = u0(x), ut(x, 0) = ui(x) and nt(x, 0) = 0, n°(x,s)= n0(x,s). (1.7)

In this article, we aim to provide a regularity result to the global attractors found in [4] for the problem (1.1)-(1.3).

2. Preliminaries

This section contains a summary of the assumptions and main results of [4]. A word a,bout notation: we will often drop the dependence on x and even t or s from the unknowns u(x,t) and nt(x,s) writing only u and rf instead. The norm in the space Lp(Q) is denoted || ■ ||p except in the common occurrence when p = 2 where we simply write the L2(Q) norm as || ■ ||. The L2(Q) product is simply denoted (■, ■). Other Sobolev norms are denoted by occurrence; in particular, since we are working with the homogeneous Dirichlet boundary conditions (1.6), in Hj(Q), we will use the equivalent norm

||u||H (fi) = ||Vu|

and in particular,

||u|| < —!=||Vu||, (2.1)

where Ai > 0 denotes the first eigenvalue of the Dirichlet-Laplacian. With D(-A) = H2(Q) n H0(Q), we are able to define, for any s > 0,

Hs := D((-A)s/2).

Given a subset B of a Banach space X, denote by ||B||X the quantity supxeB ||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.

(HI) Let a € C1(Q) be such that the space

meas{x € r : a(x) > 0} > 0,

and

Va := {^ € L2(Q) : J a(x)|V^(x)|2dx < to, = 0}, is a Hilbert space endowed with the product

(X,^)yi := / a(x)Vx(x) ■ V^(x)dx. afi

(Two examples are given in [4].) Above ^r = 0 is meant in the sense of trace which is well-defined when Va ^ Wi'i(^). In addition, we also assume the continuous embeddings hold

Hi(Q) ^va ^ L2(Q),

and also that Au := div(a(x)Vu) is a self-adjoint non-positive operator. (H2) Assume b € L^(Q) is a non-negative function and co is a constant satisfying, for all x € Q,

inf {a(x) + b(x)} > c0 > 0.

xefi

(H3) Assume g € C 1(R+) n L1(R+) satisfies, for all s > 0,

g(s) > 0 and g'(s) < —5g(s). (2.2)

We also impose on g the smallness condition

k0 := / g(s)ds < IM!"1. (2.3)

0

Remark 1. Assumption (H1) allows us to set the space for the past history function nt. Indeed, define

M0 := L2(R+; V1) = {n(x,s) : g(s)||n(x, s)||2ids < which is Hilbert with the product

(n,Z)m° := j g(s) ^ j a(x)Vg(x,s) ■ VZ(x,s)d^ds.

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 w0 := {x € R3 : a(x) = 0}, then what is only required is b(x) > 0 on any neighborhood of w0.

Equation (2.2) of assumption (H3) implies g decays to zero exponentially. Moreover, by (2.3), we have that, for all x € Q,

0 < I0 < 1 — k0a(x) (2.4)

where

I0 := 1 — k0||a||".

Now we make our final assumptions. (H4) Let f € C2(Q) and assume there exists Cf > 0 such that, for all s € R,

|f''(s)|< Cf(1 + |s|). (2.5)

(Hence, the nonlinear term is allowed to attain critical growth.) We also assume that

liminf^ >-4Ai (2.6)

s

cf. (2.1).

Remark 2. The two conditions (2.5) and (2.6) are used in [17] which treats the asymptotic behavior of a phase-field equation with memory. The assumption (2.5) 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). (2.7)

The condition (2.7) appears in many recent works on semilinear wave equations with memory (e.g. [13]) and the strongly damped wave equation (this condition refers to the subcritical setting of those problems), see for example [2, 3, 12, 21, 25, 28, 29]. By (2.6) we find that for some a € (0, A1), there exists pf > 0 so that, for all s € R, there hold

f (s)s >—l0as2 — pf (2.8)

and, for F(s) := J0 f (a)da,

F(s)>-e-fs*-pf. (2.9)

Observe though both (2.8) and (2.9) follow when (2.6) is replaced by the less general assumption,

liminf f'(s) >-loAi. (2.10)

Assumption (2.5) and condition (2.10) appear in equations with memory terms [5, 8, 11, 29].

Concerning the new regularity results described in section 3, we additionally assume the following assumptions hold along with (H1)-(H4).

(Hlr) Suppose a € C1(Q) is such that the space

V2 := {^ € L2(Q) : J a(x) (|A^(x)|2 + |^(x)|2) dx < to, ^|r = 0},

is a Hilbert space endowed with the product

(X,^)y2 := / a(x) (Ax(x)A^(x) + x(x)^(x)) dx. afi

Also, assume the continuous embedding holds

v2 ^ H0(q).

Remark 3. It should be noted that the embedding D(-A) ^ V,2, where D(-A) := H2(Q) n H0(Q), does not hold. The interested reader should see [1, Section 3] where it is shown H2(Q) ^ V^.

(H4r) Assume that there exists $ > 0 such that, for all s € R,

f'(s) > (2.11)

Remark 4. The last assumption (2.11) appears in [5, 14-16, 26]. Such a bound is commonly utilized to obtain the precompactness property for the semigroup of solution operators associated with evolution equations where the use of fractional powers of the Laplace operator present a difficulty, if they are even well-defined.

Throughout the remainder of this article, we simply denote (1.5)-(1.7) under assumptions (H1)-(H4) and (H1r) and (H4r) as problem P.

The finite energy phase-spaces we study problem P in involve the following Hilbert spaces. First,

H0 := Hi(Q) x L2(Q) x M0, endowed with the norm whose square is given by, for U = (u, v,n) € H0,

||U||Ho := ||Vu||2 + ||v||2 + ||n|Mo.

Later we also require

M1 := L2(R+; V2) = {n : jT g(s) ||n(s) ^ ds < to}

and

H1 := H2(Q) x H 1(Q) x M1,

with the norm whose square is given by, for U = (u, v, n) € H1,

IIUIlHi := IMI^n) + IIvIlHi(n) + IlnlMi.

Here H 1(Q) is normed with

IMIffi(n) = + M)1/2 ,

and concerning the H2(Q) norm above, we know by H2-elliptic regularity theory (cf. e.g. [19, section 8.4]),

IMIn2(Q) < C (||A^|| + ||^||), (2.12)

for some constant C > 0.

So that we may write problem P in an operator formulation, we also define the following spaces,

D(T) := {n € M0 : ns € M0, n(0) = 0},

where ns denotes the distributional derivative of n and the equality n(0) = 0 is meant as

lim0 11n(s) | =0,

si0

and

v € Hq(Q), n € D(Tr),

D(L) := <J U = (u, v, n) € H0

div[(1 - fc0a(x))Vu] + / g(s)div[a(x)Vn(s)]ds € L2(Q) J'

Jo

to which we observe that there holds D(L) C H1. On these spaces we defined the associated operators

Trn := —ns, for n € D(Tr),

and

v

CU :=

fM

div[(1 — k0 a(x))Vu] + g(s)div[a(x)Vn(s)]ds — b(x)v

0

V v + Tr n y

' for U € D(L).

For each t € [0,T], the equation

nt = Tr n4 + v(t) (2.13)

holds as an ODE in M0 subject to the initial condition

n0 = no € M0. (2.14)

Concerning the initial value problem (IVP) (2.13)-(2.14), we have the following proposition (cf. [27]).

Proposition 1. The operator Tr with domain D(Tr) the generator of the right-translation semigroup. Moreover, rf can be explicitly represented by

t f u(t) — u(t — s) if 0 < s < t,

n (s) = <

I n0(s — t) + u(t) — u(0) if s > t.

Next we define the nonlinear functional by

F(U) := (0,-f (u)+ h, 0). Problem P can now be written as the abstract Cauchy problem on H0,

d

-u = cu + nui t> o, (215)

U(0) = Uo = (uo,ui,no) e H0.

Later, when we are concerned with the regularity properties of problem P, we will also be interested in a more regular subspace of H0 (this is discussed further below).

Definition 1. Let T > 0 and Uo = (uo,ui,no) e H0 = H 1(Q) x L2(Q) x M0 be given. A function U e C([0, T]; H0) is called a mild solution to (2.15) on [0,T] if and only if F(U(■)) e L1(0, T; H0) and U satisfies the variation of constants formula for all t e [0, T],

U(t) = eLtUo + f eL(t-s)F(U(s))ds. Jo

The map U = (u, ut, n) is a mild solution on [0, to) (i.e., is a global mild solution) if it is a mild solution on [0, T], for every T > 0.

The notion of regular solution used in this article is given precisely in equation (3.1). A regular solution requires better data, e.g. U0 e H1 = H2(Q) x H 1(Q) x M1, and a trajectory that remains in the same space, e.g. U(t) e H1. Indeed, our notion will also include the tail spaces defined above. Here, regular solutions are mild solutions that persist in the space

H2 (Q) x H1 (Q) xT1 Vt > 0.

Concerning the spaces V^ and V^ from above, it is important to note that although the injection V1 ^ V2 is compact, it does not follow that the injection M0 ^ M1 is. Indeed, see [27] for a counterexample. Moreover, this means the embedding H1 ^ H0 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 n e M0 by, for all t > 0,

T(t; n):= J g(s)||Vn(s)||2ds.

(0,1/t )U(r,ro)

With this we set,

T1 n e M1 : ns e M0, n(0) = 0, suptT(t; n) < ^.

t >1 J

The space T1 is Banach with the norm whose square is defined by

llnllTi := llnllM + llnslMo +suptT(t; n). (2.16)

T >1

Importantly, the embedding T1 ^ M0 is compact. (We should mention that although the works [11, 18] treat PDE with an integrated past history variable, the compactness issue still applies to

models with a relative displacement history variable, such as (1.4) here. In fact, the compactness issue is more delicate in this setting; one must introduce so-called "tail functions," cf. [11, Lemma 3.1] or [18, Proposition 5.4]). Hence, let us now also define the space

K1 := H2(Q) x H1 (Q) x T1,

and the desired compact embedding K1 — H0 holds. Again, each space is equipped with the corresponding graph norm whose square is defined by, for all U = (u, v,n) € K1,

IIUIIKi := IIuIH2(q) + IIvIIHi(Q) + IInIITi.

Concerning the IVP (2.13)-(2.14), we will also call upon the following (cf. [11, Lemmas 3.6]).

Lemma 1. Let n0 € D(Tr). Assume there is p > 0 such that, for all t > 0, ||Vu(t)|| < p. Then there is a constant C > 0 such that, for all t > 0,

sup tT(t; n*) < 2 (t + 2) e-<5i sup tT(t; no) + Cp2.

T>1 T>1

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 C0-semigroup of contractions eLt in H0 (cf. [4, Lemma 3.1]) and the local Lipschitz continuity of F : H0 — H0.

Theorem 1. Given h € L2(Q) and U0 = (u0, u1, n0) € H0, problem P possesses a unique global mild .solution satisfying the regularity

u € C([0, to); H0(Q)), u € C([0, to); L2(Q)) and n* € C([0, to); M0). (2.17) If U0 = (u0,u1,n0) € D(L), the solution is regular and satisfies

U € C([0, to); D(L)).

In addition, if Zl(t) = (ul(t),u*(t),^'*), i = 1,2, are any two mild solutions to problem P corresponding to the initial data Z°,Z° € H0, respectively, where ||Z°||%o < R and ||Zq||%o < R for some R > 0, then for any T > 0 and for all t € [0, T],

||Z 1(t) — Z2(t)||ho < eQ(R)T||Z 1(0) — Z2(0)|H0 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.

E(t) := |Mt)||2 + / (1 — fc°a(x))|Vu(t)|2dx + ||nt|Mo +2 / (F(u(t)) — h(x)u(t)) dx. (2.18)

Jq JQ

Lemma 2. The energy E(t) is non-increasing along any solution U(t) = (u(t),ut(t),n*). In addition, there exists , C/h > 0, independent of U, such that for all t > 0,

E(t) < ¿°|(u(t),ut(t),n*)|Ho — Cfh.

The following is [4, Theorem 2.2].

Theorem 2. Let h € L2(Q) and U0 = (uo,Ui,no) € H0. The dynamical system (H0,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 Mn(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 H0. 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.

Corollary 1. For each R > 0 and every U0 = (u0,u1,n0) € H0 such that ||U0||%o < R, there exists a positive nondecreasing function Q(-) such that, for all t > 0,

||S(t)U0||ho < Q(R). (2.19)

In addition, there exists a function Q(-) such that

1°° (ll-^K(T)ll2 + 6\\vT\\2Mo) dr < Q(R). (2.20)

Consequently, there also holds

rm

|2

||ut(r)||2dT < Q(R). (2.21)

0

Proof. The first result is a consequence of the existence of a global/universal attractor. To show (2.20), let R > 0 be given and U0 € H0 be such that ||U0||Ho < R. Next we formally derive the "energy identity" associated with problem P by multiplying (1.5) by 2ut to then integrate over Q; this yields (cf. [4, Equation (3.7)]),

d f°° f

—E + 2 g(s) / a(x)Vrf(s) ■ Vutd,xds + 2\\ ^/b{x)ut\\2 = 0. dt J0 Jo

where E is the energy functional (2.18). Observe, thanks to (2.19), (2.4) and (2.9), we readily find C(R) > 0 such that, for all t > 0,

|E(t)| < C(R). (2.22)

Next we note that with (3.4)2 there holds,

/m r d fm d

g(s) I a(x)Vrf(s) ■ Vutdxds = j^W'Wmo + / g{s) — \\rj%}ds,

/0 JO dt J0

and applying (2.2) yields,

rm d fm rm

I 9{*)-^\\rf{*)\\2vids = ~l 9\sms)rvlds>8 Jo g(8)\\rf(8)\\2vid8. (2.23) Hence, we have

ri ,_

JtE +S^ll^o+2\\s/b^)ut\\2<0. (2.24)

Thus, integrating (2.24) over (0, t) produces (2.20).

Now we show (2.21) easily follows from (2.20). Indeed, using the Mean Value Theorem for Definite Integrals, for each t > 0, there is € Q so that

\[sMx)utf= / b(x)\Ut(T)\2dx = b(Zr)\\Ut(T) Jo

Now consider

/*oo /*oo

/ b(Cr)\\Ut(r)\\2dT= / || ^/b{x)ut{T)\\2dT,

J0 Jo

where 6(x) ^ 0, that is, where 6(x) is not identically equal to zero on Q; thus, motivated by the average value, 6({r) > 0 for each t > 0. Define 6* := infr>0 6({r) > 0. So with (2.20) we find

f ^ 1

/ \\ut(r)\\2dr<-Q(R).

Jo

The thesis (2.21) follows with hypotheses (H5). The proof is complete. □

Corollary 2. The semigroup of solution operators S(t) admits a bounded absorbing set B in H0; that is, for any subset B c H0, there exists ts > 0 (depending on B) such that for all t > ts, S(t)B c B.

Proof. The proof follows directly from the fact that the attractor A is bounded in H0; e.g., a ball in H0 of radius ||A||%° + 1 is an absorbing set in H0. □

Remark 5. Unfortunately we do not know the rate of convergence of any bounded subset in H0 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 H0 is exponential in the sense that there is a constant w > 0 such that for any nonempty bounded subset B c H0 and for all t > 0 there holds,

distH°(S(t)B, B) < Q(R)e-rot.

Here, given two subsets U and V of a Banach space X, the Hausdorff semidistance between them is

distX(U, V) := sup inf ||u — v||X.

3. Regularity

The aim of this section, and indeed the aim of this article, is to show the existence of a smooth compact subset of H0 containing the global attractor A. This is achieved by finding a suitable subset C of K1 ^ H0; hence, C is compact in H0. 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 Gronwall 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 we assume the hypotheses (H1r) and (H4r) hold in addition to (H1)-(H4).

The main result in this section is the following.

Theorem 3. Assume hypotheses (H1)-(H4), (H1r) and (H4r) hold. There exists a closed and bounded subset CcK1 and a constant w > 0 such that for every nonempty bounded subset B c H0 and for all t > 0, there holds

distH°(S(t)B,C) < Q(||B||h°)e-wi.

Consequently, the global attractor A (cf. Theorem 2) is bounded in K1 and trajectories on A are regular solutions of the form

u € C([0, to); H2(Q)), ut € C([0, to); H1 (Q)) and rf G C([0, to); T1). (3.1)

The proof of Theorem 3 requires several lemmas.

Step 1. The semigroup of solution operators is decomposed into two operators S(t) = K(t) + Z(t) for all t > 0.

Step 2. In Lemma 3 we establish the global existence for the associated operators K(t) as well as provide a uniform bound on K(t) in H0, and, rather importantly, provides various dissipation integrals for various terms. These dissipation results are key to the method for obtaining compactness.

Step 3. Next, Lemma 4 establishes the global existence for the associated operators Z(t). We also show that the operators Z(t) are uniformly decaying to zero in H0.

Step 4. Upon differentiating the problem corresponding to the operators K(t) we establish a higher-order uniform bound on dtK(t) in H0 in Lemma 5. This argument is crucial for obtaining the asymptotic compactness for the non-memory terms of the operators K(t).

Step 5. The remaining Lemma 6 and Lemma 7 establish the appropriate bounds on the memory variable to complete the asymptotic compactness of K(t). It is certainly nontrivial to establish asymptotic compactness for solution operators that involve problems with memory terms. (Indeed, recall the embedding H1 ^ H0 is not compact.)

Step 6. The proof of Theorem 3 follows. This result ultimately provides a higher-order bound on the global attractor demonstrated in the prequel.

Set

^(s) := f (s) + Ps with p > tf so that ^'(s) > 0 (3.2)

and set ^(s) := f0S(We remind the reader of (2.11).) Let U0 = (u0,u1,n0) € H0. Decompose (1.5)-(1.7) into the functions v, w, £ and Z where v + w = u and £ + Z = n satisfy, respectively,

problem V and problem W which are given by

vtt—div[(1 - k0a(x))Vv]— / g(s)div[a(x)V£i(s)]ds+6(x)vi+^(u)-^(w)=0 in Q x R+,

0

£t = -£t + vt in Q x R+,

v(x, t) = 0, £*(x,s) = 0

v(x, 0) = u0(x), vt (x, 0) = u1(x), £*(x, 0) = 0, £0(x,s) = n0(x,s)

on r x R+, at Q x {0}

(3.3)

and

wtt — div[(1-k0a(x))Vw]- / g(s)div[a(x)VZi(s)]ds+6(x)wt+^(w)=h(x)+Pu in Q x R+,

0

Zt = —ZS + wt in Q x R+,

w(x, t) = 0, Zt(x, s) = 0 on r x R+,

w(x, 0)=0, wt(x, 0)=0, Zt(x, 0)=0, Z0(x, s) = 0 at Q x {0}.

t t (3.4)

We now define the operators K(t)U0 := (w(t),wt(t),Zt) and Z(t)U0 := (v(t),vt(t),£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 1 and the regularity described in (2.17)).

The first of the subsequent lemmas shows that the operators K(t) are bounded bounded on H0. The following lemma provides an estimate that will be extremely important later in this section.

oo

Lemma 3. Assume the hypotheses of Theorem 3 hold. For each U0 = (u0, U, no) G H0 there exists a unique global weak .solution

W :=(w,wt*) G C([0, to); H0) (3.5)

to problem W. Moreover, for each R > 0 and for all U0 G H0 with ||U0||h° < R, there holds, for all t > 0,

||K(t)U01|„0 < Q(R) (3.6)

for some nonnegative increasing function Q(-). There also holds

/•те

/ iK(T)||2dT < Q(R). (3.7)

0

In addition, for every e > 0 there exists a function Q(-) such that for every 0 < s < t, R> 0 and U0 = (u0,u1,n0) G H0 with ||U0|ho < R, there holds

lk(T)||2 + II II2 + 5WfMo + |K(r)||2 + ||л/бй^(г)||2 + 5||CT||^o) dT

J (3.8) e 1

<r2(t-s) + -Q(R)-

Finally, there holds

rt+1 . __x

j[ (ll^(T)ll2 + ¿llnl^o + Цл/бЙ^МЦ2 + |И(т)||2 + ¿IICl^o) dT < Q(R). (3.9)

Proof. As we have already stated above, the existence of global mild solutions satisfying (3.5) follows by arguing as in the proof of Theorem 1. The bound (3.6) essentially follows from the existence of a global attractor for problem P (cf. Corollary 1). The dissipation property (3.7) follows by arguing exactly as in the proof of Corollary 1 keeping in mind both u(1) and u(b) make sense, and that we are able to utilize the bound (2.21) for either one.

We are now interested in establishing (3.8). Indeed, multiplying (3.4)1 by 2wt and integrating over Q, applying (3.4)2 and applying an estimate like (2.23), all with w and Z in place of u and n, respectively, and Ew denoting the corresponding functional E, produces (in place of (2.24))

r] _

jtEw+6\\C\\2M0+2\\ s/b^)wt\\2 < 2f3(u,wt). (3.10)

Since

and by (3.6)

d

2p(u, wt) = 2p(ut,w) + 2/3—(и, w)

e1.......2

2P(ut, w) < p2C(R)||ut|| < e + C so the differential inequality (3.10) becomes

r] _

~{EW - 2p(u, w)} + ¿HCl^o + 2\\y/b(xjwt\\2 < s + C£|M|2. (3.11)

In light of (2.20) and (2.21), adding ||t/,i||2 + || s/b{x)ut{T)\\2 + 5\[r]T\\'2M0 to both sides of (3.11) and integrating the result over (s,t) then applying (2.19), (3.6) and (2.22) for problem W produces the desired estimate (3.8).

To show (3.9), we now add in the bound ||ut||2 + ¿||n||M° +2IKI|2 < C(R) into (3.10), and this time estimate the right-hand side with C(R) + ||wt||2 to obtain

ri ,_

jEw + \\ut(r)\\2+6U\\2M0 + \\s/b^)wt(r)\\2 + K(r)||2 + < C(R). (3.12)

Integrating (3.12) over (t,t + 1) and applying (2.22) for problem W yields (3.9). □

Lemma 4. Assume the hypotheses of Theorem 3 hold. For each U0 = (u0,ui, no) G H0 there exists a unique global weak solution

V :=(v,vi,Zi) G C([0, to); H0) (3.13)

to problem V. Moreover, for each R > 0 and for all U0 G H0 with ||U0 ||%° < R, there exists w1 > 0 such that, for all t > 0,

||Z(t)U0||h° < Q(R)e-wli (3.14)

for some positive nondecreasing function Q(-). Thus, the operators Z(t) are uniformly decaying to zero in H0.

Proof. As we have already stated above, the existence of global mild solutions satisfying (3.13) follows by arguing as in the proof of Theorem 1. It suffices to show (3.14).

Let R > 0 and U0 = (u0, u1, n0) G H0 be such that ||U0||%° < R. Next we rewrite the term b(x)vt in equation (3.3)1 as (b(x) + 1)vt — vt. Then multiply the result in L2(Q) by vt + ev, where e > 0 will be chosen below. When we include the basic identity

(3.15)

(tp(u) - 1p(w),Vt) = —^(ip(u)-ip(w),v)--(ip'(u)v,v)^ - ((ip'(u) - 1p'(w))wt,v) + ~(ip"(u)ut,v2) to the result and use (3.3)2, we find that there holds, for almost all t > 0,

^ jlHI2 + 2e(vt,v) + jf (1 - koa(x))\Vv\2dx + + e\\Vb(x)v\\2

+2(0(u) — ^(w), v) — (^'(u)v, v)

—2e|vt||2 + 2e / (1 — fc0a(x))|Vv|2dx — g^H^s)^ds Jo ./0 a

/»oo r

+2t / g(s) / a(x)V{i(s) • Vwteds + 2||v^(a0ut||2 ./0 ./o

—2(^'(u) — ^'(w))wt, v) + (u)ut, v2) + 2e(^(u) — ^(w), v) = 0. We now consider the functional defined by

V(i) := |K(i)||2 + 2e(vt(i),v(i)) + [ (1 - k0a(x))\Vv(t)\2dx + \\e\\2Mo + e\\ s/^)v(t)\\2

Jo

+2(^(u(t)) — ^(w(t)),v(t)) — (^'(u(t))v(t),v(t))

We now will show that, given U(t) = (u(t),ut(t),n*), W(t) = (w(t),wt(t),Z1) G H0 are uniformly bounded with respect to t > 0 by some R > 0, there are constants C1, C2 > 0, independent of t, in which for all V(t) = (v(t),vt(t),£*) G H0,

Ci|V(t)|H° < V(t) < C2|V(t)|H°. (3.16)

To this end we begin by estimating the following product with (2.1),

e

2t|(i>i,v)| < t||vt||2 +t||v||2 < t||vt||2 + T-||VV||2, (3.17)

A1

and

e\\VWM2 < e\\Vb\\l\\vf < -II6IUHWII2. (3.18)

Ai

Concerning the terms in the functional V that involve the nonlinear term using (3.2), (2.5), (2.6) and the embedding H 1(Q) ^ L6(Q), and also (2.19), there holds

IW>'(u)v,v)| < C (1 + ||Vu||2) ||VvHNvN < e||Vv||2 + C£(R)|M|2, (3.19)

where the constant 0 < C£ ~ e-1. From assumption (2.11) and (3.2)

2(^(u) - ^(w),v) > 2(p - tf)||v||2. (3.20)

Hence, for p = p(e) sufficiently large, the combination of (3.19) and (3.20) produces,

2(^(u) - ^(w),v) - (u)v,v) > 2(p - tf)||v||2 - e||Vv||2 - C£(R)||v||2 > -e||Vv||2. (3.21)

With (3.17), (3.18) and (3.21) we attain the lower bound for the functional V,

V > ( 4 - ^(2 + H&U - e ) IIV^II2 + (1 - e) |H|2 +

So for a sufficiently small e > 0 fixed (which also fixes the choice of p), there is m0 > 0 in which, for all t > 0, we have that

V(t) > mo||(v(t),vt(t),Ct)|Ho.

Now by the (local) Lipschitz continuity of f, the embedding Hi(Q) ^ L2(Q), the uniform bounds on u and w, and the Poincare inequality (2.1), it is easy to check that with (2.7) there holds

2(^(u) - ^(w),v) < 2||'0(u) - ^(w)||||v|| < C(R)||Vv||2. (3.22)

Also, using (3.2), (2.5), (2.6) and the bound (2.19), there also holds

IW>'(u)v,v)| < C(R)||Vv||2. (3.23)

Thus, with (3.22), (3.23) and referring to some of the above estimates, the right-hand side of (3.16) also follows.

Moving forward, we now work on (3.15). In light of the estimates

2W(u)-f(w))wt,v)\ < C( 1 + ||Vw|| + ||Vw||)|K|||M|2 < ±\\vf+C(R)\\wtfY, (3.24)

and

\(r(u)ut,v2)\ < C( 1 + HVulDIMIMI2 < ^|M|2 + C(R)|M|2V, (3.25)

(here the constants C(R) > 0 also depend on p > 0) we see that with (3.24), (3.25), as well as (2.4), (2.2) and (3.20), the differential identity (3.15) becomes

d

—V + t||vt||2 + 2t4i||Vv||2 + ¿H^ll^o +2t f g(s) j a{x)V^{s) ■ Vvdxds + 2\[/b{x)vt\\2 + (2t(p - tf) - i) ||v||2 (3-26)

J0 J Q p

< C(R)(|ut||2 + ||wt||2)V + 3eV,

where we also added 3e||vt||2 to both sides (observe, 3e||vt||2 < 3eV). We now seek a suitable control on the product

' a(x)V{i(s) ■ Vvdx o

2e / g(s) a(x)V{i(s) ■ Vvdxds < 2e / g(s)

J0 Jo J0

= 2eJ"g(s)\(e(s),v)vla\ds < SfeM^ll^o||< + ^l|Vt;||

ds

(3.27)

For sufficiently large ¡3 > 0, we may omit the positive terms 2\\sjb(x)vt\\2 + (2e(/3 — §) — l//i)|M|2 from the left-hand side of (3.26) so that it becomes, with (3.27),

+ e\\vt\\2 + e(2£o-^) ||Vt;||2 + (5 - 2^) < C(R) (|M|2 + |K||2 + 3e) V. (3-28)

For any e > 0 sufficiently small so that

2£0 - > 0 and 5 - 2y/e > 0,

we can find a constant m1 > 0, thanks to (3.16), such that (3.28) can be written as the following differential inequality, to hold for almost all t > 0,

d

—V + emiV < C{R) (|M|2 + |K||2 + 3e) V. (3.29)

Here we recall Proposition 2 and Lemma 3. Applying these to (3.29) yields, for all t > 0,

V(t) < V(0)eQ(R)e-mit/2, (3.30)

for some positive nondecreasing function Q(-). By virtue of (3.16) and the initial conditions provided in (3.3),

V(0) < C2(R)|(v(0),vt(0),e0)|H° < C2(R) (|VU0|2 + |M|2 + |M|M°) < Q(R).

Therefore (3.30) 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 H0. In order to establish this, we prove that the operators K(t) are uniformly bounded in K1 ^ H0.

Due to the nature of the proof of the following lemma, we also need to assign the past history for the term wt. Indeed, from below we need to consider the initial condition

c°(x, s) = — C°(x, s) = —wt(x, 0 — s).

However, since u = v + w, we can write

—ut(x, 0 — s) = —vt(x, 0 — s) — wt(x, 0 — s)

and hence assume that

vt(x, 0 — s) = ut(x, 0 — s) = — n°(x,s) and wt(x, 0 — s) = 0. (3.31)

2

Lemma 5. Assume the hypotheses of Theorem 3 hold. For each R > 0 and for all U0 = (u0,u1,n0) € H0 such that ||U0||%o < R, there holds for all t > 0

||dtK(t)U0|Ho = ||Vwt(t)||2 + ||wtt(t)|2 + ||ClHMo < Q(R) (3.32)

for some positive nondecreasing function Q( ).

Proof. For all x € Q and t,s € R+, set H(x,t) := wt(x,t) and Xt := (|(s). Differentiating problem W with respect to t yields the system

Htt-div[(1 - fc0fl(x))VH]- / g(s)div[a(x)VXt(s)]ds+6(x)Ht+^/(w)H = put in Q x R+,

0

Xf = -X* + Ht in Q x R+,

H(x,t)= wt(x,t)=0, Xt(x,s)= Ct(x,s) onr x R+,

H(x, 0) = wt(x, 0) = 0, Ht(x, 0) = wtt(x, 0) = -f (0) - ui (from (3.4)) at Q x {0},

k Xt(x, 0) = wt(x,t) - wt(x,t - 0) = 0, X0(x, s) = 0 (see (3.31)) at Q x {0}.

(3.33)

Multiply equation (3.33) 1 by Ht + eH for some e > 0 to be chosen below. To this result we apply the identities

i d 1

№'(w)H,Ht) = --^(w)H,H) --{^{w)wt,H2),

<

and (here we rely on (3.33)2)

1 d

'\\A"\\M° +

/0

f f 1 d d

/ g{8) a{x)VXtWHt{t)dxd8 = ~\\Xt\\2M0+ g^-J^^ds

•J0 J Q J 0

1 d_ 2 dt

|X¡Mo -/ g/(s)yXt(s)yVids

so that together we find d

dt-{nHt\\2 + 2e(Ht,H) + J {I- k0a(x))\VH\2dx + WX*^ + (iP'(w)H,H)}

-2£\\Ht\\2+2£\\y/b^)Ht\\2+2£{b{x)Ht,H)+2£ [ {l-k0a{x))\VH\2dx+2e{tp'{w)H, H)

JQ

-2 / g^s^X4(s)|ßids + 2e / g(s) / a(x)VX4(s) -VH(t)dxds Jo a Jo jQ

= «(w)wt, H2) + 2ß (ut, Ht) + 2ße(ut, H). Next we recall (2.2) and find

-2 / g'(s)yx4(s)yVids > 20||Xt¡Mo, (3.35)

0

and

2t / g(s) / a,(x)VXt(s) • VH(t)dxds > -SWX*^ - — \\VH\\2, (3.36)

Jo ./q o

where the last inequality follows from (2.3). For all e > 0 and t > 0, define the functional I(i):=||Ht(t)||2+2e(Ht(t),H(t))+ / (1-Aoa(x))|VH(t)|2dx+HX4¡Mo+(^'(w)H(t),H(t)). (3.37)

Q

0

Thanks to (2.4) and since > 0, there is a constant C > 0, sufficiently small, so that

C (||Ht(t)||2 + l0||VH(t)||2 + ||X4) < i(t). At this point we can write (3.34)-(3.36) with (3.37) as

d _ _ .. .,9 11 f~r~. r" .. o ~ /~t / \ -r-r -r-r\ (^ /i &

^I-2£\\Ht\\2 + 2£\\y/^Ht\\2 + 2£(b(x)Ht,H) + (^e0-j^ \\VH\f

' ^ 1 ^ ^ 11 ■ /i^T^A TJ 112 IMI ./;ini , | y n — i 11 v _JL _t_ 11

S J (3.38)

+S||X||M° + 2e(^(w)H, H) < 2«(w)wt, H2) + 2£(u, Ht) + 2£e(u, H).

Next, let us rely on the uniform bounds (2.19) and (3.14) to estimate the products on the right-hand side

2|«(w)wt, H2)| < 2||<(w)wtH21|1 < 2|<(w)wt||3/2||H||6 < 2||<(w)||6||wt||||H||6

< C(R)||wt||||VH||2 < C(R)||wtp,

(3.39)

2£|(u,Ht) + e(ut,H)| < C(R)||Ht|| + C(R)||VH|| < Ce(R) + e||Ht||2 + e2||VH||2, (3.40) where C£ ~ e-1 A e-2. Also, we know

2e(^'(w)H, H) > 2e2(^ — tf)||H||2 > 0. (3.41) Thus, combining (3.38)-(3.41) yields

jl - 3s\\Ht\\2 + 2e\\^b{x)Ht\\2 + t- (24 - e Q + l)) ||2 + 5\\X< fM0

< C(R)||wt||I + Ce(R).

Since 4t||iit||2 < 4tl, adding this to (3.42) makes the differential inequality (we also omit 2e\\^b{x)Ht\\2)

jl + e\\Ht\\2 + e ^2£0 -e(] + 1)) HVjFill2 + 6WxtW2M° < °(R) (IKII + e)1 + We now find that for any e > 0 small so that

2£0 ~ £ Q + l) > 0,

then

d

-I + el < C(R) (|K|| +1) I + C£(R) dt

to which we now apply Proposition 3 and the bounds (3.8) and (3.9) to conclude that, for all t > 0, there holds

I(t) < C(R)I(0)e-£i/2 + C£(R).

Moreover, with (3.37) and the initial conditions in (3.33) we find that there is a constant C > 0 (with e > 0 now fixed) in which

||Ht(t)||2 + ||VH(t)||2 + ||X4< C(R).

This establishes (3.32) and completes the proof. □

We derive the immediate consequence of (3.4) and (3.32).

Corollary 3. Under the assumptions of Lemma 5, there holds for all t > 0,

IIZSIIm < Q(R). (3.43)

Before we continue, we derive a further estimate for Z

Lemma 6. Under the assumptions of Lemma 5, there holds for all t > 0,

IIVZILi(R+ ;L2(n)) < C. (3.44)

Proof. Formally multiplying (3.4)2 in L^(R+; L2(Q)) by — AZ4(s) and estimating the result yields the differential inequality

d fœ d

^llVZll^R+^in)) = -jo 5(s)^llVZi(s)||2ds + (VWi,VZi)L2(K+;L2(n)) = / g (s)|VZi(s)!2ds + (Vwt, VZ4)^ (R+;L2(n))

(3.45)

2 S

< -à J gmvfWfds + ^||VWi||2 + - ||VZi||!2(R+;L2(n)) S 2

Hence, applying the bound (3.32) to (3.45), we find the differential inequality which holds for almost all t > 0

^l|VZÎ|||2(R+;L2(n)) + - l|VZÎ|||2(R+;L2(n)) < CS

where 0 < C ~ S-1. Applying a straight-forward Gronwall inequality and the initial conditions in (3.4) produces the desired bound (3.44). This concludes the proof. □

Lemma 7. Under the assumptions of Lemma 5, the following holds for all t > 0,

||K(t)U0||ki < Q(R), (3.46)

for some positive nondecreasing function Q( ). Furthermore, the operators K(t) are uniformly compact in H0.

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 (3.4) as

wtt + k0Va(x) ■ Vw + (1 — k0a(x))(—A)w

f™ . f™ . (3.47)

' g(s)Va(x) -VZ4(s)ds + g(s)a(x)(—A)Z4(s)ds + b(x)wt + ^(w) = £u.

0

Next, Using the relative displacement history definition of the memory space term

Z4(s) := w(x,t) — w(x,t — s),

we rewrite the integral

t>™ t>™

/ g(s)a(x)(—A)Z4 (s)ds = k0a(x)(—A)w — g(s)a(x)(—A)w(t — s)ds. (3.48)

00

Combining (3.47) and (3.48) shows (3.4) takes the useful alternate form

r ro

wtt - Aw - / g(s)a(x)(-A)w(t - s)ds + b(x)wt + ^(w)

J0

/•ro

+k0Va(x) ■ Vw - g(s)Va(x) ■ V(t(s)ds = pu.

0

We now report six identities that will be used below:

d

(wtt,(-A)w) = —(Vwt,Vw) - ||Vwt||2, dt

r ro

- / 9{s){a{x){-A) w(t-s) , (-A)wt(t))ds

=w(t)-<:t(s)

/•ro r ro

t

(3.49)

/ g(s)(a(x)(-A)w(t), (-A)wt(t))ds + / g(s)(a(x)(-A)Zt(s), (-A) wt(t) )ds (3.50) /0 J 0 —'

=Ctt(s)+CS (s)

k0 d .. ll2 1 d .. Atll2 1 [ro , x d

1 d 1 ro d f ro

- / fif(s)(a(a;)(-A) w(t - s) , (-A)w(t))ds

=w{t)-ct{8) (3.51)

ro

= -fc0|w|22 + J g(s)(a(x)(-A)(t(s), (-A)w(t))ds, d

(6(.T)Wi, (-A)Wi) = jt{b{x)wt, (~A)w) - (b(x)wtt, (~A)w), (3.52)

d

k0(Va,(x) ■ Vw, (~A)wt) = —k0(Va(x) ■ Vw, (~A)w) - k0(Va,(x) ■ Vwt, (~A)w), (3.53)

and

f ro

g(s)(Va(x) ■ VZt(s), (-A)wt(t))ds

J0

d

roro

= "di i 9(s)(Va(x)-V<;t(s),(-A)w(t))ds + J g(s)(Va(x) ■V(tt(s),(-A)w(t))ds.

(3.54)

Next we multiply (3.49) in L2(Q) by (-A)wt + (-A)w to obtain, in light of (3.50)-(3.54), the differential identity

|{||VWt||2 + 2(Vwt,Vw) + ||Aw||2 - fco|Mly2 + 11(1^,1

-(||VWir + 2(Vwt,Vw) + ||Aw|| - fcolMlys + HCllti

f ro

+2(b(x)wt, (-A)w) + 2k0(Va(x) ■ Vw, (-A)w) - 2 g(s)(Va(x) ■ VZt(s), (-A)w

J0

i'ro d

—2||Vwt||2 + 2||Aw||2 + / ~~ ||y2ds 2fco||y2 rro 70 " " (3.55)

+2 / g(s)(a(x)(-A)Zt(s), (-A)w(t))ds - 2(6(x)wtt, (-A)w) + 2(b(x)wt, (-A)w)

0

+2(0'(w)Vw, Vwt) + 2(0(w), (-A)w) - 2k0(Va(x) ■ Vwt, (-A)w) + 2k0(Va(x) ■ Vw, (-A)w)

roro

+2 / g(s)(Va(x) ■ VZt(s), (-A)w(t))ds - 2 g(s)(Va(x) ■ VZt(s), (-A)w(t))ds

00

= 2p(Vu, Vwt) + 2p(u, (-A)w).

We now seek a constant m2 > 0 sufficiently small so that we can write the above differential identity in the following form

d

— $ + cm2$ < Q(R) (3.56)

where

$(t) := ||Vwt(t)||2 +2(Vwt(t), Vw(t)) + ||Aw(t)||2 — k0|w(t)|2a + ||Z1M

+2(b(x)wt(t), (—A)w(t)) + 2k0(Va(x) ■ Vw(t), (—A)w(t)) (3.57)

—2 g(s)(Va(x) ■VZt(s), (—A)w(t))ds.

0

The important lower bound holds

$ > C1(|Aw|2 + ||Vwt||2 + ||Z4|Mi) — C2(R) (3.58)

for some constants C1, C2(R) > 0, and essentially follows from some basic estimates, the bounds (2.19), (3.14), (3.32), (3.44), the Poincare inequality (2.1) and with the assumptions on the functions a and b. Indeed, we estimate, for all e > 0,

2\(Vwt, Vw)| < t||Vwt||2 + -||Vw||2 < t||Vwt||2 + Ce{R), (3.59)

e

—fc0||w||?,2 = — k0 / a(x)|Aw|2dx > —fc0||a||™||Aw||2, (3.60)

a Jo

2\(b(x)wt,(-A)w)\ <-\\b(x)wt\\ + e\\Aw\\2 < Ce(R) + e\\Aw\\2, (3.61)

e

k2

2ko\(Va,(x) • Vw, (—A)w)| < — ||Va(a;) • Vwll2 + t||Aw||2 < C£(R) + t||Aw||2, (3.62)

e

and

2 g(s)|(Va(x) ■VZt(s), (—A)w(t))|ds

■J0

< l°°g(s) (;||Va(.r) • VC^s)!!2 +ti|Aw(i)||2^

r-™

< - r^)HVa||Ll|VCi(S)||2dS + t- g(s)\\Aw(t)\\2ds (3"63)

e 0 0

< ;l|Va||Ll|VCi|||2(R+;L2(n)) + sk0\\Aw\\2 < C£(R)+ ek0||Aw||2. Applying (3.59)-(3.63) to (3.57) gives us the lower bound for all e > 0,

$ > (1 — e)||Vwt||2 + (I0 — (2 + k0)e)|Aw|2 + HZlM — Ce(R).

For any fixed 0 < e < min{1,l0/(2 + k0)}, we obtain (3.58). Returning to the aim of (3.56), we first add

3||Vwt||2 + 2(Vwt, Vw)

to both sides of (3.55), and also insert

™ d ™ ™

j( g(s)-||Ci(s)||2a2ds = -yo t/WU'WW^dsZS^ gWU'WW^ds =

Putting these together and using the second inequality in (2.2), (3.55) becomes the differential inequality

d

+ HVwtll2 + 2(Vwt, Vw) + 2||Aw||2 - 2k0\\w\\2vi + ¿HCl^i

ro

+2(b(x)wt, (-A)w) + 2k0(Va(x) ■ Vw, (-A)w) - 2 g(s)(Va(x) ■ VZt(s), (-A)w(t))ds

0

< 3|Vwt||2 + 2(Vwt, Vw) + 2(b(x)wtt, (-A)w) + 2*0(Va(x) ■ Vwt, (-A)w) (3.64)

roro

-2 / g(s)(a(x)(-A)Zt(s), (-A)w(t))ds - 2 g(s)(Va(x) ■ VZt(s), (-A)w(t))ds

00

-2(0'(w)Vw, Vwt) - 2(0(w), (-A)w) + 2p(Vu, Vwt) + 2p(u, (-A)w).

(We should mention that the final bound of (3.32) is now realized to control the VZt term appearing on the right-hand side.) Next we employ some basic inequalities, the assumptions on a and b, the assumptions (2.5)-(2.7), the bounds (2.19), (3.6) and (3.32), and finally even the continuous embedding V2 ^ H(Q) of (H1r) to control the right-hand side of (3.64) with the estimates

3||Vwt||2 + 2(Vwt, Vw) - 2(0'(w)Vw, Vwt) + 2p(Vu, Vwt) < C(R), (3.65)

2(&(.rMi, (-A)w) < C(R) + ^||Aw||2, (3.66)

2k0(Va(x) ■ Vwt, (-A)w) < C(E) + ^||Aw||2, (3.67)

roro

-2/ g(s)(a(x)(-A)Zt(s), (-A)w(t))ds = -2/ g(s)(Zt(s), w(t))v|ds ./0 ./0 fro 2

<2j fifWIIC^IIv^K^llv^^ ^IICl^i +

roro

-2 / g(s)(Va(x) ■ VZt(s), (-A)w(t))ds < 2 g(s)||Va(x) ■ VZtt(s)||||Aw(t)||ds

.70 ./0

ro

< 2 / g(s)||Va||ro||VZt(s)|||Aw(t)||ds J 0

rro r ro

e 0 0

= ^l|Va||LllCtllii(K+;iioi(n)) + tfco||Aw||2 < C£(R)|ZlMo + ek0||Aw||2 < C£(R) + e^0||Aw||2

and

Hence, (3.65)-(3.71) show the right-hand side of (3.64) is controlled with, for all e > 0,

2

Ce(E) + (1 + t-fc0)||AW||2 + ^fcolMlv« +

(3.68)

< -HVa||L Z^g(s)HVC^s)||2ds + t Z"00g(s)||Aw(i)||2ds (3"69)

-2№(w),(-A)w) <C(E) + ±||A™||2, (3.70)

(~A)w) < C(R) + ^||Aw||2. (3.71)

Now fixing 0 < e < min{1/fc0, and setting

m2 = ni2(ko, 5) := minil — sko, $ — t} > 0 and c = c(ko) := 2 ( 1 H—]

V eko/

we arrive at the desired estimate (3.56).

So now we integrate the linear differential inequality (3.56) and apply $(0) = 0. Thus,

||Aw(t)||2 + ||Vwt(t)||2 + HdMi < Q(R), (3.72)

for some positive nondecreasing function Q^(■) ~ ¿-1. By combining (3.72), (3.32) and the Poincare inequality (2.1), we see that, with the H2-elliptic regularity estimate (2.12), we have with uniform bounds

w(t) € H2(Q) and wt(t) € H1 (Q) V t> 0. Additionally, collecting the bounds (3.72) and (3.43) establishes that, for all t > 0,

HdM + MClMMo < Q*(R). (3.73)

Lastly, to show (3.46) holds we need to control the last term of the norm (2.16). With the bound (2.19), we apply the conclusion of Lemma 1 here in the form

supTT(t; z*) < 2 (t + 2) e-<5i sup tT(t; Zo) + C(R) < C(R). (3.74)

T>1 T>1

where the last inequality follows from the null initial condition given in (3.4)4. Together, the estimates (3.72)-(3.74) show that (3.46) holds. This completes the proof. □

We now prove the main theorem.

Proof. [Proof of Theorem 3.] Define the subset C of K1 by

C := {U = (u,v,n) GK1 : ||U||^i < Q(R)},

where Q(R) > 0 is the function from Lemma 7, and R > 0 is such that ||U0||%o < R. Let now U0 = (u0,u1,no) G B (the bounded absorbing set of Corollary 2 endowed with the topology of H0). Then, for all t > 0 and for all U0 G B, S(t)U0 = Z(t)U0 + K(t)U0, where Z(t) is uniformly and exponentially decaying to zero by Lemma 4, and, by Lemma 7, K(t) is uniformly bounded in K1. In particular, there holds

distHo(S(t)B, C) < Q(R)e-wi. The proof is finished. □

4. Conclusions

We have show that the global attractors associated with a wave equation with degenerate viscoelastic dissipation in the form of degenerate memory possesses more regularity than previously obtained in [4]. This is established under reasonable assumptions by showing the existence of a compact attracting set to which global attractor resides. Moreover, the global attractor consists of regular solutions. The main difficulties encountered here are due to the degeneracy of the dissipation term as well as obtaining compactness for the memory term.

A. Appendix

We include two frequently used Gronwall-type inequalities that are important to this paper. The first can be found in [26, Lemma 5]; the second in [22, Lemma 2.2].

Proposition 2. Let A : R+ — R+ be an absolutely continuous function satisfying

d

—A(t) + 2r?A(i) < h(t)A(t) + k, dt

where n > 0, k > 0 and fs h(r)dr < n(t - s) + m, for all t > s > 0 and .some m > 0. Then, all t > 0,

A(i) < A(0)eme""i +

Proposition 3. Let $ : [0, to) ^ [0, to) be an absolutely continuous function such that, for some e > 0,

d

-m + 2em<mm + Ht)

for almost every t € [0, to), where f and h are functions on [0, to) such that

rt ft+i

/ |f (t)|dr < a(1 + (t - s)A), sup/ |h(r)|dr < p

J s t>0 Jt

for some a, p > 0 and A € [0,1). Then

$(t) < Y$(0)e-£t + K every t € [0, to), for some 7 = 7(f, e, A) > 1 and K = K(e, A, f, h) > 0.

Acknowledgments

The author is indebted to the anonymous referees for their careful reading of the manuscript and for their helpful comments and suggestions — in particular, for the reference [1].

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