COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume, Number, January 25 pp. 1 16 WEAKLY DISSIPATIVE SEMILINEAR EQUATIONS OF VISCOELASTICITY Monica Conti Dipartimento i Matematica F.Brioschi Politecnico i Milano I-2133 Milano, Italy Vittorino Pata Dipartimento i Matematica F.Brioschi Politecnico i Milano I-2133 Milano, Italy (Communicate by Alain Miranville) Abstract. We consier an integro-partial ifferential equation of hyperbolic type with a cubic nonlinearity, in which no issipation mechanism is present, except for the convolution term accounting for the past memory of the variable. Setting the equation in the history space framework, we prove the existence of a regular global attractor. 1. The Equation. Let Ω R 3 be a boune omain with smooth bounary Ω. For u : Ω R R, we consier the hyperbolic equation with memory on the time-interval R + = (, ) u tt + αu t k() u k (s) u(t s)s + g(u) = f, (1) arising in the theory of isothermal viscoelasticity (cf. [1, 22]). Here, α, g : R R is a nonlinear term of (at most) cubic growth satisfying some issipativity conitions, f : Ω R is the external force, whereas the memory kernel k is a convex ecreasing smooth function such that k() > k( ) >. This equation is supplemente with the initial an bounary conitions u(t) = u (t), t, u t () = u 1, (2) u(t) Ω =, t R, where the values u (t) (for t ) an u 1 are prescribe ata. The terms contributing to issipation in the above equation are αu t (which is to be consiere as a ynamical friction) an the convolution integral. Inee, performing an integration by parts, it is easily seen that (1) can be transforme into [ ] u tt + αu t k( ) u k(s) k( ) ut (t s)s + g(u) = f. (3) 2 Mathematics Subject Classification. 35B4, 35L7, 37L45, 45K5, 74D99. Key wors an phrases. Hyperbolic equations with memory, ynamical systems, Lyapunov functionals, graient systems, global attractors. 1
2 M. CONTI AND V. PATA Thus, in the limiting situation when k(s) k( ) is the Dirac mass at zero (possibly multiplie by a positive constant β), we obtain u tt + αu t k( ) u β u t + g(u) = f, that is, the semilinear wave equation with weak amping an strong amping. This equation is well-known to generate a ynamical system on H 1 (Ω) L 2 (Ω) possessing a global attractor of optimal regularity, provie that either α or β (or both) are strictly positive (see [1, 2, 11, 12, 15, 17, 2]). Setting µ(s) = k (s) (µ is positive an summable) an η t (s) = u(t) u(t s), equation (1) may be more conveniently rewritten as the system of equations on the time-interval R + (cf. [8]) u tt + αu t k( ) u µ(s) η(s)s + g(u) = f, (4) η t = η s + u t. Accoringly, the initial an bounary conitions turn into u() = u, u t () = u 1, η = η, where u = u () an η (s) = u () u ( s), an, now for all t, u(t) Ω =, η t Ω =, η t () =. The complete equivalence between the two formulations (1)-(2) an (4)-(6) is iscusse in great etail in the review paper [14]. 2. Preliminary Discussion. As we will see in the next section, problem (4)-(6) generates a issipative ynamical system on a suitable phase-space, usually calle history space (cf. [14]). It is then of some interest to investigate the asymptotic properties of such a system, to see if the longterm ynamics is capture by (possibly small) subsets of the phase-space, such as boune absorbing or attracting sets, or even global or exponential attractors. Of course, the stronger is the issipation, the more are the chances that a satisfactory picture occurs. As pointe out before, here there are two terms that provie a favorable contribution: αu t an µ(s) η(s)s. As usual, goo inications on the longterm behavior are given by the associate linear homogeneous system (i.e., when g an f both vanish), which has to be exponentially stable in orer to have some hopes for the full system to possess (at least) a boune absorbing set. This exponential stability hols true, provie that µ has an exponential ecay at infinity. Most interesting, it hols true even if α =, namely, when all the issipation is carrie out by the memory. This result, prove in [9] exploiting techniques of linear semigroup theory, is far from being trivial, for in this case the issipation is extremely weak, an the memory term is responsible of the uniform ecay to zero of the whole associate energy. However, these methos, strictly relate to the linear nature of the problem, cannot be exporte to successfully attack the semilinear nonhomogeneous case. Clearly, if (5) (6)
SEMILINEAR EQUATIONS OF VISCOELASTICITY 3 α >, things are much easier, an the exponential ecay can be prove via (quite) stanar energy estimates. Going back to the full problem, existence of global an exponential attractors, along with regularity results, have been provie for α > in [7, 5, 21]. A partial attempt to inclue the case α = has been mae in [13], where the existence of a global attractor (without any further regularity) is prove, but uner quite strong assumptions (such as global Lipschitz continuity) on the nonlinearity g. The aim of this paper is to complete the analysis of the case α =, showing the existence of a global attractor of optimal regularity for a cubic nonlinearity. We want to emphasize straight away the main ifficulty encountere here. The preliminary step to emonstrate the existence of global attractors for ynamical systems is the existence of boune absorbing sets (usually obtaine by means of energy estimates). For some reason, in the present case this seems out of reach. To overcome this (apparently) insurmountable obstacle, we shall make use of a general theorem (cf. [16, 18]), that allows to prove the existence of a global attractor without appealing to the existence of a boune absorbing set, which is eventually obtaine as a byprouct. The result applies to the so-calle graient systems, that is, ynamical systems possessing a global Lyapunov functional. Let us briefly escribe the plan of this work. - In Section 3 we properly formulate the problem as a ynamical system on a suitable phase-space, an we prove the existence of a global Lyapunov functional. - In Section 4 we present the main result, whose proof is carrie out in Sections 5,6,7. - In Section 8 we iscuss some possible evelopments, whereas in Section 9 we reformulate our result in terms of trajectory attractors. - In the Appenix we recall the abstract theorem on the existence of global attractors for graient systems. Notation. We enote by an, the norm an the inner prouct on L 2 (Ω). Naming A = : D(A) = H 1 (Ω) H 2 (Ω) L 2 (Ω) L 2 (Ω), we consier, for σ R, the Hilbert spaces H σ = D(A σ/2 ), enowe with the stanar inner proucts. Then, we introuce the L 2 -weighte Hilbert spaces M σ = L 2 µ(r +, H σ+1 ), an the infinitesimal generator of the right-translation semigroup on M T = s : D(T ) = { η(s) M : η s M, η() = } M M, where s is the istributional erivative with respect to the internal variable s. Finally, we efine the prouct Hilbert spaces H σ = H σ+1 H σ M σ. Throughout the paper, c will enote a generic constant, epening only on the structural parameters of the system uner consieration (unless otherwise specifie). Also, we shall often make use without explicit mention of the Young an the Höler inequalities, as well as of the usual Sobolev embeings. We conclue the section reporting a moifie form of the Gronwall lemma, that we write in full generality for further references.
4 M. CONTI AND V. PATA Lemma 2.1. Let ψ : [, ) R be an absolutely continuous function, which fulfills for some ε > an almost every t the ifferential inequality t ψ(t) + 2εψ(t) h 1(t)ψ(t) + h 2 (t), where t τ h t+1 1(y)y m 1 +ε(t τ), for all τ [, t], an sup t h t 2 (y) y m 2, for some constants m 1, m 2. Then there exist M 1, M 2 such that ψ(t) M 1 ψ() e εt + M 2, t. Moreover, if m 2 = (that is, if h 2 ), it follows that M 2 =. Proof. Fix t >. For τ [, t], we set ω(τ) = t Then, by the Gronwall lemma, ψ(t) ψ()e ω() + t The inequality (cf. [19]) yiels the esire result. τ h 1 (y)y 2ε(t τ) m 1 ε(t τ). t e ω(τ) h 2 (τ)τ e m1 ψ() e εt + e m1 e εt e ετ h 2 (τ) τ. t e ετ h 2 (τ) τ m 2e ε e εt 1 e ε 3. The Associate Graient System. We assume hereafter the following set of hypotheses. Let f H be inepenent of time, an let g C 2 (R) satisfy the issipation an the growth conitions g (y) c(1 + y ), y R, (7) g(y) lim inf > λ 1, (8) y y where λ 1 is the first eigenvalue of A. Concerning the kernel µ(s) = k (s), we take µ W 1,1 (R + ), µ, such that µ (s) + δµ(s), s R +, for some δ >. (9) To avoi the presence of unnecessary constants, we assume the normalization conition µ(s)s = 1, an we put µ() = 1, which can always be one by rescaling µ, an changing δ accoringly. By the above assumptions, µ is ecreasing (strictly, whenever µ(s) > ) an µ(s) e δs, s R +. Then, setting for simplicity k( ) = 1, problem (4)-(6) for α = can be translate into the evolution equation in H u tt + Au + µ(s)aη(s)s + g(u) = f, t R +, η t = T η + u t, t R +, (1) (u(), u t (), η ) = (u, u 1, η ).
SEMILINEAR EQUATIONS OF VISCOELASTICITY 5 This equation (cf. [7, 21] an references therein) generates a C -semigroup or ynamical system S(t) on the phase-space H. In particular, the thir component η of the solution has the explicit representation formula { η t u(t) u(t s), < s t, (s) = (11) η (s t) + u(t) u, s > t. Note that the set S of stationary points of S(t) is mae of vectors (u,, ) H, with u solution to the elliptic problem Au + g(u ) = f. Appealing to (8), it is then clear that S is boune in H. Accoring to Definition A.1 in the Appenix, we have Proposition 3.1. S(t) is a graient system on H. Proof. We efine the function Φ C(H, R) as Φ(u, v, η) = E(u, v, η) + G(u) f, u, where E(u, v, η) = 1 ( A 1/2 u 2 + v 2 + η 2 ) M 2 is the energy functional, an u(x) G(u) = g(y)yx. On account of (7)-(8), it is reaily seen that Ω c E c Φ ce 2 + c, for some c >, possibly very small, which epens only on the value of the limit in (8). We now show that Φ is ecreasing along the trajectories of S(t). Inee, multiplying the first equation of (1) by u t in H an the secon by η in M, we get t Φ = 1 µ (s) A 1/2 η(s) 2 s. (12) 2 Finally, if Φ is constant along a trajectory of S(t), using (9) we learn that η t M = for all t. By force of (11), we conclue that (u(t), u t (t), η t ) = (u,, ) S. Hence, Φ is a Lyapunov functional. 4. The Global Attractor. Our main result reas as follows. Theorem 4.1. The ynamical system S(t) on H generate by (1) possesses a (unique) connecte global attractor A. Moreover, A is containe an boune in H 1. The existence of the global attractor A will be obtaine by applying Theorem A.3. Inee, in light of Proposition 3.1 an the bouneness of the set S, it will suffice to fin, in corresponence to any given boune set B H, a ecomposition S(t) = L(t) + N(t) as in Remark A.5. The regularity of A will be prove via a bootstrap argument envisage in [15].
6 M. CONTI AND V. PATA In orer to accomplish this program, we write (cf. [15]) g = g + g 1, where g, g 1 C 2 (R) fulfill g (y) c(1 + y ), y R, (13) g () =, (14) g (y)y, y R, (15) g 1(y) c, y R. (16) Then, we fix a boune set B H an, for z = (u, u 1, η ) B, we split the solution S(t)z = (u(t), u t (t), η t ) into the sum L(t)z + N(t)z, where L(t)z = (v(t), v t (t), ξ t ) an N(t)z = (w(t), w t (t), ζ t ) solve the problems v tt + Av + µ(s)aξ(s)s + g (v) =, ξ t = T ξ + v t, (v(), v t (), ξ ) = z, an w tt + Aw + µ(s)aζ(s)s + g(u) g (v) = f, ζ t = T ζ + w t, (w(), w t (), ζ ) =. The next sections will be evote to prove the uniform ecay of L(t) an the smoothing property of N(t). 5. Some Preliminary Lemmata. We begin to establish some estimates that shall be neee in the course of the investigation. Here, as in the sequel, we shall perform formal multiplications, which are justifie within a proper regularization scheme. Lemma 5.1. Let σ [, 1] be given, an let z H σ such that z H R for some R. Assuming that S(t)z H σ for all t >, there exist two constants m, m R > (the first inepenent of R) such that the inequality t u t, η M σ 1 ε A (σ+1)/2 u 2 1 2 Aσ/2 u t 2 + m R ε η 2 M σ m µ (s) A (σ+1)/2 η(s) 2 s + ε hols for all ε (, 1). The same estimate is vali replacing (u, u t, η) with (w, w t, ζ) an, for σ =, with (v, v t, ξ). In this latter case, without the last term ε appearing in the right-han sie. Proof. We write the proof for (u, u t, η) (the other two cases are treate similarly). Using the secon equation of (1), we have t u t, η M σ 1 = A σ/2 u t 2 u t, T η M σ 1 u tt, η M σ 1. (17) (18)
SEMILINEAR EQUATIONS OF VISCOELASTICITY 7 Integrating by parts with respect to s, in light of the ecay of µ an of the equality η() =, we obtain u t, T η M σ 1 = ( A σ/2 u t µ (s) A σ/2 u t, A σ/2 η(s) s 1 2 Aσ/2 u t 2 m ) 1/2 µ (s) A σ/2 η(s) 2 s µ (s) A (σ+1)/2 η(s) 2 s, for m = 1 2λ 1. Concerning the other term, the first equation of (1) yiels u tt, η M σ 1 = u, η M σ + µ(s)a (σ+1)/2 η(s)s 2 + g(u) f, η M σ 1. But an u, η M σ A (σ+1)/2 u A (σ+1)/2 u η M σ ε A (σ+1)/2 u 2 + 1 4ε η 2 M σ, µ(s) A (σ+1)/2 η(s) s µ(s)a (σ+1)/2 η(s)s 2 ( 2 µ(s) A η(s) s) (σ+1)/2 η 2 M σ. Finally, since σ 1 an A 1/2 u is boune (with a boun epening on R), g(u) f, η M σ 1 g(u) f µ(s) A σ η(s) s c R ε η 2 M σ + ε, for some c R >. Note that, to prove the analogous result for (v, v t, ξ), this term becomes g (v), ξ M 1, which, using the fact that g () =, is controlle by g (v), ξ M 1 c R A 1/2 v ξ M ε A 1/2 v 2 + c R ε ξ 2 M. Setting m R = c R + 5 4, we conclue that u tt, η M σ 1 ε A (σ+1)/2 u 2 + m R ε η 2 M σ + ε. Collecting the two above estimates, we reach the esire conclusion. Lemma 5.2. Assume that z H R for some R. Then, for every ε >, there exists Mε,R > such that t for every t τ. τ u t (y) 2 y ε(t τ) + M ε,r, Proof. In this proof, the generic constant c may epen on R. Define I(t) = 2Φ(t) 2ε 2 u t (t), η t M 1,
8 M. CONTI AND V. PATA where Φ is the Lyapunov functional. Notice that I is boune (with a boun epening on R). Then, collecting (12) an Lemma 5.1, an exploiting (9), t I + ε2 u t 2 ( δ 2mδε 2 2m R ε ) η 2 M + cε3 cε 3, for ε small enough. In that case, an integration on (τ, t) yiels the require inequality, which then clearly hols for every ε >. Lemma 5.3. Let σ [, 1] be given, an assume that z H σ R for some R. Then there exists CR σ such that S(t)z H σ C σ R, t. Moreover, for all ε > there exists Mε,R σ such that for every t τ. t τ A σ/2 u t (y) 2 y ε(t τ) + M σ ε,r, Proof. The result for σ = is alreay known. Therefore, consier σ (, 1]. In the following, the generic constant c may epen on R (in fact, c will epen only on the H -norm of z). For ε (, 1) an ν, we efine the functional J σ (t) = E σ (t) + F σ (t) ε 2 u t (t), η t M σ 1 + νε 2 A σ/2 u t (t), A σ/2 u(t), where E σ (t) = 1 2 ( A (σ+1)/2 u(t) 2 + A σ/2 u t (t) 2 + η t 2 M σ ) is the higher-orer energy functional, an It is clear that, for ε small enough, F σ (t) = g(u(t)), A σ u(t) f, A σ u(t). 1 2 E σ(t) c J σ (t) 2E σ (t) + c. (19) Also, using the first equation of (1), an recalling the embeings H (2+σ)/2 L 6/(1=σ) (Ω) an H 1 σ L 6/(1+2σ) (Ω), it is straightforwar to check that ( ) 1 Eσ + F σ t 2 = g (u)u t, A σ u µ (s) A (σ+1)/2 η(s) 2 s an c ( 1 + u 2 L 6/(1 σ) ) ut A σ u L 6/(1+2σ) c ( 1 + A 1/2 u A (1+σ)/2 u ) u t A (1+σ)/2 u c u t + c u t E σ, t Aσ/2 u t, A σ/2 u A σ/2 u t 2 1 2 A(σ+1)/2 u 2 + η 2 M σ + c.
SEMILINEAR EQUATIONS OF VISCOELASTICITY 9 Exploiting now (9) an Lemma 5.1, an collecting the two above inequalities, we en up with t J σ + ε 2( ν ) 2 ε A (σ+1)/2 u 2 + ε 2( 1 ) 2 ν A σ/2 u t 2 (2) ( δ ) + 2 mδε2 νε 2 m R ε η 2 M σ c u t + c u t E σ + cνε 2 + ε 3. Choosing now ν = 1 4 an ε small enough, in light of (19), the above inequality turns into t J σ + ε J σ c + c u t + c u t J σ, for some ε >. Then, from Lemma 5.2 an the generalize Gronwall Lemma 2.1 we get the first part of the thesis. To obtain the integral control, we consier again (2) with ν = an ε sufficiently small. Notice that now we know that E σ c. Therefore we are le to t J σ + ε2 2 Aσ/2 u t 2 c u t + ε 3. An integration on (τ, t), together with a further application of Lemma 5.2, complete the argument. 6. Exponential Decay of L(t). We now prove that the C -semigroup L(t) on H has an exponential ecay. Lemma 6.1. There exists κ > an an increasing positive function Γ such that whenever z H R. L(t)z H Γ(R)e κt, t, Proof. In this proof, the generic constant c may epen on R. Let us consier the functional v(x,t) J(t) = E(v(t), v t (t), ξ t ) + g (y)yx ε 2 v t (t), ξ t M 1 + ε2 4 v t(t), v(t). By virtue of (15), if ε is small enough there hols Ω 1 2 E(v(t), v t(t), ξ t ) J(t) ce(v(t), v t (t), ξ t ). Thus, on account of Lemma 5.1, an provie that ε is small enough, with calculations similar to those in the proof of Lemma 5.3, we obtain the inequality t J + 2κ RJ, for some κ R >, an the Gronwall Lemma yiels L(t)z H ce κ Rt, t. It is then stanar matter to see that κ R can be replace by, say, κ 1 (that is, a positive constant inepenent of R), upon increasing the above constant c accoringly.
1 M. CONTI AND V. PATA Incientally, if in the original system g 1 = an f =, then S(t) = L(t). In that case, we have prove the exponential stability of the whole semigroup S(t). In fact, the result can be generalize to the equation, recently consiere in [3], u tt + α u t m u t + k()au + k (s)au(t s)s + g (u) =, which, setting k( ) = 1, translates in the history space setting into the system u tt + α u t m u t + Au + µ(s)aη(s)s + g (u) =, (21) η t = T η + u t. Then we have Proposition 6.1. Let m [, 4]. If α L 6/(4 m) (Ω), an α, then the energy associate to (21) exponentially ecays to zero. Proof. We limit ourselves to give a brief outline of the argument, which parallels the one of Lemma 6.1, with minor moifications. First, the solution u is easily seen to belong to the space L m+2 (R +, L m+2 α (Ω)). As before, we consier the functional J(t) (now for (u, u t, η)). The time erivative of J prouces in the left-han sie of the ifferential inequality the extra term For any q H 1 (Ω), there hols u t m+2 + ε 2 α u Lα m+2 t m u t, η M 1 ε2 4 α u t m u t, u. α u t m u t, q α 1/(m+2) L 6/(4 m) u t m+1 L m+2 α q L 6 c u t m+1 A 1/2 q. L m+2 α Hence, we conclue that for every ρ > there is c ρ > such that α u t m u t, η M 1 1 4 α u t m u t, u c ρ u t m+2 + ρ ( A 1/2 m+2. u + η M ) L m+2 α At this stage, we alreay know that the energy is boune, with a boun epening on the norm of the initial ata, thus, the extra term is greater than or equal to (1 c ρ ε 2 ) u t m+2 cρε 2( A 1/2 u 2 + η 2 ) L m+2 M. α Therefore, fixing ρ small enough, the secon term is controlle by the analogous one appearing in the inequality, whereas the first one can be cancelle, provie we subsequently choose ε small enough. In conclusion, we recover which entails the result. t J + 2κ RJ, Remark 6.1. Proposition 6.1 extens the main result of [3]. In particular, it is worth noting that α can be unboune (for m < 4), so that, in some zones of the omain Ω, the amping can be extremely effective. This, in conformity with the physical viewpoint, plays against issipativity (think, for instance, to the simple situation of the ampe penulum). Also, observe that in in [3] the initial history is null, that is, the convolution integral is taken from to t, whereas the result hols true for nontrivial initial histories as well.
SEMILINEAR EQUATIONS OF VISCOELASTICITY 11 7. Compactness of N(t). We finally procee to establish the compactness property of N(t). First, we prove Lemma 7.1. Let σ [, 3 4 ] be fixe, an assume that z Hσ R for some R. Then there exists KR σ such that N(t)z H (1+4σ)/4 K σ R, t. Proof. Here, the generic constant c may epen on R. The proof strictly follows the arguments of Lemma 5.3, with N(t) in place of S(t) (working with the exponent σ + 1 4 ). Hence, for ε (, 1), we efine the functional Jσ N (t) = Eσ N (t)+fσ N (t) ε 2 w t (t), ζ t M (4σ 3)/4 + ε2 4 A(1+4σ)/8 w t (t), A (1+4σ)/8 w(t), where Eσ N (t) = 1 ( A (5+4σ)/8 w(t) 2 + A (1+4σ)/8 w t (t) 2 + ζ t 2 ) M 2 (1+4σ)/4 is the energy of orer σ + 1 4 of (w(t), w t(t), ζ t ), an F N σ (t) = g(u(t)) g (v(t)), A (1+4σ)/4 w(t) f, A (1+4σ)/4 w(t). Again, for ε small enough, 1 2 EN σ (t) c Jσ N (t) 2Eσ N (t) + c. On account of the first equation of (18), we get ( E N t σ + Fσ N ) 1 2 = (g (u) g (v))u t, A (1+4σ)/4 w µ ε(s) A (5+4σ)/8 ζ(s) 2 s + g (v)w t, A (1+4σ)/4 w + g 1(u)u t, A (1+4σ)/4 w ε2 8 A(1+4σ)/8 w t 2 + c A σ/2 u t 2 + c A σ/2 u t E N σ + c A 1/2 v E N σ + c. The last inequality follows from Sobolev embeings (cf. [7, 15], where analogous computations are encountere). At this stage, the above constant c epens on ε, which, however, will be eventually fixe. Inee, on account of Lemma 5.1, choosing ε small enough, we are le to t J N σ + 2ε J N σ c + h + hj N σ, for some ε >, where h(t) = c A 1/2 v(t) + c A σ/2 u t (t). Notice that from Lemma 5.3 an Lemma 6.1 we have t τ h(y)y ε (t τ) + c. Hence, the thesis follows by applying the generalize Gronwall Lemma 2.1. We are now in a position to emonstrate the existence of the global attractor. This will follow from the next lemma. Lemma 7.2. There exists a compact set K H such that N(t)B K, t.
12 M. CONTI AND V. PATA Proof. Let R = sup z B z H. From Lemma 7.1, if z B then N(t)z H 1/4 K R, t. The representation formula (11) for ζ reas { ζ t w(t) w(t s), < s t, (s) = w(t), s > t. Thus, as ζ t () =, s ζ t (s) = { w t (t s), < s t,, s > t. Since the kernel µ is ecreasing, it is then apparent that ζ t is boune in M 1/4 an s ζ t is boune in M 3/4, uniformly as z B an t. Moreover sup t sup sup A 1/2 ζ t (s) <. s R + z B Hence, from the compactness lemma [21, Lemma 5.5], ζ t lies in a compact subset of M. Since (w(t), w t (t)) is boune in H 5/4 H 1/4 H 1 H, we are one. Summarizing what we have seen so far, the ynamical system S(t) on H fulfills the hypotheses of the abstract Theorem A.3. Therefore, there exists the global attractor A. Moreover (see Remark A.5) A is a boune subset of H 1/4. To complete the proof of Theorem 4.1, we are left to show that A is a boune subset of H 1. But if z A, exploiting the full invariance property of the global attractor, we apply again Lemma 6.1 an Lemma 7.1, for ata z A, an we iscover that in fact A is boune in H 1/2. Repeating twice this bootstrap argument (cf. [15]), we en up to prove the esire bouneness in the space H 1. 8. Further Developments. As remarke at the beginning, the main ifficulty here was to fin a compact attracting set without having at our isposal a boune absorbing set. Once this obstacle has been remove, the asymptotic analysis can be pushe much further the sole existence of the global attractor. As a matter of fact, we can even consier, in the spirit of [6, 7], a one-parameter family of equations. Namely, u tt + Au + µ ε (s)aη(s)s + g(u) = f, η t = T η + u t, where, for ε (, 1], µ ε (s) = 1 ( s ) ε 2 µ. ε The limiting situation ε = correspons to the strongly ampe wave equation u tt + Au + Au t + g(u) = f. Then, for every ε [, 1], we have a ynamical system S ε (t) acting on the phasespace { Hε H 1 H L 2 µ = ε (R +, H 1 ), ε >, H 1 H, ε =. Following the ieas of [6, 7], it is possible to prove the existence of a family E ε of exponential attractors for S ε (t) of (uniformly) boune fractal imension, which is stable for the singular limit ε, with respect to the symmetric Hausorff
SEMILINEAR EQUATIONS OF VISCOELASTICITY 13 istance. In particular, this means that the global attractor A of the original problem has finite fractal imension. Also, requiring aitional regularity on f an g, along with some compatibility conitions for g, A can be prove to be a boune subset of H m, with m N as big as f an g permit (cf. [6]). We will possibly iscuss these issues in more etail in forthcoming papers. 9. An Alternative Point of View. V.V. Chepyzhov an A. Miranville consiere in [5] equation (1), for α >. They introuce the space E = { (u(t), u t (t)) : u C b (R, H 1 ), u t C b (R, H ) }, where R = (, ] an C b stans for continuous an boune, an efine the C -semigroup Σ(τ) acting on E by the formula (Σ(τ)u )(t) = u(t + τ) t, where u(t) is the solution to (1) with initial ata u E. Then, they prove Theorem 9.1. Σ(τ) has a (unique) connecte global attractor A E. Moreover, A is boune in C b (R, H 1 H 2 ). In their terminology, the global attractor A is a compact set in C loc (R, H 1 H ) (i.e., the space of continuous functions enowe with the local uniform convergence topology), boune in E, strictly invariant uner the action of Σ(τ), which attracts any boune set B E in the topology of C loc (R, H 1 H ), namely, for every T >, [ ] lim ist C([ T,],H τ 1 H )(Σ(τ)B, A ) =, where ist enotes the usual Hausorff semiistance. This approach, in some sense more natural, can actually be recovere within the history framework. Inee, the recent paper [4] proves (for a wie class of equations, incluing this one) that if the associate semigroup S(t) acting on the history phase-space H possesses a global attractor A, then Σ(τ) possesses a global attractor A, given by A = { (u(t), η t ), t R : (u(), η ) A }. Hence, in light of our results, an exploiting the full invariance of the attractor, we have Theorem 9.2. The semigroup Σ(τ) corresponing to (1) with α = possesses a (unique) connecte global attractor (in the sense of Chepyzhov-Miranville) A E. In aition, A is boune in L (R, H 2 H 1 ). Remark 9.1. In fact, since S(t) is a ynamical system also on the phase-space H 1, A is boune in C b (R, H 2 H 1 ). Appenix: The Abstract Result. Let S(t) : X X be a C -semigroup on a Banach space X. Definition A.1. A function Φ C(X, R) is calle a Lyapunov functional if - Φ(u) if an only if u X ; - Φ(S(t)x) is nonincreasing for any x X ; - if Φ(S(t)x) = Φ(x) for all t >, then x is a stationary point for S(t). If there exists a Lyapunov functional, then S(t) is calle a graient system.
14 M. CONTI AND V. PATA Remark A.2. The existence of a Lyapunov functional ensures that boune sets have boune orbits. Recall that the Kuratowski measure of noncompactness α(b) of a boune set B X is efine by α(b) = inf { : B has a finite cover of balls of X of iameter less than }. We now state the result, which can be foun in [16, 18]. However, for the reaer convenience, we inclue here a self-containe rather simple proof. Theorem A.3. Assume the following conitions. (i) There exists a Lyapunov functional Φ. (ii) The set S of stationary points is boune in X. (iii) For every nonempty boune set B X we have that lim t α(s(t)b) =. Then S(t) possesses a connecte global attractor A that consists of the unstable manifol of the set S. Proof. Let B X be a nonempty boune set. Then, on account of (iii), the ω- limit set ω(b) is nonempty, compact, positively invariant, an attracts B. Let C be the boune an positively invariant set efine by { C = x X : Φ(x) < sup u S } Φ(u) + 1. Observe that there exists t = t (B) such that S(t )ω(b) C. Inee, for every x X, ist X ( S(t)x, S ), an by the continuity of S(t), for every k ω(b) there exists a neighborhoo U k of k an a time t k > such that S(t)U k C, for every t t k. Extract then a finite subcover U k1,..., U kn of ω(b), an set t = max { t k1,..., t kn }. The attraction property of ω(b) implies the existence of a positive function ψ(t) vanishing at infinity such that, for every x B, we can write S(t)x = k(t) + q(t), with k(t) ω(b) an q(t) X ψ(t). Thus, for a given x B, we have S(t + t)x = S(t )k(t) + S(t ) [ k(t) + q(t) ] S(t )k(t). Notice that S(t )k(t) S(t )ω(b) C. Moreover, the continuity S(t ) C(X, X ) implies the uniform continuity of S(t ) in a neighborhoo of the compact set ω(b). Thus, S(t ) [ k(t) + q(t) ] S(t )k(t) X 1, provie that ψ(t) is small enough, that is to say, provie that t is large enough. We conclue that the ball B of X centere at zero of raius R = sup v C v X + 1 is an absorbing set for S(t). Then, applying (iii) with B in place of the generic boune set B, the thesis follows by stanar arguments of the theory of ynamical systems. Remark A.4. In [16], Φ is require to be boune below. However, this conition oes not seem to be necessary. On the contrary, the requirement that if Φ(u) then u X (there overlooke) is crucial to guarantee the bouneness of orbits of boune sets. Remark A.5. Conition (iii) of the theorem is usually verifie by fining, for any given boune set B X, a ecomposition S(t) = L(t) + N(t) such that [ ] L(t)x X =, lim t sup x B
SEMILINEAR EQUATIONS OF VISCOELASTICITY 15 an N(t)B K(t), where K(t) is a compact set epening on t an B. It can also happen that K(t) = K (i.e, inepenent of t). In that case, the global attractor A belongs to the compact set K associate with B. Acknowlegments. We thank the referee for careful reaing an valuable comments. REFERENCES [1] J. Arrieta, A.N. Carvalho, J.K. Hale, A ampe hyperbolic equation with critical exponent, Commun. Partial Differential Equations, 17 (1992), 841 866. [2] A.V. Babin, M.I. Vishik, Regular attractors of semigroups an evolution equation, J. Math. Pures Appl., 62 (1983), 441 491. [3] S. Berrimi, S.A. Messaoui, Exponential ecay of solutions to a viscoelastic equation with nonlinear localize amping, Electron. J. Differential Equations, no.88 (24), 1 pp. [4] V.V. Chepyzhov, S. Gatti, M. Grasselli, A. Miranville, V. Pata, Trajectory an global attractors for evolution equations with memory, submitte. [5] V.V. Chepyzhov, A. Miranville, Trajectory an global attractors of issipative hyperbolic equations with memory, Commun. Pure Appl. Anal., to appear. [6] M. Conti, V. Pata, M. Squassina, Singular limit of ifferential systems with memory, Iniana Univ. Math. J., to appear. [7] M. Conti, V. Pata, M. Squassina, Singular limit of issipative hyperbolic equations with memory, submitte. [8] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (197), 297 38. [9] M. Fabrizio, B. Lazzari, On the existence an asymptotic stability of solutions for linear viscoelastic solis, Arch. Rational Mech. Anal., 116 (1991), 139 152. [1] M. Fabrizio, A. Morro, Mathematical problems in linear viscoelasticity, SIAM Stuies Appl. Math. 12, SIAM, Philaelphia, 1992. [11] J.M. Ghiaglia, A. Marzocchi, Longtime behaviour of strongly ampe wave equations, global attractors an their imension, SIAM J. Math. Anal., 22 (1991), 879 895. [12] J.M. Ghiaglia, R. Temam, Attractors for ampe nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273 319. [13] C. Giorgi, J.E. Muñoz Rivera, V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 26 (21), 83 99. [14] M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups an Functional Analysis (A. Lorenzi an B. Ruf, Es.), pp.155 178, Progr. Nonlinear Differential Equations Appl. no.5, Birkhäuser, Boston, 22. [15] M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3 (24) 849 881. [16] J.K. Hale, Asymptotic behavior of issipative systems, Amer. Math. Soc., Provience, 1988. [17] A. Haraux, Two remarks on issipative hyperbolic problems, in Séminaire u Collège e France, J.-L. Lions E., Pitman, Boston, 1985. [18] O.A. Layzhenskaya, Fining minimal global attractors for the Navier-Stokes equations an other partial ifferential equations, Russian Math. Surveys, 42 (1987), 27 73. [19] V. Pata, G. Prouse, M.I. Vishik, Traveling waves of issipative non-autonomous hyperbolic equations in a strip, Av. Differential Equations, 3 (1998), 249 27. [2] V. Pata, M. Squassina, On the strongly ampe wave equation, Comm. Math. Phys., to appear. [21] V. Pata, A. Zucchi, Attractors for a ampe hyperbolic equation with linear memory, Av. Math. Sci. Appl., 11 (21), 55 529.
16 M. CONTI AND V. PATA [22] M. Renary, W.J. Hrusa, J.A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific & Technical; Harlow John Wiley & Sons, Inc., New York, 1987. Receive December 24; revise January 25. E-mail aress: monica@mate.polimi.it E-mail aress: pata@mate.polimi.it