EXPONENTIAL AND POLYNOMIAL DECAY FOR FIRST ORDER LINEAR VOLTERRA EVOLUTION EQUATIONS

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1 EXPONENTIAL AND POLYNOMIAL DECAY FOR FIRST ORDER LINEAR VOLTERRA EVOLUTION EQUATIONS EDOARDO MAININI AND GIANLUCA MOLA 2, Abstract. We consier, in an abstract setting, an instance of the Coleman- Gurtin moel for heat conuction with memory, that is the Volterra integroifferential equation t u(t) β u(t) k(s) u(t s)s =. We establish new results for the exponential an polynomial ecay of solutions, by means of conitions on the convolution kernel which are weaker than the classical ifferential inequalities.. Introuction Given a real Hilbert space (H,,, ), we consier the following linear homogeneous Volterra integro-ifferential equation of the first orer t (.) t u(t) + βau(t) + k(s)au(t s)s =, t >, u() = u H. Here β > an A is a strictly positive selfajoint operator on H with omain D(A). The memory kernel k is a piecewise smooth convex ecreasing function on R + = (, ), summable along with its (istributional) erivative. In the concrete formulation (.2) H = L 2 (Ω), A =, D(A) = H (Ω) H 2 (Ω), (.) escribes the evolution of heat flow in an isotropic rigi heat conuctor occupying a boune smooth volume Ω R 3. The unknown function u is the absolute temperature in the boy, accoring to the so-calle Coleman-Gurtin conuction law [5]. Problem (.) has been stuie by many authors, both for the sake of wellposeness an stability issues, an it is well-known (see e.g. [3, 8,, 2, 4]) that, for every u H, it possesses a unique weak solution u C([, ), H). Energy issipation. The main task of this paper is to establish, uner ifferent requirements on the memory kernel k, uniform ecay properties of the system energy, efine by 2 E(t) = u(t) 2 k (s) u(y)y s, The author was supporte by the Postoctoral Fellowship of the Japan Society for the Promotion of Sciences (No. PE667). t s V

2 2 EDOARDO MAININI AND GIANLUCA MOLA having set u(t) = for t <, V = D(A /2 ) an V = A /2. As we will show later, this is the actual form for the energy, where the integral term accounts for past values of the variable u. The issipativity of (.) lies in the fact that E(t) is a ecreasing function. To be more precise, we introuce the following Definition.. Let Λ : [, ) [, ) be a ecreasing vanishing function. We say that E(t) has a (uniform) ecay rate Λ if there exists an increasing positive Q such that E(t) Q(E())Λ(t). In particular, we shall investigate the following instances: Λ(t) = e εt for some ε > [exponential stability], Λ(t) = ( + t) p for some p > [polynomial stability of rate p]. From now on, the stability of problem (.) is unerstoo to correspon to the one of E, accoring to the above efinition. Moreover, Q will always enote a generic positive constant, possibly varying within the same formula, an (increasingly) epening only on E(). Energy ecay results have alreay been achieve in linear viscoelasticity with memory, both for the infinite elay [8] an for the Volterra [6] equation, uner very general assumptions on the memory kernel. We point out that investigations therein exploite rely on the so-calle past history setting. Such an approach, originally raise in [7], is necessary in orer to treat the infinite elay case in the framework of the semigroup of operators, on a suitable phase-space. Moreover, this tool seems to be very useful even for a Volterra-type equation, where the semigroup generation cannot be expecte (an then non-exponential uniform ecay is possible). In either problems [8] an [6], without requiring the kernel to fulfill any ifferential inequality, exponential stability is achieve, as well as polynomial one in the Volterra case. We stress that in both the above quote examples, such results can be establishe also when all the issipation of the system is carrie out by the memory term solely, provie that an aitive constraint on the so-calle flatness rate of the kernel (cf. [8]) is assume. Therefore, for the sake of our problem, we introuce the history space setting, which inee enables us to achieve ecay results removing ifferential inequalities on the kernel k. Nevertheless, a complete parallel to the secon orer problem [6] oes not hol. That is, we are not yet able to treat the limiting case β =, which seems to require the introuction of some new, eep technique. Thus, in this paper we shall work uner the restriction β >, with the only exception of the last section, where we shall in fact investigate polynomial stability assuming k to fulfill a ifferential inequality. Memory kernel hypotheses. We now briefly iscuss some conitions an results alreay available in literature. As the infinite-elay counterpart of (.) generates a linear semigroup, in [] it is shown that, whenever k(s) is smooth an µ(s) = k (s) fulfills the ifferential inequality (.3) µ (s) + δµ(s), s >,

3 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 3 for some positive δ, then the semigroup is exponentially stable. It is immeiate to see that (.3) is equivalent to (.4) µ(t + s) e δt µ(s), s >, t. On the other han, (.4) is weakene in [3], where the assumption (first introuce in [8]) (.5) µ(t + s) Ce δt µ(s), t, for some C, δ > an for a.e. s >, is shown to be necessary an sufficient. Since (.5) implies the existence of γ > such that (.6) k (s) + γk(s), for a.e. s >, we have a necessary conition for the exponential ecay of the semigroup in terms of the kernel k. We recall that (.) is only a particular instance, obtaine by taking null values of u(t) for negative times, of the infinite elay equation, so that its energy E(t) ecays whenever the semigroup oes. Nevertheless, our aim is the one to take avantage of the Volterra finite time elay in orer to stuy the stability uner more general assumptions than (.6). To this purpose, Fabrizio an Polioro in [9] make use of the so-calle exponential ecay property (.7) k(s)e δs s <, for some δ >. They show the necessity of (.7) for a particular exponential stability property (which we iscuss in the sequel), uner the further assumption of square summability for k, an for the concrete formulation (.2). There is inee a gap between conitions (.5)-(.6) an (.7), as we will show in Section 3 with a counterexample. For what concerns polynomial stability, we introuce the polynomial ecay property for the kernel k: (.8) ( + s) r k(s)s <, r >. Again, in [9] a result concerning the necessity of such a conition for polynomial ecay is establishe. We will return on these issues with some etail later in Section 3. The main theorem. In the present paper, uner some basic assumptions that guarantee well-poseness an energy issipation, we will show that uniform ecay properties for E(t) hol assuming (.7) or (.8). Our main result, which will be state rigorously in Section 3, reas as follows: provie that the kernel k satisfies the exponential (resp. polynomial) ecay property, then problem (.) is exponentially (resp. polynomially) stable.

4 4 EDOARDO MAININI AND GIANLUCA MOLA Plan of the paper. We conclue our introuction by outlining the rest of the paper. In Section 2 we state in a precise way the basic assumptions on the memory kernel, then we translate the problem in the so-calle history space setting an recall from [3] a well-poseness result. Moreover, we establish some general ecay properties. In Section 3, the main theorem is rigorously state uner some issipativity assumptions on k (corresponing to (.7) an (.8)), an the necessity of such assumptions is also iscusse. Sections 4 an 5 are eicate to prove the main result in the exponential an polynomial case, respectively. Finally, in Section 6 the polynomial stability is revisite, uner the further requirement that the kernel fulfills a ifferential inequality, even in the egenerate case β =. 2. Semigroup generation an general ecay properties In orer to procee in our investigation, as alreay remarke, it is convenient to view our problem as an orinary ifferential equation in a proper Hilbert space accounting for the past history of the variable u (cf. [7]). We first state the basic assumptions on the kernel, which are a refinement of the ones outline in the introuction. Basic assumptions. We suppose k to be a (nonnegative) ecreasing convex summable function, an we require its a.e. erivative k to be summable as well. Let an κ = µ(s) = k (s) µ(s)s <. Corner points for k are allowe, but with the following restriction. We assume the set of jump points of µ to be a strictly increasing sequence {s n } n R +, either finite (possibly empty) or converging to s (, ], such that µ is absolutely continuous on each interval I n = (s n, s n+ ). If s <, we also require the absolute continuity on the interval (s, ), whereas in this case s may or may not be a jump point. Uner such hypotheses, µ is ecreasing an possibly singular for s +, µ is efine almost everywhere, an k is a piecewise C function. Remark 2.. In the rest of the paper, these basic assumptions are always unerstoo to hol. The history space setting. We exten the solution u to negative times, setting u(t) = for t <, an we introuce the following auxiliary past history variable η t (s) = t s u(r)r, t, s >. Note immeiately that η (s) = for all s >. Accoring to the above efinition, the integro-ifferential equation of problem (.) reas (cf. [3]) t u(t) + βau(t) + µ(s)aη t (s)s =, t >.

5 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 5 The past history variable η is the unique mil solution (in the sense of [2, 4]) of an abstract Cauchy problem in the µ-weighte space M = L 2 µ(r +, V ), that is { t η t = T η t + u(t), t > η =, where, as a consequence of the basic assumptions (see [2]), the linear operator T is the infinitesimal generator of the right-translation C -semigroup on M, efine as { } T η = η, D(T ) = η M : η M, lim η(s) = in V. s + Here the superpose prime symbol enotes the istributional erivative with respect to the internal variable s. Consequently, we efine the Hilbert space norme by with H = H M, (u, η) 2 H = u 2 + η 2 M, η 2 M = µ(s) η(s) 2 V s. From now on, we let u = u() an η = η. In this setting (cf. [3, Section 3]) it is possible to show that a function u is a weak solution to (.) if an only if the vector z = (u, η) is a mil solution of the orinary ifferential equation in H z(t) = Lz(t), t >, (2.) t z() = (u, ). The operator L is efine by ( ( L(u, η) = A βu + D(L) = { z H : u V, βu + ) ) µ(s)η(s)s, T η + u, } µ(s)η(s)s D(A), η D(T ). Moreover, in our present case the explicit representation formula (cf. [9]) for η reas s u(t r)r, < s t, (2.2) η t (s) = u(t r)r, s > t. As shown in [3], we have the following well-poseness result Theorem 2.2. The equation in problem (2.) generates a C -semigroup of contractions S(t) on H. Remark 2.3. We stress that Theorem 2.2 hols on the whole space H, not just on H {}.

6 6 EDOARDO MAININI AND GIANLUCA MOLA As a consequence, for any given (u, η ) H, we have z(t) = (u(t), η t ) = S(t)(u, η ). The energy of the semigroup is efine as Moreover (see [3] again), there hols E(t) = E(z(t)) = S(t)(u, η ) 2 H. Theorem 2.4. For every initial ata (u, η ) D(L), the corresponing solution S(t)(u, η ) belongs to C ([, ), D(L)), an the associate energy fulfills the ifferential ientity (2.3) with t E(t) + 2β u(t) 2 V J[η t ] = n µ (s) η t (s) 2 V s + J[η t ] =, [µ(s n ) µ(s + n )] η t (s n ) 2 V, where the sum inclues the value n = if s <. In particular, E(t) is a ecreasing function of t. From now on we shall restrict our investigation to initial ata of the form (u, ), to take avantage of the one-to-one corresponence between problems (.) an (2.). Also, the associate energy is E(t) = S(t)(u, ) 2 H. In view of representation formula (2.2), this is inee the energy efine in the introuction. Since an initial atum of the form (u, ) H {} belongs to D(L) if u V, it follows that E(t) satisfies equality (2.3) for any u V. Remark 2.5. In the sequel, we shall always assume the initial atum to belong to V {}. Thanks to the semigroup continuity property, our ecay results will be unerstoo to exten by ensity to the whole space H {}, where equality (2.3) is still fulfille, provie that the time erivative of E(t) is intene in the sense of istributions. General ecay properties. We first notice that, without introucing new assumptions, it is possible to obtain a uniform ecay for the energy. To this purpose, efine so that (2.2) turns out to be U(t) = u(y)y, (2.4) η t (s) = U(t) U(t s), t, s >, where it is unerstoo that U(t) = for t. We have the following Theorem 2.6. E(t) is polynomially stable of rate.

7 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 7 Proof. We introuce the following energy functionals, alreay encountere in [3], Ψ (t) = U(t), u(t), 2 Ψ 2 (t) = 2 k(s) η t (s) U(t) 2 V s. These functionals are well efine since (2.4) hols an U(t s) = for s t. Due to the continuity of k(s), there exist s >, C such that k(s) Cµ(s) for all s (, s]. Then s k(s) U(t) 2 V s 2 2C s s k(s) η t (s) 2 V s + 2 µ(s) η t (s) 2 V s + 2 2C η t 2 M + 2Ψ 2 (t). s s so, letting Q = 2( s k(s)s) max{, C} we en up with (2.5) U(t) 2 V Q [ η t 2 M + Ψ 2 (t) ]. k(s) U(t s) 2 V s k(s) U(t s) 2 V s A straightforwar computation (see [3]) shows that Ψ(t) = 2Ψ (t) + Ψ 2 (t) fulfills the ifferential equality t Ψ(t) = 2 ηt 2 M + u(t) 2. We now efine, for M >, the further functional J (t) = ME(t) + Ψ(t). Therefore, recalling the energy ientity (2.3) an the Poincaré inequality λ 2 2 V (corresponing to the continuous inclusion V H), we obtain, for M large enough, (2.6) J (t) + εe(t), t for some ε >. Since η (s), we have J () = ME(). Integrating inequality above on (, t) we get then an (2.7) Ψ 2 (t) + ε J (t) + ε Ψ(t) + ε E(τ)τ ME(), E(τ)τ ME(), E(τ) t ME() U(t), u(t) ME() + U(t) E().

8 8 EDOARDO MAININI AND GIANLUCA MOLA By means of (2.5), with the Young an Poincaré inequalities we have (2.8) U(t) E() Q 4λ E() + ηt 2 M + Ψ 2 (t). Letting K = ε ( + M + Q ) we obtain 4λ E(τ)τ KE(). In particular, being E(t) ecreasing, we conclue so that te(t) E(τ)τ, E(t) KE(). t Since E(t) is boune, the thesis follows. Remark 2.7. Changing the constants in the Young inequality use in (2.8), we fin U(t) E() Q 2λ E() + 2 ηt 2 M + 2 Ψ 2(t), so from (2.7) it is easily seen that, letting R be a generic positive constant, Ψ 2 (t) RE(). This hols also for U(t) 2 V, thanks to (2.5). Moreover, back to (2.4), we infer η t (s) 2 V RE(), uniformly in t an s. This is the main consequence of the Volterra equation finite elay structure, an in the sequel it will play an important role. 3. The main result We will state our main result uner the following reformulation of conitions (.7) an (.8). Dissipativity assumptions. Let p (, ]. Suppose there exists C such that (3.) k(s) CΛ p (s), s >, where e δs (δ > ), if p = ; Λ p (s) =, if p <. ( + s) p Notice that, being k a ecreasing function, (3.) with p = is equivalent to the exponential ecay property (.7), up to replacing δ in (.7) with an arbitrarily chosen γ > δ. Also, the polynomial ecay property (.8) implies (3.) with p = r+. Conversely, if (3.) hols for p (, ), (.8) follows with r = p ε, for any (small) ε >. Therefore, we have the corresponence between (3.) an (.7)-(.8).

9 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 9 Remark 3.. Conition (3.) with p = is weaker than the ifferential inequality (.6), even for convex kernels. A counterexample can be constructe as follows. For n N, efine Then the kernel k n (s) = e n2 s + ( + n 2 )e n2. k(s) = max n N k n(s) fulfills (3.) for p =, with C = δ =, but easy computations show that (.6) cannot hol. We immeiately prove a lemma which yiels a useful estimate to be use in the sequel. Lemma 3.2. Let k enjoy the issipativity conition (3.). (/(p + ), ) there exists a positive c = c(p, ω, κ) such that [µ(s)] ω s c <. Then, for any ω Proof. Since µ is ecreasing, (3.) can be equivalently state as follows. There exist C an s such that (3.2) µ(s) CΛ p+ (s), s > s, where s can be chosen to be zero if µ is not singular in the origin. Let A = {s (, s ) : µ(s) > } an B = {s (, s ) : µ(s) }. Then, exploiting (3.2), we have [µ(s)] ω s = [µ(s)] ω r + [µ(s)] ω s + where c = A B [Λ p+ (s)] ω s = s e ωδs s [µ(s)] ω s κ + s + C ω c,, if p = ; ωδ ( + s ) ω(p+)+, if p <. ω(p + ) The thesis is then achieve, provie that we choose c = κ + s + C ω c. Remark 3.3. We stress that, in the case p =, the lower boun on ω in lemma above is to be consiere as. Moreover, in this case, the constant c epens on δ also. We now state the main result of this paper. Theorem 3.4. Assume conition (3.). Then if p =, then E(t) is exponentially stable; if p (2, ), then E(t) is polynomially stable of rate r, for any r < p.

10 EDOARDO MAININI AND GIANLUCA MOLA Necessary conitions. Before proceeing with the proof, we iscuss the main result in comparison with the necessary conitions obtaine by Fabrizio an Polioro (see [9]) an mentione in the introuction. Inee, in [9], for the particular case of the concrete formulation (.2), the authors prove (by means of Laplace transform methos) that if the solution u satisfies (3.3) e αt u(t) 2 V t < for some α > an the kernel k is square integrable, then (.7) hols. To be consistent, until the en of this section, we restrict to formulation (.2), an we also assume k L (R + ). Then, the conition (3.) with p = is easily shown to be necessary as well. In fact, starting from an exponentially ecaying energy an recalling ientity (2.3), we have E(t) Qe εt, e γτ t E(τ)τ + 2β e γτ u(τ) 2 V τ, for some fixe γ >. An integration by parts yiels e γt E(t) E() γ e γτ E(τ)τ + 2β e γτ u(τ) 2 V τ, so that conition (3.3) is fulfille, provie that we choose γ < ε. The exponential ecay property for k follows by [9, Theorem 2.5]. Concerning polynomial stability, we first recall the necessary conition of [9]: efining { } p = sup r : ( + s) r k(s)s <, { } q = sup r : ( + t) 2r u(t) 2 V t <, result [9, Theorem 4.2] reas p q uner the hypothesis q >. That is, if { } (3.4) sup r : ( + t) r u(t) 2 V t < = p for some p >, then (3.5) sup { r : } ( + s) r k(s)s < p 2. Remark 3.5. Such a necessary conition, as remarke also in [6], is somehow unsatisfactory, although ifficult to improve. Inee, for large values of p the rate loss is rather gross. Now let p > an E(t) Q ( + t) p.

11 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS We can show how (3.5) is still necessary. To this purpose, introuce γ (, p); back to (2.3) once again, we are le to ( + τ) γ t E(τ)τ + 2β ( + τ) γ u(τ) 2 V τ, so that, by means of an integration by parts, we euce ( + t) γ E(t) E() γ ( + τ) γ E(τ)τ + 2β The first three summans are boune, so ( + τ) γ u(τ) 2 V τ < ( + τ) γ u(τ) 2 V τ. for any γ < p. Then, (3.4) hols, an we can apply the result of [9] to obtain (3.5). Summing up, we prove the following Theorem 3.6. Assume (.2) to hol an k L (R + ); if E(t) is exponentially stable, then (3.) hols with p =, if E(t) is polynomially stable of rate r >, then (3.) hols for any p < r Exponential stability In this section we will prove Theorem 3.4 in the case p =. We begin efining the following functionals Υ(t) = Θ(t) = k(s) η t (s) 2 V s, e δs η t (s) 2 V s, being δ as in (3.). The next set of lemmas provie essential estimates involving the above functionals, which will be crucial in the sequel. Along the rest of this paper let c be a (positive) generic constant. Lemma 4.. The functionals Υ an Θ are well efine, an fulfill the inequality (4.) Υ(t) cθ(t). Proof. Inequality (4.) follows irectly by assumption (3.). Moreover, using the bouneness of η t (s) V (see Remark 2.7), the well efinition of Θ is achieve, whereas the one for Υ is a consequence of (4.). Lemma 4.2. The following ifferential inequalities hol (4.2) (4.3) t Υ(t) + 2 ηt 2 M c u(t) 2 V, t Θ(t) + δ 2 Θ(t) c u(t) 2 V.

12 2 EDOARDO MAININI AND GIANLUCA MOLA Proof. Notice previously that, as µ is nonincreasing, s µ(r)r = s [µ(r)] 2/3 [µ(r)] /3 r [µ(s)] 2/3 [µ(r)] /3 r, for any s >. Therefore, choosing ω = /3 in Lemma 3.2, we are le to (4.4) k(s) c[µ(s)] 2/3, s >. By means of the equation for the past history variable, a irect calculation leas to the following ifferential equality for Υ(t) (4.5) t Υ(t) + ηt 2 M = 2 Concerning the right han sie, we have ( k(s) η t (s), u(t) V ν k(s) η t (s), u(t) V s. ) 2 k(s) η t (s) V s + ν u(t) 2 V, for some ν (, ). As a consequence of (4.4) an again Lemma 3.2 with ω = /3, we get Therefore k(s) η t (s) V s c 2 [µ(s)] 2/3 η t (s) V s ( /2 c [µ(s)] s) /3 η t M c η t M. k(s) u(t), η t (s) V 2νc 2 η t 2 M + 2 ν u(t) 2 V, which, back to equality (4.5), yiels (4.2), provie that we choose ν small enough. Analogously to (4.5), we see that Θ(t) fulfills the ifferential equality (4.6) Θ(t) + δθ(t) = 2 t It is immeiate to realize that e δt η t (s), u(t) V s δ 4 Θ(t) + δ which, in (4.6), conclues the proof. e δt η t (s), u(t) V s. ( ) e δs s u(t) 2 V, We en the section with the proof of the exponential stability result. Proof of Theorem 3.4 (case p = ). We efine the further functional L(t) = ME(t) + Υ(t) + Θ(t), for some M to be suitably chosen in the sequel. Inequality (4.) implies that (4.7) E(t) + Θ(t) L(t) c [E(t) + Θ(t)]. Summing up inequalities (2.3), (4.2) an (4.3), an recalling the Poincaré inequality, we euce t L(t) + Mβλ u(t) ηt 2 M + δ 2 Θ(t) + (Mβ c) u(t) 2 V.

13 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 3 Provie that M is large enough, by setting ε = c min{mβλ, /2, δ/2}, we finally reach L(t) + εl(t). t By means of (4.7), invoking the Gronwall Lemma, we get which conclues the proof. E(t) L(t) ME()e εt = Qe εt, 5. Polynomial stability In this section we will prove theorem 3.4 in the case p (2, ). Remark 5.. For the case p 2 we aress the reaer to Theorem 2.6. Again, the proof requires the introuction of suitable functionals. We let Υ be as in the previous section; also, we efine Θ p (t) = ( + s) p ηt (s) 2 V s. Analogously to the case p =, we have Lemma 5.2. The functionals Υ an Θ p are well efine, an fulfill the inequalities ( ) /q (5.) Υ(t) cθ p (t) C q ( + s) p+ ηt (s) 2 V s, for any q > p/(p ), where C q is a positive constant epening only on q an (increasingly) on E(). Proof. Left han sie of inequality (5.) follows irectly by assumption (3.). By means of the Höler inequality, we have, for some < σ < p, Θ p (t) = ( + s) σ ( + s) p σ ηt (s) 2 V s ( ) +σ ( ( + s) σ(p+)/(+σ) ηt (s) 2 p+ V s ) p σ ( + s) p+ ηt (s) 2 p+ V s. By virtue of Remark 2.7, we have η t (s) 2 V Q, so the first factor is well efine for σ > /p. Therefore (5.2) [Θ p (t)] q C q q for q > p/(p ), being ( C q = ( + s) ( + s) p+ ηt (s) 2 V s ) q s (pq p )/(q ) q Q. This proves the right han sie inequality in (5.). Moreover, by the uniform boun on η t (s) V, we infer the well efinition of Θ p (t), an as a consequence of the left han sie inequality in (5.), also of Υ(t).

14 4 EDOARDO MAININI AND GIANLUCA MOLA Remark 5.3. It is worth to point out that the constant C q above is well efine only for q > p/(p ) an tens to infinity for q p/(p ). Lemma 5.4. The following ifferential inequalities hol (5.3) t Υ(t) + 2 ηt 2 M c u(t) 2 V, (5.4) t Θ p(t) + k q [Θ p (t)] q c u(t) 2 V, for any q > p/(p ), being k q = Cq q. Proof. Proof of inequality (5.3) goes exactly like the one of (4.2), noticing that ω in Lemma 3.2 can be chosen to be /3 for all p > 2. Concerning (5.4), a straightforwar computation shows that (5.5) t Θ p(t) + p ( + s) p+ ηt (s) 2 V s = 2 Again by the Höler inequality we infer ( p 2 ( + s) p ηt (s), u(t) V s ( + s) (p+)/2 ( + s) (p )/2 ηt (s) V u(t) V s ) /2 ( ( + s) p+ ηt (s) 2 V s ( ( + s) p+ ηt (s) 2 V s + 2p ( + s) p ηt (s), u(t) V s. ) /2 ( + s) p u(t) 2 V s s ( + s) p ) u(t) 2 V, Notice that, as p > 2, the last term is well efine. Therefore, from (5.5) we euce the inequality t Θ p(t) + p 2 ( + s) p+ ηt (s) 2 V s c u(t) 2 V. By means of the right han sie inequality in (5.), this will imply (5.4), provie that we choose k q = Cq q. Remark 5.5. As a consequence of Remark 5.3, the constant k q efine above tens to for q p/(p ). Again, we en the section with the proof of the polynomial stability result. Proof of Theorem 3.4 (case 2 < p < ). We efine, for M, the functional (5.6) L p (t) = ME(t) + Υ(t) + Θ p (t). Inequality (5.) implies that (5.7) E(t) + Θ p (t) L p (t) c [E(t) + Θ p (t)].

15 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 5 Let q be fixe, with q > p/(p ). Collecting the ifferential inequalities (2.3), (5.3) an (5.4), we are le to t L p(t) + Mβλ u(t) ηt 2 M + k q [Θ(t)] q + (Mβ c) u(t) 2 V, where λ is the constant appearing in the Poincarè inequality. Being q fixe, we have C q Q an k q Q. Therefore, provie that M is large enough, we euce t L p(t) + Q {E(t) + [θ p(t)] q }. Since Υ(t) cθ p (t), an since E(t) E() Q, we have, for any q >, [L p (t)] q Q{E(t) + [Θ p (t)] q }, so that, by the ifferential inequality above, we eventually en up with t L p(t) + Q [L p(t)] q, which, by means of (5.7), implies the estimate E(t) L p (t) Q( + t) r. From the limitation q > p/(p ), we immeiately infer r < p, an we stress that the constant Q epens on q an iverges for q p/(p ). Remark 5.6. If the kernel has the form /( + s) r, conition (3.) is verifie with p = r, so we fin a polynomial ecay of the energy of rate r ε; we also have the constraint r > Polynomial ecay for a kernel satisfying a ifferential inequality Another task of our investigation is the polynomial stability whenever the kernel fulfills a ifferential inequality of the type (6.) k (s) + δ[ k (s)] ω, s >, for some δ > an ω >. Some results uner assumptions of this kin can be foun in [, 5, 6, 7], for the linear secon orer problem in viscoelasticity. Remark 6.. The techniques exploite in this note are not fit to tackle the more ifficult case of problem (.) with β =. Inee, in that case the issipation is ue to the memory term solely. Nevertheless, uner hypothesis (6.), we are able to provie a result both for β > an β =. We also mention that in the latter situation, problem (.) is an abstract version of the so-calle Gurtin-Pipkin moel for heat conuction (see [3]). Also, in the special case of an exponential kernel k(s) = e s, from the Gurtin-Pipkin law we recover the Maxwell-Cattaneo moel (cf. [2]). We point out that, also for the case β =, we have the corresponence between problems (.) an (2.). Moreover, uner the basic assumptions of Section 2, Theorem 2.2 still hols an the energy equality is given again by (2.3).

16 6 EDOARDO MAININI AND GIANLUCA MOLA It is convenient to state (6.) in terms of µ, namely (6.2) µ (s) + δ[µ(s)] ω, s > ; this is the polynomial corresponing of the exponential conition (.3). We will show that a stronger result than Theorem 3.4 can be provie uner (6.2). Namely, the rate of ecay is improve by one. Let us give the precise statement. Theorem 6.2. Let β in problem (.). For any fixe p > 2 assume that µ fulfills, for some δ >, the ifferential inequality (6.3) µ (s) + δ[µ(s)] p+ p, s >. Then E(t) is polynomially stable with rate r, for any r < p. Notice immeiately that assumption (6.3) implies that there exist C an s such that there hols C µ(s) ( + s), s > s. p Therefore, arguing as in Lemma 3.2, we easily see that [µ(s)] ω s <, for any ω (/p, ). The proof of the case β = requires the introuction of the further functional (6.4) Φ(t) = κ ψ(s) u(t), η t (s) s. Here, in orer to hanle the (possible) integrable singularity of µ at, following [8], we have efine, for any fixe s ν (, s ), the function ψ = ψ sν : (, ) [, ), as ψ(s) = µ(s ν )χ (,sν](s) + µ(s)χ [sν, )(s), where χ I enotes the inicator function of any interval I (, ). Remark 6.3. In orer to recover Theorem 2.6 in the case β =, we nee to introuce an hypothesis on the so calle flatness rate of the kernel (see [3, 6]; see also [8]). One efines (for any measurable P R + ) the probability measure m µ (P) = µ(s)s, κ an the flatness set of µ as the flatness rate of µ is the quantity F µ = {s R + : µ(s) >, µ (s) = }; P R µ = m µ (F µ ). If β =, the functional J efine in the proof of Theorem 2.6 oes not fulfill (2.6). Nevertheless, in [3] it is shown that the moifie functional J (t) = ME(t) + Φ(t) + aψ(t), where M an a are suitable positive constants, oes satisfy (2.6), uner the further restriction R µ < /2. If such a conition hols, since J () = J (), we can

17 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 7 integrate (2.6) an repeat the conclusion of the proof of Theorem 2.6 with J in place of J. Now, assuming (6.3), it is easily seen that R µ =, so that Theorem 2.6 hols. In particular we get η t (s) V Q (see Remark 2.7). Next lemma provies a basic estimate on the erivative of Φ. Lemma 6.4. Let β = an (6.3) hol. Then, for every ν (, ), there exist s ν (, s ) an positive constants c an c 2 epening only on ν (possibly unboune as ν ) such that the functional Φ(t) fulfills the inequality (6.5) t Φ(t) ( ν) u(t) 2 + c [µ(s)] p+ p ( ) c 2 µ (s) η t (s) 2 V s J[η t ]. Proof. Immeiate computation yiels (6.6) t Φ(t) = ψ(s) u(t), t η t (s) s κ κ η t (s) 2 V s ψ(s) t u(t), η t (s) s. Arguing exactly like in [8, Lemma 4.], it is possible to prove the following control κ ψ(s) u(t), t η t (s) s ( ν) u(t) 2 c 2 ( ) µ (s) η t (s) 2 V s J[η t ]. Concerning the secon summan in right han sie of (6.6), we have κ ψ(s) t u(t), η t (s) s ( ) ψ(s) µ(ζ) η t (s), η t (ζ) V ζ s = κ ( µ(s) η t (s) V s κ κ ( [µ(s)] p p ) 2 ) ( s [µ(s)] p+ p c [µ(s)] p+ p η t (s) 2 V s, ) η t (s) 2 V s having use the fact that (p )/p > /p. The thesis is then achieve by collecting both of the above inequalities. Lemma 6.5. Let β =. For M > large enough, the functional efine by fulfills the inequalities (6.7) B(t) = ME(t) + Φ(t) K E(t) B(t) KE(t),

18 8 EDOARDO MAININI AND GIANLUCA MOLA an (6.8) t B(t) + C u(t) 2 + C 2 for some positive constants K, K, C an C 2. [µ(s)] p+ p η t (s) 2 V s, Proof. Inequalities (6.7) follow immeiately by the efinition of Φ(t). In orer to prove (6.8), we collect the energy ientity (2.3) an (6.5), an by means of assumption (6.3), we have (for M large enough) t B(t) ( ν) u(t) 2 + c [µ(s)] p+ p η t (s) 2 V s ( ) + (M c 2 ) µ (s) η t (s) 2 V s J[η t ] ( ν) u(t) 2 + c [µ(s)] p+ p η t (s) 2 V s δ(m c 2 ) [µ(s)] p+ p η t (s) 2 V s (M c 2 )J[η t ]. Inequality (6.8) then follows, provie that we choose M > c 2 + c /δ. Remark 6.6. In the case β >, simply set B(t) = E(t). Then, by virtue of (6.3) along with the Poincaré inequality, from the energy ientity (2.3) we easily get B(t) δ t so that (6.7) an (6.8) follow. [µ(s)] p+ p η t (s) 2 V s 2βλ u(t) 2, Proof of Theorem 6.2. Let σ (, ). The Höler inequality entails η t 2 M = ( [µ(s)] σ [µ(s)] σ η t (s) 2 V s [µ(s)] σ(p+) +σp η t (s) 2 V s ) +σp p+ ( [µ(s)] p+ p ) p σp η t (s) 2 p+ V s. Using once more the bouneness of η t (s) V, which hols after Remark 6.3, uner the restriction σ(p + )/( + σp) > /p, we infer (6.9) η t 2q M Cq q [µ(s)] p+ p η t (s) 2 V s for any fixe q > p/(p ). The constant C q plays the same role as in Section 5: it epens increasingly on E() an singularly on q. So let us fix q (p/(p ), );

19 DECAY IN LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 9 then, by means of (6.7) an (6.9), an since q >, we are le to B(t) q K q E(t) q Q [ ] u(t) 2q + η t 2q M [ ] Q u(t) 2 + [µ(s)] p+ p η t (s) 2 V s. Therefore, from (6.8) we obtain t B(t) + Q B(t)q, which, together with (6.7), yiels E(t) Q( + t) r. From the limitation q > p/(p ), we immeiately infer r < p. Again, we have a singular epenence of Q on r. Acknowlegments. The authors wish to thank Professor Vittorino Pata to have raise the problem an for many fruitful iscussions. References [] F. Ammar-Khoja, A. Benaballah, J.E. Muñoz Rivera, R. Racke, Energy ecay for Timoshenko systems of memory type, J. Differential Equations 94, 82-5 (23) [2] C. Cattaneo, Sulla conuzione el calore, Atti Sem. Mat. Fis. Univ. Moena 3, 83 (948) [3] V.V. Chepyzhov, E. Mainini, V. Pata, Stability of abstract linear semigroups arising from heat conuction with memory, Asymp. Anal , (26) [4] V.V. Chepyzhov, V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46, (26) [5] B.D. Coleman, M.E. Gurtin, Equipresence an constitutive equations for rigi heat conuctors, Z. Angew. Math. Phys. 8, (967) [6] M. Conti, S. Gatti, V. Pata, Uniform ecay properties of linear Volterra integro-ifferential equations, Math. Moels Methos Appl. Sci. (to appear) [7] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, (97) [8] M. Fabrizio, G. Gentili, D.W. Reynols, On rigi linear heat conution with memory, Int. J. Eng. Sci. 36, (998) [9] M. Fabrizio, S. Polioro, Asymptotic ecay for some ifferential systems with faing memory, Appl. Anal. 8, (22) [] C. Giorgi, G. Gentili, Thermoynamic properties an stability for the heat flux equation with linear memory, Quart. Appl. Math. 5, (993) [] C. Giorgi, M.G. Naso, V. Pata, Exponential stability in linear heat conuction with memory: a semigroup approach, Comm. Appl. Anal. 5, 2 (2 34) [2] 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.55 78, Progr. Nonlinear Differential Equations Appl. no.5, Birkhäuser, Boston (22) [3] M.E. Gurtin, A.C. Pipkin, A general theory of heat conuction with finite wave spees, Arch. Rational Mech. Anal. 3, 3-26 (968) [4] R.K. Miller, An integroierential equation for rigi heat conuc- tors with memory, J. Math. Anal. Appl. 66, (978) [5] J.E. Muñoz Rivera, E. Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous n-imensional viscoelastic equation with polynomially ecaying kernels, Commun. Math. Phys. 77, (996)

20 2 EDOARDO MAININI AND GIANLUCA MOLA [6] J.E. Muñoz Rivera, M.G. Naso, E.Vuk, Asymptotic behaviour of the energy for electromagnetic systems with memory, Math. Methos Appl. Sci. 27, (24) [7] J.E. Muñoz Rivera, R. Racke, Magneto-thermo-elasticity - large-time behavior for linear systems, Av. Differential Equations 6, (2) [8] V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 64, (26) [9] V. Pata, A. Zucchi, Attractors for a ampe hyperbolic equation with linear memory, Av. Math. Sci. Appl., (2) [2] A. Pazy, Semigroup of linear operators an application to partial ifferential equations, Springer-Verlag, New York 983 Classe i Scienze Scuola Normale Superiore Piazza ei Cavalieri 7, I-5626 Pisa, Italy aress: eoaro.mainini@sns.it 2 Dipartimento i Matematica F.Brioschi Politecnico i Milano Via Bonari 9, I-233 Milano, Italy an Department of Applie Physics, Grauate School of Engineering, Osaka University, Suita, Osaka , Japan aress: gianluca.mola@polimi.it