ON THE WAVE EQUATION WITH A TEMPORAL NON-LOCAL TERM

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1 Dynamic Sysems and Applicaions 16 (7) ON THE WAVE EQUATION WITH A TEMPORAL NON-LOCAL TERM MOHAMED MEDJDEN AND NASSER-EDDINE TATAR Universié des Sciences e de la Technologie, Houari Boumedienne, Faculé de Mahémaiques, 1613 Bab Ezzouar, Alger. Algeria (mmedjden@yahoo.fr) King Fahd Universiy of Peroleum and Minerals, Deparmen of Mahemaical Sciences, Dhahran, 3161 Saudi Arabia (aarn@kfupm.edu.sa) ABSTRACT. This paper is concerned wih he asympoic behavior for an inegro-differenial equaion which appears in viscoelasiciy. I is proved ha he energy of he sysem decays exponenially o zero as ime goes o infiniy provided ha he kernel in he memory erm is also exponenially decaying. New assumpions are discussed. Key words and phrases: Exponenial decay, memory erm, relaxaion funcion, viscoelasiciy AMS subjec classificaions: 35L, 35B4, 45K5 1. INTRODUCTION We shall consider he following wave equaion wih a emporal non-local erm and a weak inernal damping u + au = u h( s) u(s)ds, in R + (1) u =, on Γ R + u(x, ) = u (x), u (x, ) = u 1 (x), in where is a bounded domain in R n wih smooh boundary Γ =. The funcions u (x) and u 1 (x) are given iniial daa and he relaxaion funcion h() will be specified laer on. This problem models some phenomena in viscoelasiciy, see [16,4] for a discussion on how hese models arise. Similar linear problems as well as some nonlinear versions have been sudied by many auhors [6,16,3,4,17,1] see also he references in [16] and [4]. For similar problems wih singular inegrable and non-inegrable kernels we refer he reader o [1,14,5,7,8,9,15,11] for insance. In [1], his problem was considered on sar-shaped domains and wih a nonlinear dissipaion g(u ). The case where a localized viscoelasic dissipaion acing on a cerain subdomain is complemened by a weak inernal dissipaion on he oher par of he domain is considered in []. We obain resuls of he same naure considering differen approaches. The problem in [] is nonlinear and herefore requires a more delicae reaemen. We noe here ha, Received January 3, $15. c Dynamic Publishers, Inc.

2 666 M. MEDJDEN AND N. TATAR in all he previous works, he following resricion on he kernel was imposed. h () ηh(), I is he purpose of his paper o give an explici rae of convergence for soluions of problem (1). I will be shown ha he soluion is exponenially asympoically sable provided ha he kernel appearing in he memory erm is also exponenially decaying o zero. Moreover, we replace he above frequenly used assumpion by he condiions h () and e α h() L 1 (, ) for some α >. No oher condiion on he derivaive of h() is imposed. Our argumen has wo feaures: i is simple (no heavy machinery is needed) and i covers some kernels which were no reaed previously. An ineresing family of examples may be consruced by aking nonincreasing regular funcions below e α for some α > and which are consans on some small inervals. On hese inervals where he funcion is consan he condiion h () ηh(), is no saisfied for any η >. One can avoid his siuaion by saring he argumen of he exponenial decay from a poin on he righ hand side of hese inervals bu in his case we may consider an infinie (bu counable) sequence of inervals as in he following second family of examples. Anoher ype of examples consiss in choosing (almos everywhere regular) funcions of he form (1 + n n )e n n e n, on [n, n + 1 ] n h() = e 1 + (n + 1) n+6 e n n+6 e n, on [n + 1, n ) n+3 e, everywhere else for n 3. We can easily check ha + h(s)ds Ce which implies ha + h(s)e αs ds for < α < 1. On he oher hand, lim sup h() = 1/e and hence lim sup h()e δ = + for any δ >. Consequenly, he condiion h () ηh(), canno be saisfied. To his end we esablish a new Lyapunov funcional. Indeed, we modify he energy associaed o he sysem by an addiional, suiably chosen, erm which will cancel ou some undesirable erms. We also discuss he case a =, ha is wihou he inernal dissipaion. In his case we will show ha he inegral erm induces a weak damping which alone is capable of driving he sysem o res also in an exponenial manner. For well posedness and regulariy we refer he reader o he aforemenioned works and o [13]. Theorem 1. Assume ha h is a coninuous funcion and (u, u 1 ) H 1 () L (). Then here exiss a unique soluion o problem (1) such ha u L (, ; H 1 ()), u L (, ; L ()), u L (, ; L ()).

3 WAVE EQUATION WITH TEMPORAL NON-LOCAL TERM 667 The assumpion on h() we replaced is no needed o prove he exisence, uniqueness or coninuous dependence resuls. asympoic behavior quesion. In his paper, we will concenrae only on he The plan of he paper is as follows: in he nex secion we consider he case of an inernal damping (a > ) and in Secion 3, we discuss he case a =.. ASYMPTOTIC BEHAVIOR In his secion we sae and prove our resul. Firs, we suppose ha he kernel h() is a C 1 (R +, R + ) funcion saisfying (A1) h () for all R +, (A) 1 h(s)ds = l >, (A3) e α h() L 1 (R + ) for some α >. Nex, we define he energy of (1) by E() = 1 ( u + u ) dx. In his secion we rea he case a >. Wihou loss of generaliy we may suppose ha a = 1. Theorem. Assume ha he hypoheses (A1) (A3) hold. Then he energy of (1) decays o zero exponenially, ha is, here exis posiive consans C and β > such ha E() Ce β,. Proof: A differeniaion of E() wih respec o yields de() () = u dx + u h( s) u(s)ds dx. d Seing, i is easy o see ha (3) 1 d Then, defining (h u)() = h( s) u() u(s) ds dx, (h u)() = d {( u h( s) u(s)ds dx + 1 u)() (h + 1 d ) } h(s)ds d u dx 1h() u dx. e() := 1 u dx + 1 we obain from () and (3), (4) e () = u dx 1 h() ( ) 1 h(s)ds u dx + 1 (h u)(), u dx + 1 (h u)().

4 668 M. MEDJDEN AND N. TATAR Observe ha by assumpion (A1) we have e (),. definiions of e(), (h u)() and (A), here exiss M > such ha (5) E() Me(),. Nex, we inroduce wo funcionals Φ() = and where and α is as in (A3). Ψ() = u u dx H α ( s) u(s) ds dx + H α () = e α h(s)e αs ds Using equaion (1) 1 of our problem, we obain dφ() = u dx + u u dx d = u dx u u dx u dx + Moreover, from he u h( s) u(s)ds dx. Clearly, u h( s) u(s)ds dx 1 u dx + 1 ( h( s) u(s)ds) dx 1 u dx + 1 h( s)ds h( s) u(s) ds dx 1 u dx + 1 l h( s) u(s) ds dx. Therefore (6) dφ() d u dx + 1 u u dx u dx + 1 l u dx h( s) u(s) ds dx. A differeniaion of Ψ() wih respec o using Leibniz rule yields (7) dψ() ( + ) = h(s)e αs ds u dx h( s) u(s) ds dx αψ(). d Le us inroduce he funcional V () = e() + εφ() + ηψ() wih < ε < 1 and η >. From he above relaions (4), (6) and (7) we infer ha V () = e () + εφ () + ηψ ()

5 or (8) WAVE EQUATION WITH TEMPORAL NON-LOCAL TERM 669 u dx 1 h() u dx + 1 u)() + ε u (h dx ε u u dx ε u dx + ε u ε(1 l) dx + h( s) u(s) ds dx ( + ) + η h(s)e αs ds u dx η h( s) u(s) ds dx αηψ() [ ε V () (1 ε) ( u dx η + h(s)e ds)] αs u dx ( ) + 1 u)() εφ() αηψ() η ε(1 l) h( s) (h u(s) ds dx. If we choose α such ha + h(s)e αs ds < 1, hen we can selec η such ha 1 l ε(1 l) < η < ε ( + h(s)e αs ds) 1. Consequenly, he coefficiens of u dx and are negaive. h( s) u(s) ds dx in (8) Le us add and subrac µ(h u)() o he righ hand side of (8), hen using he esimaion we ge (h u)() = h( s) u() u(s) ds dx (1 l) u dx + h( s) u(s) ds dx [ ( ε ) η + h(s)e αs ds (1 l)µ] u dx (1 ε) u dx εφ() αηψ() µ(h u)() V () ( η ε(1 l) µ) h( s) u(s) ds dx. Finally, we choose µ small enough so ha he coefficiens are posiive. Therefore, here exiss a posiive consan β > such ha We deduce ha V () βv (),. V () V ()e β,. From he definiions of V () and e() we conclude he asserion of our heorem. The proof is complee.

6 67 M. MEDJDEN AND N. TATAR 3. THE UNDAMPED CASE a = If h saisfies he addiional condiion ha h () e σ L 1 (, ) for some σ > hen i is possible o prove an exponenial decay resul wihou he inernal dissipaion. Tha is for he case a =. This siuaion is less favorable han he firs one since we are lef only wih he memory erm. We can show ha his memory erm alone is enough o produce a weak dissipaion which is able o drive he sysem o res in an exponenial manner. The proof is essenially similar o he proof of he previous heorem. We need only o make clear how we compensae he erm u dx in () which we will lose by aking a =. To his end we inroduce firs he funcional Λ 1 () := u h( s) (u() u(s)) ds dx wih an appropriae coefficien. Noice ha by he inequaliy ab δa + 1 4δ b, we have Λ 1 () δ 1 C p (1 l) ( ) u dx + 1 l 1 4 δ u δ dx +(1 l)c p δ h( s) u(s) ds dx, where C p is he Poincaré consan. This implies ha, adding his erm o V (), we obain a new funcional which is again bounded below by a posiive consan imes e(). Therefore he exponenial decay of his new funcional will imply he exponenial decay of e(). On he oher hand he derivaive of Λ 1 () wih respec o is equal o (9) dλ 1 () d = u h( s) ( u(s) u()) ds dx ) h( s) ( u(s) u()) ds dx + ( ) ( h( s) u(s)ds + u ) ( s) (u(s) u()) ds dx h(s)ds h u dx. ( ) If >, hen his derivaive will provide us wih a h(s)ds u dx. The only erm in he righ hand side of he relaion (9) which canno be readily conrolled ( is he hird one. We have u ( s) (u(s) u()) ds dx h = u ( s)u(s)ds dx ( s)ds u h h u dx and (1) δ 3 u dx + Cp u 4δ 3 h ( s)u(s)ds dx h (s) ds h ( s) u(s) ds dx. For he second erm in he righ hand side of (1) we need o inroduce he funcional Λ () = H σ ( s) u(s) ds dx where + H σ () = e σ h (s) e σs ds.

7 WAVE EQUATION WITH TEMPORAL NON-LOCAL TERM 671 h ( s) u(s) ds dx, see The derivaive of Λ () will provide us wih a (7). Acknowledgmen: The second auhor is graeful for he financial suppor and he faciliies provided by King Fahd Universiy of Peroleum and Minerals, Deparmen of Mahemaical Sciences. The auhors would like also o hank he anonymous referee for his/her valuable commens and suggesions. REFERENCES [1] M. M. Cavalcani, M. Aassila and J. A. Soriano, Asympoic sabiliy and energy decay raes for soluions of he wave equaion wih memory in a sar-shaped domain, SIAM J. Conrol Op., 38 (5) (), [] M. M. Cavalcani and H. P. Oquendo, Fricional versus viscoelasic damping in a semilinear wave equaion, SIAM J. Conrol Opim. 4 (3), No. 4, [3] H. Engler, Weak soluions of a class of quasilinear hyperbolic inegrodifferenial equaions describing viscoelasic maerials, Arch. Ra. Mech. Anal., 113 (1991), [4] M. Fabrizio and A. Morro, Mahemaical Problems in Linear Viscoelasiciy, SIAM Sud. Appl. Mah., Philadelphia 199. [5] K. B. Hannsgen and R. L. Wheeler, Behavior of he soluions of a Volerra equaion as a parameer ends o infiniy, J. Inegral Eqs., 7 (1984), [6] W. J. Hrusa, Global exisence and asympoic sabiliy for a nonlinear hyperbolic Volerra equaion wih large iniial daa, SIAM J. Mah. Anal., 16 (1985), [7] W. J. Hrusa and M. Renardy, On wave propagaion in linear viscoelasiciy, Quar. Appl. Mah., 43 (1985), [8] W. J. Hrusa and M. Renardy, On a class of quasilinear parial inegrodifferenial equaions wih singular kernels, J. Diff. Eqs., 64 (1986), 195. [9] W. J. Hrusa and M. Renardy, A model equaion for viscoelasiciy wih a srongly singular kernel, SIAM J. Mah. Anal., 19 (1988), [1] S. O. Londen, An exisence resul for a Volerra equaion in a Banach space, Trans. Amer. Mah. Soc., 35 (1978), [11] J. Miloa, J. Nečas and V. Mah. Univ. Carolinae, 31 No 3 (199), Šverák, On weak soluions o a viscoelasiciy model, Commen. [1] V. Paa and A. Zucchi, Aracors for a damped hyperbolic equaion wih linear memory, Adv. Mah. Sci. Appl., 11 (1) [13] J. Prüss, Evoluionary Inegral Equaions and Applicaions, Birkhaüser Verlag, Basel, [14] M. Renardy, Some remarks on he propagaion and non-propagaion of disconinuiies in linearly viscoelasic liquids, Rheol. Aca, 1 (198), [15] M. Renardy, Coercive esimaes and exisence of soluions for a model of one-dimensional viscoelasiciy wih a noninegrable memory funcion, J. Inegral Eqs. Appl., 1 (1988), [16] M. Renardy, W. J. Hrusa and J. A. Nohel, Mahemaical Problems in Viscoelasiciy, in Piman Monographs and Surveys in Pure and Applied Mahemaics No. 35, John Wiley and Sons, New York [17] Q. Tiehu, Asympoic behavior of a class of absrac inegrodifferenial equaions and applicaions, J. Mah. Anal. Appl., 33 (1999),