 Advances in Pure Mathematics, 2011, 1, 276-279 doi:10.4236/apm.2011.15049 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Exponential Decay Rate of the Perturbed Energy of the Wave Equation with Zero Order Term Hamchi Ilhem Department of Mat hematics, University of Batna, Algeria E-mail: hamchi_ilhem@yahoo.fr Received April 30, 2011; revised May 26, 2011; accepted June 10, 2011 Abstract In this paper, we consider the wave equation with zero order term. We use the compactness uniqueness ar-gument and some result of I. Lasiecka and D. Tataru in  to prove, directly, the exponential decay rate of the perturbed energy. Keywords: Perturbed Energy, Compactness Uniqueness Argument 1. Introduction Consider the boundary feed-back system  0101=0 in ,=0 on ,=0 on ,0=,0= in .ttttyyqy Qyybyyyy y  (1) Here is a bounded domain of with smooth boundary and  is a partition of *nn01, such that and 0 01=.  , >0T=0,QT, =0,ll T . 1=0,1l:b,qq and are two positiv e bounded fu nctions, that is there exists four positive constants  such that for all and for all :q00h01=:.0 and =:.0.xhxh h For all >0, we define the perturbed energy of the system (1) by for all Here is the usual energy defined by   =,Et Ett0.tE2221=d2tEtyqyy x and ()= d,ttyMyx where =2 .1.Myhyny  In the ca se of V. Komornick and E. Zuazua in  have shown that =0,qE decays exponentially. When 0q, there is some difficulty to obtain this result since we have a lower order term with respect to the energy in some multiplier estimate (see (3.5) in ). The purpose of this paper is to overcome this kind of difficulty, where we prove an useful estimate then by the compactness uniqueness argument we absorb the lower order term. Finally, we employ some result of I. Lasiecka and D. Tataru in . In all this paper, is a generic positive constant in-dependent of the initial data and it may change from line to line. C2. Exponential decay rate of E Using the multiplier method we can show that the energy of the system (1) is a decreasing function. That is, for all we have Et0,21d== d0.dttEEt byt (2) The proof of the main theorem involves two lemmas. Lemma 1 For all two positive constants and S such that 0T>TST, where 0 is some sufficiently large positive constant, we have for T sufficiently small  203d.4TSCTE TESCyQ (3) Here d=dd.QxtProof. First we have from the definition of E 11EtE tEt , (4) where  is the constant verifying H. ILHEM277  =dttyMyxEt. On the other hand , we have (s ee (3.5) in )  2d.tEtCEtCqy x  Integrate over ,ST use (4) with  sufficiently small and by the decreasing of , we find E 201d.1TSESETESCTE TCyQ   Therefore  201d.1TSCTE TESCyQ With  sufficiently small we find the desired result. To absorb the lower order term 2dTSyQ from the estimate (3) we apply the compactness uniqueness argu-ment (see for example ). Lemma 2 For , where is sufficiently large, we have 0>TST0T221dTTtSSyQCby d, (5) where d=dd.tProof. First, we have from (3)  200d,TSCCETESy QTT with sufficiently large, we obtain 0T 20d.TSCETESy QT (6) On the other hand , by (2)  2121=d=1d.1TTttSSTtSESETE ttETbyET byd Then, with  sufficiently small, we find   2121111d142d,3TtSTtSES ESET byET by   by (6), we find  2201ddTTtSSCESESCy QCbyT,  with sufficiently large, we find 0T221ddTTtSSESCbyy Q . (7) Now, we come back to the proof of (5). It is sufficient to prove (see ] that, for some large enoug h, we have 0T0022001ddTTtyQCby. We argue by contradiction. There exists a sequence of solutions ky of system (1) such that 0201limd= 0Tktkby  (8) and 020d=1 for all ,TkyQ k (9) where represents the energy of . kEkyLet, for all , k =kk kEt Ett, where =dkkttyMykx. If we apply (7) with and 0, we obtain by (8), (9) and (4) that  =,kEt Et=0S=TT0kE is bounded and therefore there exists a subsequence ky such that 100weakly in0,;,kyy LTH and 20weaklyin 0,.kyy LT Using (8) and passing to the limit, we obtain 00001=0 in 0,,=0 on 0,,==0 on 0,. tttyyqyTyTyby T Hence we find If then =0y020limd=0.TkkyQ  This contradicts (9). If 0y, then is solution of =tzy0=0 in0,, ttzzqzT Copyright © 2011 SciRes. APM H. ILHEM 278 001=0 on 0,,=0 on 0,. zTzT Then, for sufficiently large, . So, is a solution of 0T=0zy00001=0 in 0,,=0 on 0,,=0 on 0,. yqy TyTyT If we multiply the first equation by , integrate over and use the first Green’s formula we find yQ0220d=0,TyqyQ thus . =0yWe give, now, the proof of the main result. Theorem 3 For any initial data 01,yyD, the en-ergy perturbed E is exponentially stable. Proof. If we insert (5) in (3) we find  2013d,4TtSCTE TESCby  But   21d= d=1.1TTttSSbyEttE SETES   Then  011CCTE TES If we choose sufficiently large we find such that 0T0< <1r 1,ET ESr (10) for all 0This imply that .TST 0for all .ET ESTST (11) If we apply (10) repeatedly on the intervals , , we get 00,1mT mT=0,1,m0000111=1 .rEm TEmTrEm TEmTr 1 Put 1=rps sr and 0=m11.mmmsps s Using lemma 3.3 in  to obtain for all m ,msSm where St is the solution of the system  1=0= 0.tSt rStSE0 Here we have used 1=StI pStSt  11=1rIIStr,0rSt. The last system have the solution then 1=0rtSt eE100 for all .rmEmT eEm Let then 0,tT00= where 0,tT mTT0 this imply that 000= where =2.tmTT T T Then by (11) 00=.Et EmTEmT  So 201100000tTtrrTTEt EmTeEeE   . Thus 00 for ,tEt cEetT 21=cer and 01=.rT where Remark 4 We can treat exactly in the same way the situation of the second order hyperbolic equation with variable coefficients, linear zero order term and polyno-mial growth of the nonlinear feedback near the origin. In this case, we use the Riemann geometric approach to handle the case of the variable coefficients principal part (see ] to show, directly, that we have an exponential or polynomial decay rate of the perturbed energy func-tional defined for all >0 by 12=,Et EttEt where  depends on the behavior of non linear d amping at the origin. 3. References ,sEmT to find  S. Feng and X. Feng, “Nonlinear Internal Damping of Copyright © 2011 SciRes. APM H. ILHEM Copyright © 2011 SciRes. APM 279Wave Equations with Variable Coefficients,” Acta Mathematica Sinica, Vol. 20, No. 6, 2004, pp. 1057-1072. doi:10.1007/s10114-004-0394-3  Y. Guo and P. F. Yao, “Stabilization of Euler-Bernoulli Plate Equation with Variable Coefficients by Nonlinear Boundary Feedback,” Journal of Mathematical Analysis and Applications, Vol. 317, No. 1, 2006, pp. 50-70. doi:10.1016/j.jmaa.2005.12.006  V. Komornick and E. Zuazua, “A Direct Method for Boundary Stabilization of the Wave Equation,” Journal de Mathématiques Pures et Appliquées, Vol. 69, 1990, pp. 33-54.