Asymptotic Behaviour Of Solutions For Some Weakly Dissipative Wave Equations Of p-laplacian Type

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1 Alied Matheatics E-Notes, (), c IN 67-5 Available free at irror sites of htt:// Asytotic Behaviour Of olutions For oe Weakly Dissiative Wave Equations Of -Lalacian Tye Nour-Eddine Aroun and alia Miouni y Received 7 March Abstract In this aer we study decay roerties of soe weakly dissiative wave equations of -Lalacian tye. Introduction We consider the initial boundary roble for the nonlinear wave equation of -Lalacian tye with a weak nonlinear dissiation of the tye utt u (t)(u t j u t j u t ) ; u(x; ) u (x); u () (x; ) u (x) in : where u div(jr x uj r x u),, is a ositive function and is a bounded doain of R n (n ), with a sooth For the roble (), when and, Messaoudi [7] showed that, for any initial data (u ; u ) H () L (), the roble has a unique global solution with energy decaying exonentially. In the case when g(u t ) ju t j u t, Nakao [9] showed that () has a unique global weak solution if (n 3); n 3 and a global unique strong solution if > (n ); n 3 (of course if n or then the only requireent is ). In addition to global existence the issue of the decay rate was also addressed. In both cases it has been shown that the energy of the solution decays algebraically if > and decays exonentially if. This iroves an earlier result by Nakao [], where he studied the roble in an abstract setting and established a theore concerning decay of the solution energy only for the case (n ); n 3. Our urose in this aer is to give an energy decay estiates of the solutions to the roble () for a weak nonlinear dissiation, we extend the results obtained by Ye [6], also we rove in soe cases an exonential decay when > and the dissiative ter is not necessarily suerlinear near the origin. We use a new ethod recently introduced by Martinez [6] (see also []) to study the decay rate of solutions to the wave equation u x ug(u ) in R, where Matheatics ubject Classi cations: 35B4, 35L7. y Laboratory of Matheatics, Djillali Liabes University, idi Bel Abbes, Algeria 75

2 76 Weakly Dissiative Wave Equations is a bounded doain of R n. This ethod is based on new integral inequality that generalizes a result of Haraux [4]. Throughout this aer the functions considered are all real valued. We oit the sace variable x of u(x; t), u t (x; t) and sily denote u(x; t), u t (x; t) by u(t), u (t), resectively, when no confusion arises. Let l be a nuber with l. We denote by k : k l the L l nor over. In articular, L nor is denoted k : k. ( : ) denotes the usual L inner roduct. We use failiar function saces W ;. Preliinaries and Main Results The function (t) satis es the following hyotheses: (H) : R! R is a nonincreasing function of class C on R satisfying () d : We de ne the energy associated to the solution of () by the following forula E(t) ku k kr xuk : We rst state two well known leas, and then we state and rove a lea that will be needed later. LEMMA (obolev-poincaré inequality). Let q be a nuber with q < (n ) or q n n (n ), then there is a constant c c(; q) such that kuk q c kruk for u W ; (): LEMMA ([5]). Let E : R! R be a non-increasing function and assue that there are two constants q and A > such that E q (t) A E()q E(); < : Then we have and q q E(t) E() 8t ; if q > qat E(t) E()e At 8t ; if q : LEMMA 3 ([6]). Let E : R! R be a non-increasing function and : R! R an increasing C function such that () and (t)! as t! :

3 N. E. Aroun and. Miouni 77 Assue that there exist q and A > such that Then we have and E(t) q (t) (t) A E()q E(); < : q q E(t) E() 8t ; if q > qa(t) E(t) ce()e!(t) 8t ; if q : PROOF. Let f : R! R be de ned by f(x) : E( (x)), (we reark that has a sense by the hyotheses assued on ). The function f is non-increasing, f() E() and if we set x : (t) we obtain (T ) () f(x) q dx (T ) () E (x) q dx E(t) q (t) A E()q E() A E()q f(()); < T < : etting s : () and letting T!, we deduce that s f(x) q dx A E()q f(s); s < : Thanks to Lea, we deduce the desired results. Now we recall the following global existence, which can be established by using the arguent in [9]. THEOREM. Assue that (u ; u ) W ; () L (). Then the roble () adits a unique strong solution on [; ) in the class C([; [; W ; ()) \ C ([; ); L ()): Our ain result is the following. THEOREM. Let (u ; u ) W ; L (); < n (n ) and suose that (H) holds. Then the solution u(x; t) of the roble () satis es the following energy decay estiates. () If, then there exists a ositive constant! such that t E(t) C(E())ex! () d 8t > :

4 78 Weakly Dissiative Wave Equations () If >, then there exists a ositive constant C(E()) deending continuously on E() such that E(t)! R C(E()) t () d ; 8t > : EXAMPLE. ) uose that (t) t ( ), by alying Theore we obtain E(t) C(E())e!t if [; ); ; E(t) C(E())t ( ) if < ; > and E(t) C(E())(ln t) if ; l < : ) uose that (t) t ln t ln t ln k, where k is a ositive integer and t ln (t) ln(t); ln k (t) ln(ln k (t)); by alying Theore, we obtain E(t) C(E())(ln k t) if ; > ; E(t) C(E())t ( ) (ln t ln t ln k t) if < ; > : 3) uose that (t) t (ln t), by alying Theore, we obtain 8 >< >: ( ) E(t) C(E())t (ln t) if < ; > ; E(t) C(E())(ln t) ( ) if ; < ; > ; E(t) C(E())(ln t) if ; ; > : 3 Proof of Theore First we have the following energy identity for the roble (). LEMMA 4 (Energy identity). Let u(t; x) be a global solution to the roble () on [; ) as in Theore. Then we have E(t) t (s)u (s)g(u (s))dsdx E() for all t [; ) and where we set g(u ) u ju j u.

5 N. E. Aroun and. Miouni 79 Now, we shall derive the decay estiate for the solutions in Theore. For this we use the ethod of ultiliers. We denote by c various ositive constants which ay be di erent at di erent occurrences. We ultily the rst equation of () by E q u, where is a function satisfying all the hyotheses of Lea 3. We obtain E q E q E q u u u (t)(u ju j u ) dx uu dx (t)u(u ju j E q u udx u )dx T E q uu dx (qe E q E q ) uu dx E q ju j dx E q jruj dx We deduce that E q De ne E q E q (t)u(u ju j E q E q T uu dx (t) u dx u )dx: (qe E q E q ) uu dx E q jruj dx (t)u(u ju j u )dx () t (s)ds: It is clear that is a non decreasing function of class C on R. Hyothesis (H) ensures that (t)! as t! : (3) ince E is non-increasing, is a bounded non-negative function on R (and we denote by its axiu), we nd that h E(t)q uu dx i ce q T ce()q ; 8t ; (qe E q E q ) uu dx c ( E (t))e(t) q

6 8 Weakly Dissiative Wave Equations c E(t) q ( ce() q ; ) E q u dx E q (t) C E q () E q (t)e (t) (t)(u ju j )dx where we have also used the Hölder and obolev-poincaré inequalities. Using these estiates we conclude fro () that E(t) q (t) ce() q c E() q E q (t)u(u ju j u )dx: (4) Now, we estiate the ters of the right-hand side of (4) in order to aly the results of Lea 3: E q (t)u(u ju j u )dx E q E q ju j> (t)u(u ju j (t)u(u ju j We estiate the rst ter, we get E q (t)u(u ju j u )dx t E q (t)juu jdx E q s E q u )dx u )dx (t)juu j ju j dx (t)juu jdx: (5) Using the Hölder and obolev Poincaré inequalities and the energy identity fro Lea 4, we get E q (t)juu jdx

7 N. E. Aroun and. Miouni 8 E q (t) E q (t) kuk L C() E q (t) C() E q (t) C() C ()" C ()" t s juj dx " dx E q E (t) (t)! ju j ju j! dx dx # (u g(u )) ( ) dx ( ) E (q ) C () " ( E )! u g(u )dx E (q ) C E(): (6) " We choose q such that (q ) q, thus we nd q ( ). Using the Hölder inequality and the obolev ibedding, we obtain E q (t)ug(u )dx E q (t) c E q (t) c juj dx ju j> E q (t)( E ) : juj> u g(u )dx Alying Young s inequality, we obtain E q (t)ug(u ) dx et " " C()" C()" ( )( ) E () ( )( ) E() jg(u )j! E q () (t) C()! dx " E q C() " ( E ) E(): (7). Choosing " and " sall enough, we deduce fro (4), (6) and

8 8 Weakly Dissiative Wave Equations (7) that E(t) q q C q C ( )( ) E() C ( ) E() C q C q ( )( ) C ( ) E() q! E() q E() where C; C ; C ; C ; C are di erent ositive constants indeendent of E(). Hence, we deduce fro Lea 3 that q q E(t) C q C q q t ( )( ) C q ( ) (s) ds q : References [] J. Ball, Rearks on blow u and nonexistence theores for nonlinear evolutions equations, Quart. J. Math. Oxford, 8(977), [] A. Benaissa and. Mokedde, Global existence and energy decay of solutions to the Cauchy roble for a wave equation with a weakly nonlinear dissiation, Abs. Al. Anal., (4), [3] V. Georgiev and G. Todorova, Existence of solutions of the wave equation with nonlinear daing and source ters, J. Di. Eq., 9(994), [4] A. Haraux, Two rearks on dissiative hyerbolic robles, Research Notes in Matheatics, Pitan, 985, [5] V. Koornik, Exact Controllability and tabilization, The Multilier Method, Masson-John Wiley, Paris, 994. [6] P. Martinez, A new ethod to decay rate estiates for dissiative systes, EAIM Control Oti. Calc. Var., 4(999), [7]. A. Messaoudi, Decay of the solution energy for a nonlinearly daed wave equation, Arab. J. for cience and Engineering, 6(), [8] M. Nakao, On solutions of the wave equations with a sublinear dissiative ter, J. Di. Eq., 69(987), 4 5. [9] M. Nakao, Rearks on the existence and uniqueness of global decaying solutions of the nonlinear dissiative wave equations, Math.., 6(99), [] M. Nakao, Decay of solutions of soe nonlinear evolution equations, J. Math. Anal. Al., 6(977), [] M. Nakao and K. Ono, Global existence to the Cauchy roble of the seilinear wave equation with a nonlinear dissiation, Funkcialaj Ekvacioj, 38(995),

9 N. E. Aroun and. Miouni 83 [] L. Payne and D. H. attinger, addle oints and instability on nonlinear hyerbolic equations, Israel J. Math., (975), [3] D. H. attinger, On global solutions for nonlinear hyerbolic equations, Arch. Rational Mech. Anal., 3(968), [4] G. Todorova, table and unstable sets for the Cauchy roble for a nonlinear wave equation with nonlinear daing and source ters, J. Math. Anal. Al., 39(999), 3 6. [5] G. Todorova, Cauchy roble for a nonlinear wave with nonlinear daing and source ters, C. R. Acad ci. Paris er. I., 36(998), [6] Y. J. Ye, On the decay of solutions for soe nonlinear dissiative hyerbolic equations, Acta Math. Al. in. Engl. er., (4), 93.