On Behaviors of the Energy of Solutions for Some Damped Nonlinear Hyperbolic Equations with p-laplacian Soufiane Mokeddem

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1 International Journal of Advanced Research in Mathematics ubmitted: IN: , Vol. 6, pp 13- Revised: doi:1.185/ Accepted: cipress Ltd., witzerland Online: On Behaviors of the Energy of olutions for ome Damped Nonlinear Hyperbolic Euations with p-laplacian oufiane Mokeddem Laboratory of Biomathematics, Djillali Liabes University, P. B. 89, idi Bel Abbes, Algeria s mokeddem@yahoo.com Keywords: Hyperbolic euations with p-laplace, Decay estimate of energy. Abstract. In this paper we are concerned with nonlinear damped hyperbolic euation with p-laplace of the form u tt p u+σtu t u t +ω u m u = u r u. Used the multiplier techniues combined with a nonlinear integral ineualities given by Martinez we established a decay rate estimate for the energy. Introduction This paper deals with the decay rate estimate for the energy of the problem u tt p u + σtu t u t + ω u m u = u r u in [, + [, P ux, t = on [, + [, ux, = u x, u t x, = u 1 x on, where p u = div x u p x u and p is real number, is a bounded domain in R n with smooth boundary and the real numbers ω, m and r satisfy appropriate conditions to be made precise in the seuel. everal authors have studied the global existence and asymptotic behavior of solutions related to the problem P see for instance [], [4], [5], [1], [19] and []. In all this above cited papers the damping term played an important role in order to give energy decay estimates. In the case where σ 1 with considering α u t instead of damping term σtu t u t in the probleme P, Gao and Ma [5] obtained global existence results by means of the Faedo-Galerkin approximations. Further they shown the asymptotic behavior of solutions through the use the integral ineuality given by Nakao [11]. However, it will be difficult to proceed by this method with more general functions σ. Also, Chen, Yao and hao [4] investigated the global existence and uniueness of a solution to an initial boundary problem u tt p u u t + gx, u = fx. There they established a polynomial decay of energy under certain assumptions on g where p < n. ee also Ye [14, 15], Ma and oriano [7] for related results. It is worth mentioning some other papers in connection with asymptotic behavior of solutions to the nonlinear hyperbolic euation with dissipative effects, e.g., [1], [3], [9], [13], [16] and the references therein. Inspired by [4], we investigated in this paper the decay rate estimate for the energy of the global solutions to the problem P. For our purpose, we use the multiplier method combined with a nonlinear integral ineualities given by Martinez [8] which depends on the construction of a special weight function that depends on the behavior of σ. The paper is organized as follows. In the next section, we present some notations and material needed for our work. The statement and the proof of our main result will be given in section 3. To simplify notation, we often write ut instead ux, t and u t t instead u t x, t. The norm in Lebesgue space L p is denoted by p, in particular denotes L. We also write euivalent norm. p instead of W 1,p norm W 1,p and throughout this paper the functions considered are all real valued. cipress applies the CC-BY 4. license to works we publish:

2 14 Volume 6 Preliminaries First, suppose that σ : R + R + is a non increasing positive function of class C 1 on R +, satisfying + στ dτ = +. 1 We denote the total energy functional associated to the solutions of the problem P by Et = 1 u t + 1 p u p p + F x, udx, for u W 1,p, t and F x, u = u fx, sds and fx, u = ω u m u r u. Before stating our main result, we briefly recall the following result on the existence of a solution of the problem P. Theorem 1. Assume that u, u 1 W 1,p L, then the problem P admits a solution ut in the class u C[, ; W 1,p C 1 [, ; L. 3 This result can be established by using Faedo-Galerkin method. The proof closely follows the argument presented in [4], [6] and [13]. We now present some useful lemmas which will be used later. Lemma. Let ux, t be a global solution to the problem P on [,. Then we have, for all t [,. d dt Et = σt u t + u t. This Lemma can be easily proved by multiplying the both sides of the first euation of P by u t, integrating over and then using integration by parts. Lemma 3 obolev-poincaré ineuality. Let p N and r R with r < + n p or r np n p n p + 1. Then there is a constant c = c, p, r such that u r c u p for u W 1,p. The case p = r = gives the known Poincaré s ineuality. In order to solve the energy decay of the problem P, we use the following lemma. Lemma 4 [8]. Let E : R + R + be a non increasing function and ϕ : R + R + an increasing C function such that ϕ = and ϕt + as t +. Assume that there exist and γ > such that + Et +1 ϕ t dt 1 γ E E, < +. 4 Then we have if =, Et E e 1 γϕt, t, if >, Et E, t. 1 + γ ϕt

3 International Journal of Advanced Research in Mathematics Vol Main results and proof We are now ready to state and prove our main result. Theorem 5. Let u, u 1 W 1,p L and n > p >. uppose that 1 holds. Assume further that p r p and r < m < np. Then there exists a positive constant ce depending n p continuously on E such that the solution ux, t of the problem P satisfies the following energy decay estimate p Et t ce στ dτ p t >. 5 Proof. Multiplying by E ϕ t u on both sides of the first euation of P and integrating over [T, ], we obtain that = E ϕ u [u tt p u + σtu t u t + fx, u] dx dt, where T + and ϕ is a function satisfying all the hypotheses of Lemma 4. By an integration by parts we see that T = E ϕ uu t E E 1 ϕ + E ϕ uu t dxdt E ϕ u t dxdt + E ϕ u p dxdt + E ϕ σt u u t u t dxdt + E ϕ fx, u dxdt. Hence from the definition of energy and a simple computation we get T p E +1 ϕ dt = E ϕ uu t + E E 1 ϕ + E ϕ uu t dxdt + p + 1 E ϕ u t dxdt E ϕ σt u u t u t dxdt 6 + E ϕ p F u u fx, u dx dt. We must estimate every terms of right-hand side of 6 to arrive at a similar ineuality as 4. Define, t ϕt = στ dτ,

4 16 Volume 6 so that ϕ is a nondecreasing function of class C on R + and the hypothesis 1 ensures that ϕt + as t +. 7 Exploiting Cauchy-chwartz ineuality, obolev-poincaré ineuality and the definition of energy we get uu t dx u u t c u p u t cet 1 1 p Et. Using nonincreasing property of E and the fact that ϕ is a bounded non negative function on R + we denote by µ its maximum we obtain that T E ϕ uu t dx 1 cµe+ + 1 p, 8 here and from now on, c denotes a positive constant which can be different from line to line. imilarly we have E E 1 ϕ + E ϕ uu t dxdt T cµ E E p dt + ce p ϕ dt cµe p. 9 On the other hand, from Lemma we have that 1 + p E ϕ We also need to estimate 1 + p T u t dxdt 1 + p E ϕ E ϕ E t dt ce +1. σt E t ϕ pf u ufudx dt. ut + u t dxdt 1 From obolev-poincaré ineuality, there exists r > such that r u p p u p p u W 1,p. As fx, u = ω u m u r u, with F x, u = u fx, sds we have, F u = ω u m m We also notice that there exists ω > where ω > ω so that, u r r. 11 ω m u m ω p u p + F u u R. 1

5 International Journal of Advanced Research in Mathematics Vol Thus, we obtain from 11 that E t ϕ p F u u fx, u dx dt = E t ϕ and taking into account 1, we have E t ϕ r p u r dx dt = r E t ϕ E t ϕ p F u u fx, u dx dt E t ϕ E t ϕ r p u r r rm p m u m dx dt ω r p m u m F u dx dt, ω r p p u p + F u F u dx dt ω r p p u p + F u dx dt Using the definition of the energy Et we see that ω p u p + F u dx cet Conseuently, E t ϕ pf u ufudx dt cr p E Et ϕ dt cr p E +1 t ϕ dt. 13 The remaining term of the right hand side of 6 can be estimate as follows, E ϕ σt uu t u t dxdt = E ϕ σt = E ϕ σt uu t dxdt uu t dxdt + E ϕ σt E ϕ σt u u t dxdt u u t dxdt E ϕ σt u u t dxdt We received from Hölder ineuality and obolev-poincaré ineuality that E ϕ σt u u t dxdt E ϕ σt u p u t p dt. p 1

6 18 Volume 6 We also have This gives u t p p 1 E ϕ σt = c p c p ut c p p u u t dxdt c E + 1 p ϕ σt 1 1 E t E t σt 1 E + 1 p E t ϕ σt σt dt.. 1 dt Further, by Young ineuality, we have for ε > E ϕ σt u u t dxdt c ε = c ε E + 1 p ϕ tσt dt + E + 1 p ϕ t σt dt + c ε E. Thus takes, + 1 = + 1, so that = p /p. p Then substituting the estimates 8, 9, 1, 14 and 13 into 6, we get E +1 ϕ t dt ce p + c E +1 + c E ce p + c E + c E E, E c ε E where c, c and c are different positive constants independent of E. Let T +, we have from 15 that + E +1 ϕ t dt Thus we receive from Lemma 4 that Et ce p + c E + c ce p + c E + c E E,, E where ce = ce p + c E + c As = p /p, we have The proof is thus finished. Et ce p p t 1+ t σs ds σs ds 1 = ce 1 t σs ds is a positive constant depending on E. p p, t [, +. 1,

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