Research Article | Open Access
Soufiane Mokeddem, Khaled Ben Walid Mansour, "The Rate at Which the Energy of Solutions for a Class of -Laplacian Wave Equation Decays", International Journal of Differential Equations, vol. 2015, Article ID 721503, 5 pages, 2015. https://doi.org/10.1155/2015/721503
The Rate at Which the Energy of Solutions for a Class of -Laplacian Wave Equation Decays
We will investigate the decay estimate of the energy of the global solutions to the p-Laplacian wave equation with dissipation of the form under suitable assumptions on the positive function . For this end we use the multiplier method combined with nonlinear integral inequalities given by Martinez; the proof is based on the construction of a special weight function that depends on the behavior of .
In this paper we are concerned with the energy decay rate of the -Laplacian type wave equation of the form where is a bounded domain in with smooth boundary , , are real numbers, and is a positive function satisfying some conditions to be specified later.
Problem can be considered as a system describing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Voigt model. The bibliography of works in this direction is so long. We mention, for instance, the works of Andrews , Andrews and Ball , Ang and Dinh , and Kawashima and Shibata .
In , Benaissa and Messaoudi considered the following problem: where , . They showed that, for suitably chosen initial data, the problem has a global weak solution, which decays exponentially even if . Further they proved the global existence by using the potential well theory introduced by Sattinger .
Similar results have been established by Ye [7, 10]. In these works the author used the Faedo-Galerkin approximation together with compactness criteria and difference inequality introduced by Nakao .
In , with considering instead of the damping term , we have obtained global existence result by using the argument in  combined with the concepts of so-called stable sets due to Sattinger . We have also shown the asymptotic behavior of global solutions through the use of the integral inequality given by Komornik .
The purpose of this paper is to give an energy decay estimate of the solution of problem . Our proof is based on the multiplier method combined with nonlinear integral inequalities given by Martinez .
The plan of the paper is as follows. In the next section we present some assumptions, technical lemmas, and main result. Then in Section 3 we are devoted to the proof of decay estimate.
For simplicity of notation, we denote by the Lebesgue space norm. In particular denotes and the inner product of . We also write equivalent norm instead of norm As usual, we write, respectively, and instead of and . Furthermore, throughout this paper the functions considered are all real valued.
2. Preliminaries and Main Result
We state, without proof, a global existence result for the problem . For more details we refer the reader to .
We now present the following well-known lemmas which will be needed later.
Lemma 3 (Sobolev-Poincaré inequality). Let and with or . Then there is a constant such that The case gives the known Poincaré’s inequality.
Lemma 4 (see ). Let be a nonincreasing function and an increasing function such that Assume that there exist and such thatThen one has
Now we are in position to state and prove our main result.
Theorem 5. Let and Suppose that (2) holds. Then the solution of the problem satisfies the following energy decay estimates.(1)If , then there exists a positive constant such that(2)If , then there exists a positive constant depending continuously on such that
3. Proof of Main Result
From now on, we denote by various positive constants depending on the known constants and they may be different at each appearance.
Multiplying by on both sides of the first equation of and integrating over , where is a function satisfying all the hypotheses of Lemma 4 and , we obtain that By an integration by parts we see that Hence from the definition of energy and a simple argument we can obtainNow we must estimate both sides of (14) to arrive at a similar inequality as (8).
Define It is clear that is a nondecreasing function of class on and hypothesis (2) ensures thatSince is nonincreasing, is a bounded nonnegative function on (we denote by its maximum) and, using the definition of energy, Cauchy-Schwartz inequality, and Sobolev-Poincaré inequality, we havewhere the above estimate follows from the fact thatAgain, exploiting Cauchy-Schwartz inequality, Sobolev-Poincaré inequality, the definition of energy, and (18), we obtainwhere the fact that is nonincreasing is used.
Furthermore, by using Lemma 2, we haveWe then estimate the last term in (14) as follows: which implies that Using Hölder’s and Sobolev-Poincaré’s inequalities, we see that We also have This gives Further, by Young inequality, we have for We choose such that Thus
Letting , this yields the following estimate: and we conclude from Lemma 4 that It is clear that, for , we have and the energy associated with the solution of the problem satisfies the decay property in (10).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
- G. Andrews, “On the existence of solutions to the equation ,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980.
- G. Andrews and J. M. Ball, “Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity,” Journal of Differential Equations, vol. 44, no. 2, pp. 306–341, 1982.
- D. D. Ang and P. N. Dinh, “Strong solutions of a quasilinear wave equation with nonlinear damping,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 337–347, 1988.
- S. Kawashima and Y. Shibata, “Global existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992.
- A. Benaissa and S. A. Messaoudi, “Exponential decay of solutions of a nonlinearly damped wave equation,” Nonlinear Differential Equations and Applications, vol. 12, no. 4, pp. 391–399, 2005.
- Y. Zhijian, “Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms,” Mathematical Methods in the Applied Sciences, vol. 25, no. 10, pp. 795–814, 2002.
- Y.-J. Ye, “On the decay of solutions for some nonlinear dissipative hyperbolic equations,” Acta Mathematicae Applicatae Sinica, vol. 20, no. 1, pp. 93–100, 2004.
- Y. Ye, “Exponential decay of energy for some nonlinear hyperbolic equations with strong dissipation,” Advances in Difference Equations, vol. 2010, Article ID 357404, 12 pages, 2010.
- D. H. Sattinger, “On global solution of nonlinear hyperbolic equations,” Archive for Rational Mechanics and Analysis, vol. 30, no. 2, pp. 148–172, 1968.
- Y. Ye, “Existence of global solutions for some nonlinear hyperbolic equation with a nonlinear dissipative term,” Journal of Zhengzhou University, vol. 29, no. 3, pp. 18–23, 1997.
- M. Nakao, “A difference inequality and its application to nonlinear evolution equations,” Journal of the Mathematical Society of Japan, vol. 30, no. 4, pp. 747–762, 1978.
- N.-E. Amroun and S. Mimouni, “Asymptotic behaviour of solutions for some weakly dissipative wave equations of p-Laplacian type,” Applied Mathematics E-Notes, vol. 11, pp. 175–183, 2011.
- A. Benaissa and S. Mokeddem, “Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type,” Mathematical Methods in the Applied Sciences, vol. 30, no. 2, pp. 237–247, 2007.
- S. Mokeddem and K. B. W. Mansour, “Asymptotic behaviour of solutions for p-Laplacian wave equation with m-Laplacian dissipation,” Zeitschrift für Analysis und ihre Anwendungen, vol. 33, no. 3, pp. 259–269, 2014.
- Y. Ye, “Global existence and asymptotic behavior of solutions for some nonlinear hyperbolic equation,” Journal of Inequalities and Applications, vol. 2010, Article ID 895121, 10 pages, 2010.
- V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Research in Applied Mathematics, John Wiley, Chichester, UK, 1994.
- P. Martinez, “A new method to decay rate estimates for dissipative systems,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 4, pp. 419–444, 1999.
Copyright © 2015 Soufiane Mokeddem and Khaled Ben Walid Mansour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.