Journal of Applied Nonlinear Dynamics
Strict Decay Rate for System of Three Nonlinear Wave Equations Depending on the Relaxation Functions
Journal of Applied Nonlinear Dynamics 11(2) (2022) 309321  DOI:10.5890/JAND.2022.06.004
Hiba Abouatia$^1$, Amar Guesmia$^2$, Khaled Zennir$^{1,3}$
$^1$ Laboratoire de Math\'ematiques Appliqu\'ees et de Mod\'elisation,
D\'epartement de Math\'ematiques, Universit\'e 8
Mai 1945 Guelma. B.P. 401 Guelma 24000 Alg\'erie
$^2$ Department of Mathematics, University 20 Aout 1955, Skikda, Algeria
$^3$ Department of Mathematics, College of Sciences
and Arts, Qassim University, ArRass,
Saudi Arabia
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Abstract
The main aim of this article is to study the decay rate of a system of three semilinear wave equations with strong external forces in $\mathbb{R}^n$, including damping terms of memory type with past history which is very important problem from the point of view of application in sciences and engineering. We work in a weighted phase spaces where the problem is well defined and deduce a decay result depending on the relaxation functions. Using the FaedoGalerkin method and some energy estimates, we prove the existence of global solution owing to to the weighted function. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincar\'e's inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in [1] and [2] for a single equation and [3] for coupled system to the case of a system of three equations. The work is relevant in the sense that the problem is more complex than what can be found in the literature. However, the techniques involved in order to study this generalization is a combination of the techniques used in [1] in order to deal with the memory and weighted spaces with standard techniques in order to deal with coupled system with nonlinearities.
Acknowledgments
The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.
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