Strict Decay Rate for System of Three Nonlinear Wave Equations Depending on the Relaxation Functions - Abouatia, Hiba; Guesmia, Amar; Zennir, Khaled (Journal of Applied Nonlinear Dynamics, 2026)
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 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 Faedo-Galerkin 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.