Wissem Boughamsa, Amar Ouaoua

Global Existence and General Decay of Solution for a Nonlinear Wave Equation with Variable Exponents and Memory Term

In this paper, we consider the following wave equation: \[ u_{tt}-\Delta u-\Delta u_{tt}+\int\limits_{0}^{t}g(t-s) \Delta u( s)\,ds+|u_{t}|^{m( x) -2}u_{t}=b|u| ^{p( x)-2}u. \]
First, we prove that the equation has a unique local solution for a suitable conditions by using Faedo--Galerkin methods, and we also prove that the local solution is global in time. Finally, we demonstrate that the solution with positive initial energy decays exponentially.

Mathematics Subject Classification: 35B40, 35L70, 35L10

Key words and phrases: Wave equation, variable exponents, memory term, global existence, general decay