**Wissem Boughamsa, Amar Ouaoua**

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

**abstract:**

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