Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study

Abstract

In this work, we consider a viscoelastic wave equation with boundary damping and variable exponents source term. The damping terms and variable exponents are localized on a portion of the boundary. We first, prove the existence of global solutions and then we establish optimal and general decay estimates depending on the relaxation function and the nature of the variable exponent nonlinearity. Finally, we run two numerical tests to demonstrate our theoretical decay results. This study generalizes and enhances existing literature results, and the acquired results are thus of significant importance when compared to previous literature results with constant or variable exponents in the domain.

Keywords

• variable exponent
• lebesgue and sobolev spaces
• boundary feedback
• viscoelasticity
• relaxation functions
• general decay
• finite difference method