Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations

Abstract

In this work, we consider an initial-boundary value problem related to the nonlinear coupled viscoelastic equations \[ \left\{ \begin{array}{c} \left\vert u_{t}\right\vert ^{j}u_{tt}-\Delta u_{tt}-div\left( \left\vert \nabla u\right\vert ^{\alpha -2}\nabla u\right) -\Delta u+\int\limits_{0}^{t}g\left( t-s\right) \Delta uds+\left\vert u_{t}\right\vert ^{m-1}u_{t}=f_{1}\left( u,v\right) ,\text{ } \\ \left\vert v_{t}\right\vert ^{j}v_{tt}-\Delta v_{tt}-div\left( \left\vert \nabla v\right\vert ^{\beta -2}\nabla v\right) -\Delta v+\int\limits_{0}^{t}h\left( t-s\right) \Delta vds+\left\vert v_{t}\right\vert ^{r-1}v_{t}=f_{2}\left( u,v\right) .\text{ } \end{array} \right. \] We will show the exponential growth of solutions with positive initial energy.