Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations

Authors : Erhan PİŞKİN, Şeyhmus ALTINDAĞ

Pages : 70-78

Doi:10.32323/ujma.484532

View : 18 | Download : 17

Publication Date : 2019-06-28

Article Type : Research Paper

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