Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations

Yıl 2019, Cilt: 2 Sayı: 2, 70 - 78, 28.06.2019

Erhan Pişkin , Şeyhmus Altındağ

https://doi.org/10.32323/ujma.484532

Öz

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.

Anahtar Kelimeler

Viscoelastic wave equations, Exponential growth, Nonlinear damping term, Viscoelastic wave equations

Kaynakça

• [1] W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal., 71 (2009) 2257-2267.
• [2] B.S. Houari, Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, Z. Angew. Math. Phys., 62 (2011) 115-133.
• [3] X. Han , M. Wang, Global existence and blow-up solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal., 71 (2009) 5427-5450.
• [4] S.A. Messaoudi , B.S. Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010) 277-287.
• [5] E. Pişkin, A lower bound for the blow up time of a system of viscoelastic wave equations with nonlinear damping and source terms, J. Nonlinear Funct. Anal., 2017 (2017) 1-9.
• [6] E. Pişkin, Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms, Malaya Journal of Matematik, 3(2) (2015) 168-174.
• [7] B.S. Houari, S.A. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wavw equations, Nonlinear Differ. Equ. Appl., 18 (2011) 659-684.
• [8] Y. Zhao , Q. Wang, Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms, Boundary Value Problems, 35 (2018) 1-13.
• [9] J. Hao, S. Niu, H. Men, Global nonexistence of solutions for nonlinear coupled viscoelastic wave equations with damping and source terms, Boundary Value Problems, 250 (2014) 1-11.
• [10] L. Fei, G. Hongjun, Global nonexistence of positive initial energy solutions for coupled nonlinear wave equations with damping and source terms, Abst. Appl. Anal., 2011 (2011) 1-14.
• [11] J. Hao, L. Cai, Global existence and blow up of solutions for nonlinear coupled wave equations with viscoelastic terms, Math. Meth. Appl. Sci., 39 (2016) 1977-1989.