Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms

Abstract

The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ \( {{\mathbb{R}}^n} \) , n = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result on the solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain Ω of \( {{\mathbb{R}}^n} \) , n ≥ 1, the initial energy is positive, and the strongly nonlinear functions f₁ and f₂ satisfy the appropriate conditions. The main tool of the proof is based on the methods used by Vitillaro and developed by Said-Houari.