Global non‐existence of solutions of a class of wave equations with non‐linear damping and source terms

In this paper we consider the non‐linear wave equation$${u_{tt}}-\Delta u_t-{\rm div}({\vert\nabla u\vert^{\alpha-2}\nabla u})-{\rm div}({\vert\nabla u_t\vert^{\beta-2}\nabla u_t})+{a\vert u_t\vert^{m-2}u_t=b\vert u\vert^{p-2}u}$$a,b >0, associated with initial and Dirichlet boundary conditions. We prove, under suitable conditions on α,β,m,p and for negative initial energy, a global non‐existence theorem. This improves a result by Yang ( Math. Meth. Appl. Sci. 2002; 25 :825–833), who requires that the initial energy be sufficiently negative and relates the global non‐existence of solutions to the size of Ω. Copyright © 2004 John Wiley & Sons, Ltd.