Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

Abstract

Here a; b?0 and p?1, m?1. In case of IBVP, in a bounded domain ⊂Rn with Dirichlet boundary conditions, the following results are known: 1. When a=0, it is proved (see [1, 3, 8, 14, 16]) that the solution blows up in nite time for su ciently large initial data. 2. When b=0; Haraux and Zuazua [5] and Kopackova [7] prove the global existence result for large initial data. The behavior of the solution of Eq. (1.1) with nonlinear source and linear damping (case m=1) in an abstract setting was considered by Levine in [9]. More precisely, he showed that the solutions with negative initial energy are nonglobal.