Existence for a nonlinear wave equation with damping and source terms

Abstract

utt −∆u+Q(t, x, ut) = f(x, u), u(0) = u0, ut(0) = u1, (1.1) in (0, T ) × Rn, where 0 < T ≤ ∞, n ≥ 1. The data u0 and u1 are in the energy space and are compactly supported in Rn. The terms Q and f are nonlinear damping and source terms respectively. Problems like (1.1) have been studied in many papers. For the case Q ≡ 0 see [3], [7], [11], [12], [20], [30]; for the case f ≡ 0 see [8], [10]; when both terms Q and f interact, see [6], [9], [13], [14], [15], [18], [19], [25], [26], [27], [31], [32], [33]. For the sake of simplicity we illustrate our results by considering the model power–like equation with damping term