Blowup and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms

Abstract

This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation utt - Δu + |u|kj′(ut) = |u|p-1</sup>u in Ω x (0, T), where p > 1 and j′ denotes the derivative of a C1 convex and real value function j. We prove that every weak solution is asymptotically stability, for every m is such that 0 < m < 1, p < k + m and the initial energy is small; the solutions blow up in finite time, whenever p > k + m and the initial data is positive, but appropriately bounded.