Blow-up of Positive Initial Energy Solutions for A System of Nonlinear Wave Equations with Supercritical Sources

Abstract

The goal of this paper is to investigate the finite time blow-up of solutions with supercritical boundary/interior sources and nonlinear boundary/interior damping. First, we prove that if the interior and boundary sources dominate their corresponding damping terms, then every weak solution blows up in finite time with positive initial energy. Second, without any restriction on the boundary source, we prove the finite time blow-up of solutions, provided that the interior sources dominate both interior and boundary damping and the initial energy is nonnegative. A similar result has been shown when the boundary source is absent. Moreover, in the absence of the interior sources, we prove that the solution grows as an exponential function.