Global Existence and Decay of Energy to Systems of Wave Equations with Damping and Supercritical Sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H(Ω) × L(∂Ω) with boundary data from L(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H(Ω) into L(Ω) or L(∂Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.