Zeitschrift für angewandte Mathematik und Physik

, Volume 64, Issue 3, pp 621–658

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

Article

DOI: 10.1007/s00033-012-0252-6

Cite this article as:
Guo, Y. & Rammaha, M.A. Z. Angew. Math. Phys. (2013) 64: 621. doi:10.1007/s00033-012-0252-6

Abstract

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 H1(Ω) × L2(∂Ω) with boundary data from L2(∂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 H1(Ω) into L2(Ω) or L2(∂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.

Mathematics Subject Classification (2010)

Primary 35L0535L20Secondary 58J45

Keywords

Wave equationsDamping and source termsPotential wellNehari manifoldGlobal existenceEnergy decay ratesBlow up

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska–LincolnLincolnUSA