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

Guo, Y, Rammaha, MA. (2013). Global existence and decay of energy to systems of wave equations with damping and supercritical sources . ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, 64(3), 621-658. 10.1007/s00033-012-0252-6

cited authors

  • Guo, Y; Rammaha, MA

authors

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 H 1(Ω) × L 2(∂Ω) with boundary data from L 2(∂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 1(Ω) into L 2(Ω) or L 2(∂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. © 2012 Springer Basel AG.

publication date

  • June 1, 2013

published in

Digital Object Identifier (DOI)

start page

  • 621

end page

  • 658

volume

  • 64

issue

  • 3