Existence, Uniqueness and Blow-Up of Solutions to Wave Equations with Supercritical Boundary/Interior Sources and Damping
by
Lorena Bociu
University of Nebraska-Lincoln
Coauthors: Irena Lasiecka

We consider finite energy solutions of a wave equation with supercritical nonlinear sources and nonlinear damping. A distinct feature of the model under consideration is the presence of the double interaction of source and damping, both in the interior of the domain and on the boundary. Moreover, we consider nonlinear sources on the boundary driven by Neumann boundary conditions. Since Lopatinski condition fails to hold (unless the dim( W) = 1), the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves strong interaction between the source and the damping. We provide positive answers to the questions of local existence and uniqueness of weak solutions and moreover we give complete and sharp description of parameters corresponding to global existence and blow-up of solutions in finite time.

Date received: July 13, 2008