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December  2008, 22(4): 835-860. doi: 10.3934/dcds.2008.22.835

Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping

Lorena Bociu 1, and  Irena Lasiecka 1,

1. 

Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States, United States

Received  June 2007 Revised  September 2007 Published  September 2008

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 nonlinear sources on the boundary driven by Neumann boundary conditions. Since Lopatinski condition fails to hold (unless the $\text{dim} (\Omega) = 1$), the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves the strong interaction between the source and the damping. Thus, it is not surprising that existence theory for this class of problems has been established only recently. However, the uniqueness of weak solutions was declared an open problem. The main result in this work is uniqueness of weak solutions. This result is proved for the same (even larger) class of data for which existence theory holds. In addition, we prove that weak solutions are continuously depending on initial data and that the flow corresponding to weak and global solutions is a dynamical system on the finite energy space.
Keywords: boundary source, critical exponents, nonlinear damping, uniqueness, interior source, wave equation.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35L20.
Citation: Lorena Bociu, Irena Lasiecka. Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 835-860. doi: 10.3934/dcds.2008.22.835
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