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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 835 - 860, Volume 22, Issue 4, December 2008      doi:10.3934/dcds.2008.22.835

 
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Lorena Bociu - Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States (email)
Irena Lasiecka - Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States (email)

Abstract: 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:  wave equation, uniqueness, nonlinear damping, boundary source, interior source, critical exponents
Mathematics Subject Classification:  Primary: 35L05; Secondary: 35L20

Received: June 2007;      Revised: September 2007;      Published: September 2008.