June  2010, 28(2): 649-658. doi: 10.3934/dcds.2010.28.649

Existence of solutions for a semilinear wave equation with non-monotone nonlinearity

1. 

Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States

Received  February 2010 Revised  April 2010 Published  April 2010

For double-periodic and Dirichlet-periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with asymptotically linear nonlinearity, no resonance, and non-monotone nonlinearity when the forcing term is not flat on characteristics. The solutions are in $L^{\infty}$ when the forcing term is in $L^{\infty}$ and continous when the forcing term is continuous. This is in contrast with the results in [4], where the non-enxistence of continuous solutions is established even when forcing term is of class $C^{\infty}$ but is flat on a characteristic.
Citation: Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649
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