Existence of solutions for asemilinear wave equation with non-monotone nonlinearity

Alfonso Castro; Benjamin Preskill

doi:10.3934/dcds.2010.28.649

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2010, Volume 28, Issue 2: 649-658. Doi: 10.3934/dcds.2010.28.649 open access

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

Alfonso Castro^1, and
Benjamin Preskill^1,

1.
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711

Received: February 2010

Revised: April 2010

Published: July 2010

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Abstract

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 isflat on a characteristic.

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Mathematics Subject Classification: Primary: 35L71; Secondary: 47H09, 47H10.

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