Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity

José Caicedo; Alfonso Castro; Rodrigo Duque; Arturo Sanjuán

doi:10.3934/dcdss.2014.7.1193

Article Contents

2014, Volume 7, Issue 6: 1193-1202. Doi: 10.3934/dcdss.2014.7.1193

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Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity

1.
Departmento de Matemáticas, Universidad Nacional de Colombia, Bogotá
2.
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711
3.
Department of Mathematics, Universidad Distrital Francisco José de Caldas, Bogotá

Received: March 31, 2013

Revised: October 31, 2013

Published: November 2014

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Abstract

For Dirichlet-periodic and double periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with large forcing terms not flat on characteristics. The nonlinearity is assumed to be non-monotone, asymptotically linear, and not resonanant. We prove that the solutions are in $L^{p}$, $(p\geq 2)$, when the forcing term is in $L^{p}$. This is optimal; even in the linear case there are $L^p$ forcing terms for which the solutions are only in $L^p$. Our results extend those in [9] where the forcing term is assumed to be in $L_{\infty}$, and are in contrast with those in [6] where the non-existence of continuous solutions is established for $C^{\infty}$ forcing terms flat on characteristics. 200 words.

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Mathematics Subject Classification: Primary: 35L05, 35L-70; Secondary: 35P30.

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References

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