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

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

Pages: 1193 - 1202, Volume 7, Issue 6, December 2014      doi:10.3934/dcdss.2014.7.1193

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José Caicedo - Departmento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia (email)
Alfonso Castro - Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States (email)
Rodrigo Duque - Departmento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia (email)
Arturo Sanjuán - Department of Mathematics, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia (email)

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.

Keywords:  Semilinear wave equation, resonance, flat on characteristic.
Mathematics Subject Classification:  Primary: 35L05, 35L-70; Secondary: 35P30.

Received: April 2013;      Revised: November 2013;      Available Online: June 2014.