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

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Article - postprint


Harvey Mudd College

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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 Lp, (p≥2), when the forcing term is in Lp. This is optimal; even in the linear case there are Lp forcing terms for which the solutions are only in Lp. Our results extend those in [9] where the forcing term is assumed to be in L∞, and are in contrast with those in [6] where the non-existence of continuous solutions is established for C∞ forcing terms flat on characteristics.

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© 2014, Discrete and Continuous Dynamical Systems - Series S