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

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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 $L^{p}$, $(p\geq 2)$, 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.
    Mathematics Subject Classification: Primary: 35L05, 35L-70; Secondary: 35P30.


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