Existence of Lp-solutions for a semilinear wave equation with non-monotone nonlinearity
Article - postprint
Harvey Mudd College
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  where the forcing term is assumed to be in L∞, and are in contrast with those in  where the non-existence of continuous solutions is established for C∞ forcing terms flat on characteristics.
© 2014, Discrete and Continuous Dynamical Systems - Series S
(With J. Caicedo, R. Duque and A. Sanjun) “Existence of Lp-solutions for a semilinear wave equation with non-monotone nonlinearity", Discrete Contin. Dyn. Syst. Ser. S 7 (2014), no. 6, 1193-1202