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We prove the existence of infinitely many classical periodic solutions for a class of semilinear wave equations with periodic boundary conditions. Our argument relies on some new estimates for the linear problem with periodic boundary conditions, by combining Littlewood-Paley techniques, the Hausdorff-Young theorem, and a variational formulation due to Rabinowitz. We also develop a new approach to the regularity of the distributional solutions, by employing Gagliardo-Nirenberg estimates.

Multiplicity and regularity of large periodic solutions with rational frequency for a class of semilinear monotone wave equations