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

TL;DR

This paper establishes the existence of infinitely many classical periodic solutions with rational frequency for certain semilinear wave equations, using advanced harmonic analysis and variational methods.

Contribution

It introduces new estimates for linear periodic problems and a novel approach to regularity of distributional solutions in semilinear wave equations.

Findings

Proves existence of infinitely many classical periodic solutions

Develops new estimates combining Littlewood-Paley and Hausdorff-Young techniques

Employs Gagliardo-Nirenberg estimates for regularity analysis

Abstract

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.