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

TL;DR

This paper establishes the existence of infinitely many large, regular periodic solutions with rational frequency for certain monotone 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 this context.

Findings

Proves existence of infinitely many classical periodic solutions.

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

Provides a new regularity approach for distributional solutions.

Abstract

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