Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping

TL;DR

This paper proves the existence of finite energy solutions for a semilinear wave equation with interior damping and supercritical source terms, addressing a challenging open problem in nonlinear wave equations.

Contribution

It demonstrates the existence of solutions for super-supercritical source terms in three dimensions, advancing understanding of nonlinear wave equations with high-order nonlinearities.

Findings

Existence of finite energy solutions established for supercritical sources.

Addresses super-supercritical source terms of order |u|^p with p ≥ 5 in 3D.

Contributes to the theory of nonlinear wave equations with high nonlinearities.

Abstract

In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations.