Regular generalized solutions to semilinear wave equations

TL;DR

This paper establishes the existence and uniqueness of generalized solutions for semilinear wave equations with small nonlinearities using Colombeau algebras, ensuring global solutions with negligible errors.

Contribution

It introduces a novel approach to solving semilinear wave equations with arbitrary growth nonlinearities via Colombeau algebras and fixed point methods.

Findings

Unique solutions in Colombeau algebra for small nonlinearities

Global classical solutions with rapidly vanishing errors

Applicable in space dimensions 1, 2, 3

Abstract

The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.