The Multidimensional Damped Wave Equation: Maximal Weak Solutions for Nonlinear Forcing via Semigroups and Approximation

TL;DR

This paper develops a semigroup and eigenfunction approximation framework to analyze maximal weak solutions of the multidimensional damped nonlinear wave equation with nonlinear forcing, extending classical methods to nonlinear and operator forcing scenarios.

Contribution

It introduces a novel approach using semigroup theory and eigenfunction approximation to establish maximal weak solutions for nonlinear damped wave equations with operator forcing.

Findings

Established boundedness of the semigroup on the generator domain for all t > 0.

Extended the semigroup convolution formula to weak solutions over arbitrary time intervals.

Developed a maximal solution framework incorporating nonlinear and operator forcing without fixed point methods.

Abstract

The damped nonlinear wave equation, also known as the nonlinear telegraph equation, is studied within the framework of semigroups and eigenfunction approximation. The linear semigroup assumes a central role: it is bounded on the domain of its generator for all time t > 0. This permits eigenfunction approximation within the semigroup framework as a tool for the study of weak solutions. The semigroup convolution formula, known to be rigorous on the generator domain, is extended to the interpretation of weak solution on an arbitrary time interval. A separate approximation theory can be developed by using the invariance of the semigroup on eigenspaces of the Laplacian as the system evolves. For (locally) bounded continuous L^2 forcing, this permits a natural derivation of a maximal solution, which can logically include a constraint on the solution as well. Operator forcing allows for the incorporation of concurrent physical processes. A significant feature of the proof in the nonlinear case is verification of successive approximation without standard fixed point analysis.