On the strongly damped wave equation with constraint

TL;DR

This paper introduces a weak formulation for a semilinear strongly damped wave equation with constraints, utilizing duality techniques to establish a global existence result under finite energy initial conditions.

Contribution

It develops a novel weak formulation using duality in Sobolev-Bochner spaces for the constrained wave equation, enabling proof of global existence.

Findings

Existence of solutions under finite energy initial data

Use of duality techniques for constraint relaxation

Framework applicable to semilinear strongly damped wave equations

Abstract

A weak formulation for the so-called "semilinear strongly damped wave equation with constraint" is introduced and a corresponding notion of solution is defined. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable "relaxation" of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite "physical" energy.