Control of blow-up singularities for nonlinear wave equations

TL;DR

This paper demonstrates that, despite the general impossibility of boundary control for blow-up nonlinear wave equations, it is possible to control small data solutions to blow up on a specific set away from the boundary using Fuchsian reduction.

Contribution

It introduces a novel control method for nonlinear wave equations with blow-up, utilizing Fuchsian reduction to prescribe blow-up sets for small data solutions.

Findings

Control of blow-up solutions on prescribed sets is achievable.

Fuchsian reduction effectively constructs solutions with desired blow-up behavior.

Method has potential for broader application in nonlinear wave control.

Abstract

While the global boundary control of nonlinear wave equations that exhibit blow-up is generally impossible, we show on a typical example, motivated by laser breakdown, that it is possible to control solutions with small data so that they blow up on a prescribed compact set bounded away from the boundary of the domain. This is achieved using the representation of singular solutions with prescribed blow-up surface given by Fuchsian reduction. We outline on this example simple methods that may be of wider applicability.