Control of blow-up singularities for nonlinear wave equations

Satyanad Kichenassamy

doi:10.3934/eect.2013.2.669

Article Contents

2013, Volume 2, Issue 4: 669-677. Doi: 10.3934/eect.2013.2.669

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Control of blow-up singularities for nonlinear wave equations

Satyanad Kichenassamy^1,

1.
Laboratoire de Mathématiques, Université de Reims Champagne-Ardenne, Moulin de la Housse, B.P. 1039, F-51687 Reims Cedex 2

Received: October 2012

Revised: August 2013

Published: December 2013

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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.

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Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 35L71.

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