Instability of coupled systems with delay

Reinhard Racke

doi:10.3934/cpaa.2012.11.1753

Article Contents

2012, Volume 11, Issue 5: 1753-1773. Doi: 10.3934/cpaa.2012.11.1753

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Instability of coupled systems with delay

Reinhard Racke^1,

1.
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz

Received: January 2011

Revised: July 2011

Published: September 2012

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Abstract

We consider linear initial-boundary value problems that are a coupling like second-order thermoelasticity, or the thermoelastic plate equation or its generalization (the $\alpha$-$\beta$-system introduced in [1, 26]). Now, there is a delay term given in part of the coupled system, and we demonstrate that the expected inherent damping will not prevent the system from not being stable; indeed, the systems will shown to be ill-posed: a sequence of bounded initial data may lead to exploding solutions (at any fixed time).

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Mathematics Subject Classification: Primary: 35 B, 35K20, 35K55, 35L20, 35Q, 35R25; Secondary: 80A20.

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