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Eigenvalue problems for the wave equation with strong damping

Published online by Cambridge University Press:  14 November 2011

Pedro Freitas
Pedro Freitas
Affiliation:
Departamento de Matemática, Instituto Superior Tácnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal e-mail: [email protected]
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Abstract

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 755 - 771
Copyright
Copyright © Royal Society of Edinburgh 1997

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