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A note on linear hyperbolic evolution equations

Published online by Cambridge University Press:  14 November 2011

Hans Grabmüller
Hans Grabmüller
Affiliation:
Institut für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstrasse 3, D-8520 Erlangen, B.R.D.
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Synopsis

Given the linear hyperbolic evolution equation (P0) on a reflexive Banach space, we present a new method for an existence proof of unbounded solutions admitting an exponential growth rate as time tends to infinity. Utilizing abstract Wiener—Hopf techniques, an operational calculus is developed for the construction of the resolving operator associated with the problem under consideration. The results are based upon the fundamental hypothesis that the spectral set of the time-independent mapping A is contained in the interior of a parabola. The distance of the focus from the vertex of this parabola turns out to be a measure for the growth rate. Applicability of the results is shown in the case where A is a non-symmetric perturbation of a self-adjoint partial differential operator.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 3-4 , 1981 , pp. 305 - 318
Copyright
Copyright © Royal Society of Edinburgh 1981

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