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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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References

1Agmon, S.. Lectures on elliptic boundary value problems (Princeton, N. J.: Van Nostranad, 1965).Google Scholar
2Balakrishnan, A. V.. Fractional powers of closed operators and the semigroup generated by them. Pacific J. Math. 10 (1960), 419–437.Google Scholar
3Carleman, T.. Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. Nat. Kl. 88 (1936), 119–132.Google Scholar
4DaPrato, G. and Giusti, E.. Una caratterizzazione dei generatori di funzioni coseno astratte. Boll. Un. Mat. Ital. 22 (1967), 357–362.Google Scholar
5DaPrato, G. and Grisvard, P.. Sommes d'opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl. 54 (1975), 305–387.Google Scholar
6Fattorini, H.. Ordinary differential equations in linear topological spaces, I. J. Differential Equations 5 (1968), 72–105.Google Scholar
7Feldman, I. A.. The radiative transport equation and operator Wiener-Hopf equations. Funkcional Anal, i Priložen. 5 (1971), 106–108, and Functional Anal. Appl. 5 (1971), 262-264.Google Scholar
8Grabmüller, H.. Hyperbolic integro-differential equations of convolution type. J. Integral Equations and Operator Theory 2/3 (1979), 302–343.Google Scholar
9Hille, E. and Phillips, R. S.. Functional analysis and semi-groups. Amer. Math. Soc. Coll. Publ., vol. 31. (Providence: Amer. Math. Soc., 1957).Google Scholar
10Krein, M. G.. Integral equations on a half line with kernel depending upon the difference of the arguments. Uspehi Mat. Nauk 13 (1958), 3–120, and Amer. Math. Soc. Transl. 22 (1962), 163-288.Google Scholar
11Sova, M.. Cosine operator functions. Rozprawy Mat. 49 (1966), 1–47.Google Scholar
12Taylor, A. E.. Introduction to functional analysis (New York: Wiley, 1958)Google Scholar
13Travis, C. C. and Webb, G. F.. Second order differential equations in Banach space. In Nonlinear equations in abstract spaces, Ed. Lakshmikantham, V., 331–361 (New York: Academic Press 1978).Google Scholar
14Yosida, K.. An operator-theoretical integration of the wave equation. J. Math. Soc. Japan 8 (1956), 79–92.Google Scholar
15Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1966).Google Scholar