Abstract
Making use of the theory of Wiener-Hopf operators in the scale of abstract Krein spaces, we prove existence and uniqueness of unbounded solutions for the linear hyperbolic integrodifferential equation (Po). We extend herewith results obtained in [8] for hyperbolic evolution equations, where the convolution integral was absent. The method utilizes Dunford's functional calculus and permits thus a constructive existence proof for solutions exhibiting an exponential growth rate when time increases. Our approach bases upon the fundamental hypothesis that the spectrum of the time-independent mapping -A shows a parabolic condensation along the negative real axis. This condition completely determines the admissible geometry of the spectral set of the convolution integral operator, and a fortiori the magnitude of the exponential growth rate. The theory works in arbitrary reflexive Banach spaces.
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References
Agmon, S.: Lectures on elliptic boundary value problems. Princeton, New Jersey, Van Nostrand 1965.
Carleman, T.: Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. nat. Kl.88 (1936), 119 - 132.
Firl, R.: Monotone Integrodifferentialoperatoren und variationelle Lösungen für eine Klasse nichtlinearer Integrodifferentialgleichungen. Dissertation. Essen, Universität 1979.
Gerlach, E.: Zur Theorie einer Klasse von Integrodifferentialgleichungen. Dissertation. Berlin, Technische Universität 1969.
Gochberg, I.Z. and I.A. Feldman: Faltungsgleichungen und Projektionsverfahren zu ihrer Lösung. Basel, Birkhäuser 1974.
Grabmüller, H.: On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem. Proc. Royal Soc. Edinburgh76A (1976), 119 - 137.
Grabmüller, H.: Singular perturbation techniques applied to integro-differential equations. Research Notes in Mathematics vol.20. London, Pitman Publ. 1978.
Grabmüller, H.: A note on linear hyperbolic evolution equations. To appear.
Hille, E. and R.S. Phillips: Functional analysis and semigroups. Amer. Math. Soc. Coll. Publ. vol. 31, 2nd ed. Providence, AMS 1957.
Krein, M.G.: Integral equations on a half-line with kernel depending upon the difference of the arguments. Uspehi Mat. Nauk (N.S.)13 (1958), 3 - 120 (= Amer. Math. Soc. Transl. Ser. 2,22 (1962), 163 – 288).
Miller, R.K. and R.L. Wheeler: Asymptotic behavior for a linear Volterra integral equation in Hilbert space. J. Differential Equations23 (1977), 270 - 284.
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Grabmüller, H. Hyperbolic integro-differential equations of convolution type. Integr equ oper theory 2, 302–343 (1979). https://doi.org/10.1007/BF01682673
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DOI: https://doi.org/10.1007/BF01682673