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Abstract wave equations with acoustic boundary conditions. (English) Zbl 1109.47035
The paper is devoted to the study of wave equations equipped with time-dependent acoustic (or absorbing) boundary conditions in bounded domains in \(\mathbb{R}^n\). The author discusses the proper setting for the problem, proves a well-posedness result for the abstract initial value problem and deals with some spectral properties of an operator matrix involved. Concrete problems are also discussed.

47D06 One-parameter semigroups and linear evolution equations
47H20 Semigroups of nonlinear operators
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
34G10 Linear differential equations in abstract spaces
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35P05 General topics in linear spectral theory for PDEs
Full Text: DOI arXiv
[1] , , and , Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics Vol. 96 (Birkhäuser, Basel, 2001). · Zbl 0978.34001
[2] Beale, Indiana Univ. Math. J. 25 pp 895– (1976)
[3] Beale, Bull. Amer. Math. Soc. (N. S.) 80 pp 1276– (1974)
[4] Wave propagation in the ice-covered ocean wave guide and operator polynomials, in: Proceedings of the Second ISAAC Congress, edited by H. G. W. Begehr, R. P. Gilbert, and J. Kajiwara (Kluwer Academic Publishers, Dordrecht, 2000), pp. 1319-1333. · Zbl 1163.86304
[5] Casarino, Integral Equations Operator Theory 47 pp 289– (2003)
[6] Casarino, Discrete Contin. Dyn. Syst. Ser. B 12 pp 761– (2005)
[7] and , Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1 (Springer-Verlag, Berlin, 1988).
[8] and , Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 2 (Springer-Verlag, Berlin, 1990).
[9] Engel, Semigroup Forum 58 pp 267– (1999)
[10] and , One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics Vol. 194 (Springer-Verlag, Berlin, 2000). · Zbl 0952.47036
[11] Second Order Linear Differential Equations in Banach Spaces, Mathematics Studies Vol. 108 (North- Holland, Amsterdam, 1985).
[12] Ph. D. thesis, in preparation.
[13] Gal, J. Evol. Equ. 3 pp 623– (2003)
[14] and , Linear and semilinear boundary conditions: the analytic case, in: Semigroup Theory and Evolution Equations, Proceedings of the Second International Conference, Delft, 1989, edited by Ph. Clément, E. Mitidieri, and B. de Pagter, Lecture Notes in Pure and Applied Mathematics Vol. 135 (Marcel Dekker, 1991), pp. 193-211.
[15] Perturbation Theory for Linear Operators, Classics in Mathematics (Springer-Verlag, Berlin, 1995).
[16] , and , Semigroups for initial-boundary value problems, in: Evolution Equations 2000: Applications to Physics, Industry, Life Sciences and Economics (Proceedings Levico Terme 2000), edited byM. Iannelli and G. Lumer, Progress in Nonlinear Differential Equations Vol. 55 (Birkhäuser, Basel, 2003), pp. 277-297.
[17] Krasil’nikov, J. Appl. Math. Mech. 25 pp 1134– (1961)
[18] and , Non-homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren der mathematischen Wissenschaften Vol. 181 (Springer-Verlag, Berlin, 1972).
[19] and , Non-homogeneous Boundary Value Problems and Applications. Vol. II, Grundlehren der mathematischen Wissenschaften Vol. 182 (Springer-Verlag, Berlin, 1972).
[20] Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications Vol. 16 (Birkhäuser, Basel, 1995).
[21] Partial Differential Equations (Mir Publishers, Moscow, 1978).
[22] and , Theoretical Acoustics (McGraw-Hill, New York, 1968).
[23] Propst, J. Integral Equations Appl. 8 pp 99– (1996)
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