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Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping†

Published online by Cambridge University Press:  14 February 2012

John P. Quinn  and
David L. Russell
John P. Quinn
Affiliation:
Department of Mathematics, University of Toronto
David L. Russell
Affiliation:
Department of Mathematics and Mathematics Research Center, University of Wisconsin
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Synopsis

This report deals with the asymptotic behaviour of solutions of the wave equation in a domain Ω ⊆Rn. The boundary, Γof Ωft consists of two parts. One part reflects all energy while the other part absorbs energy to a degree. If the energy-absorbing part is non-empty we show that the energy tends to zero as t→∞. With stronger assumptions we are able to obtain decay rates for the energy. Certain relationships with controlability are discussed and used to advantage.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 1-2 , 1977 , pp. 97 - 127
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Courant, R. and Hilbert, D.Methods of mathematical physics II. Partial differential equations (New York: Interscience, 1962).Google Scholar
2Dunford, N. and Schwartz, J. T.Linear operators II. Spectral theory (New York: Interscience, 1963).Google Scholar
3Holmgren, E.Über systeme von lineargen partiellen differential-gleichungen. Öfvers. K. Vetenskaps Akad. Förh. 58 (1901), 91–103.Google Scholar
4Hale, J. K.Dynamical systems and stability. J. Math. Anal. Appl. 26 (1969), 39–59.Google Scholar
5Hörmander, L.Linear partial differential operators (Berlin: Springer, 1963).CrossRefGoogle Scholar
6John, F.On linear partial differential equations with analytic coefficients—unique continuation of data. Comm. Pure Appl. Math. 2 (1949), 209–253.CrossRefGoogle Scholar
7Lax, P. D., Morawetz, C. S. and Phillips, R. S.Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle. Comm. Pure Appl. Math. 16 (1963), 477–486.CrossRefGoogle Scholar
8Lax, P. D. and Phillips, R. S.Scattering theory (New York: Academic Press, 1967).Google Scholar
9Lions, J. L. and Magenes, E.Problèmes aux limites non-homogenes et applications I (Paris: Dunod, 1968).Google Scholar
10Lions, J. L. and Magenes, E.Problèmes aux limites non-homogenes et applications II (Paris: Dunod, 1968).Google Scholar
11Morawetz, C. S.The decay of solutions of the exterior initial-boundary value problem for the wave equation Comm. Pure Appl. Math. 14 (1961), 561–568.CrossRefGoogle Scholar
12Morawetz, C. S.Exponential decay of solutions of the wave equation. Comm. Pure Appl. Math. 19 (1966), 439–444.Google Scholar
13Phillips, R. S.Dissipative hyperbolic systems. Trans. Amer. Math. Soc. 86 (1957), 109–173.Google Scholar
14Rabinowitz, P. H.Periodic solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math. 20 (1967), 145–204.CrossRefGoogle Scholar
15Ralston, J. V.Solutions of the wave equation with localized energy. Comm. Pure Appl. Math. 22 (1969), 807–823.Google Scholar
16Rauch, J. and Taylor, M.Penetrations into shadow regions and unique continuation properties in hyperbolic mixed problems. Indiana Univ. Math. J. 22 (1972), 277–285.Google Scholar
17Rauch, J. and Taylor, M.Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974), 49–86.CrossRefGoogle Scholar
18Russell, D. L.Boundary value control of the higher dimensional wave equation. SIAM J.Control. 9 (1971), 29–42.CrossRefGoogle Scholar
19Russell, D. L.Boundary value control of the higher dimensional wave equation II. SIAM J. Control 9 (1971), 401–419.Google Scholar
20Russell, D. L.A unified boundary value controllability theory for hyperbolic and parabolic partial differential equations. Studies in Appl. Math. 52 (1973), 189–211.CrossRefGoogle Scholar
21Russell, D. L. Exact boundary value controllability theorems for wave and heat processes in star–complemented regions. In Differential games and control theory. Eds. Roxin, E. O., Liu, P.–T. and Sternberg, R. L. (New York: Dekker, 1974).Google Scholar
22Russell, D. L.Decay rates for weakly damped systems in Hilbert space obtained with controltheoretic methods. J. Differential Equations, 19 (1975), 344–370.Google Scholar
23Slemrod, M. Stabilization of boundary control systems. J. Differential Equations, to appear.Google Scholar