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Decaying and nondecaying properties of the local energy of an elastic wave outside an obstacle. (English) Zbl 0696.73017

The decaying mode of the local energy of the solution of an initial boundary value problem has been investigated for the region exterior to an obstacle for the equations of elasticity for large times. Two types of boundary conditions are considered. One is Neumann boundary condition and the other is Robin boundary condition. It is shown that Rayleigh’s surface wave prevents the exponential decay of the local energy in the case of Neumann boundary condition and the local energy blows up at the rate of the square of time variable in the case of Robin boundary conditions.
Reviewer: P.Puri

MSC:

74J99 Waves in solid mechanics
74J15 Surface waves in solid mechanics
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[1] F. Asakura, On the Green’s function for {\(\Delta\)}2 with the boundary condition of the third kind in the exterior domain of a bounded obstacle. J. Math. Kyoto Univ.,18 (1978), 615–625. · Zbl 0389.35014 · doi:10.1215/kjm/1250522512
[2] R. Gregory, The propagation of Rayleigh waves over curved surfaces at high frequency. Proc. Cambridge Philos. Soc.,70 (1971), 103–121. · Zbl 0218.73036 · doi:10.1017/S0305004100049720
[3] H. Iwashita and Y. Shibata, Exponential decay of the local energy for the elastic wave in the exterior domain outside a strictly convex cavity. (To appear)
[4] V. Kupradze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North Holland, Amsterdam, 1979.
[5] S. Mizohata, The Theory of Partial Differential Equations. Cambridge Univ. Press, Cambridge, 1973. · Zbl 0263.35001
[6] C. Morawetz, Exponential decay for solutions of the wave equation. Comm. Pure Appl. Math.,19 (1966), 439–444. · Zbl 0161.08002 · doi:10.1002/cpa.3160190407
[7] P. Morse and H. Feshbach, Methods of Theoretical Physics. McGrow-Hill. New York, 1953. · Zbl 0051.40603
[8] F. Olver, Asymptotics and Special Functions. Academic Press, New York and London, 1974. · Zbl 0303.41035
[9] M. Taylor, Rayleigh waves in linear elasticity as a propagation of singularities phenomenon. Proc. Conf. on PDE and Geometry, Marcel Dekker, New York, 1979, 273–291.
[10] T. Tokita, Exponential decay of solutions for the wave equation in the exterior domain with spherical boundary. J. Math. Kyoto Univ.,12 (1972), 413–430. · Zbl 0242.35049 · doi:10.1215/kjm/1250523528
[11] C. Wilcox, The domain of dependence inequality and initial-boundary value problems for symmetric hyperbolic system. MRC Technical Summary Report #333, Univ. Wisconsin, 1962.
[12] K. Yamamoto, Theorems on singularities of solutions to systems of differential equations. (To appear)
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