Local energy decay for the damped plate equation in exterior domains

Jiang, Song

doi:10.1090/qam/1162275

Author: Song Jiang
Journal: Quart. Appl. Math. 50 (1992), 257-272
MSC: Primary 35C15; Secondary 35L20, 43A50, 73F15, 73K10
DOI: https://doi.org/10.1090/qam/1162275
MathSciNet review: MR1162275
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish a rate of local energy decay for solutions of the damped plate equation with variable coefficients in exterior domains by using the spectral analysis to the corresponding stationary problem.

Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
H. Iwashita and Y. Shibata, On the analyticity of spectral functions for some exterior boundary value problems, Glas. Mat. Ser. III 23(43) (1988), no. 2, 291–313 (English, with Serbo-Croatian summary). MR 1012030
Song Jiang, $L^p$–$L^q$ estimates for solutions to the damped plate equation in exterior domains, Results Math. 18 (1990), no. 3-4, 231–253. MR 1078420, DOI https://doi.org/10.1007/BF03323168
S. Klainerman and Gustavo Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085, DOI https://doi.org/10.1002/cpa.3160360106
Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971
Tosinobu Muramatu, On Besov spaces and Sobolev spaces of generalized functions definded on a general region, Publ. Res. Inst. Math. Sci. 9 (1973/74), 325–396. MR 0341063, DOI https://doi.org/10.2977/prims/1195192564
Gustavo Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985), no. 5, 399–418. MR 785713, DOI https://doi.org/10.1016/0362-546X%2885%2990001-X
Reinhard Racke, Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math. 412 (1990), 1–19. MR 1078997, DOI https://doi.org/10.1515/crll.1990.412.1
Yoshihiro Shibata, On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain, Tsukuba J. Math. 7 (1983), no. 1, 1–68. MR 703667, DOI https://doi.org/10.21099/tkbjm/1496159662
Yoshihiro Shibata and Yoshio Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191 (1986), no. 2, 165–199. MR 818663, DOI https://doi.org/10.1007/BF01164023
B. R. Vaĭnberg, The analytic properties of the resolvent for a certain class of bundles of operators, Mat. Sb. (N.S.) 77 (119) (1968), 259–296 (Russian). MR 0240447

B. R. Vainberg, On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behaviour for large time of solutions of nonstationary problems, Math. USSR-Sb. 21, 221–239 (1973)

B. R. Vaĭnberg, The short-wave asymptotic behavior of the solutions of stationary problems, and the asymptotic behavior as $t\rightarrow \infty $ of the solutions of nonstationary problems, Uspehi Mat. Nauk 30 (1975), no. 2(182), 3–55 (Russian). MR 0415085
Calvin H. Wilcox, Scattering theory for the d’Alembert equation in exterior domains, Lecture Notes in Mathematics, Vol. 442, Springer-Verlag, Berlin-New York, 1975. MR 0460927