Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A damped hyperbolic equation on thin domains
HTML articles powered by AMS MathViewer

by Jack K. Hale and Geneviève Raugel
Trans. Amer. Math. Soc. 329 (1992), 185-219
DOI: https://doi.org/10.1090/S0002-9947-1992-1040261-1
PDF | Request permission

Abstract:

For a damped hyperbolic equation in a thin domain in ${{\mathbf {R}}^3}$ over a bounded smooth domain in ${{\mathbf {R}}^2}$, it is proved that the global attractors are upper semicontinuous. It is shown also that a global attractor exists in the case of the critical Sobolev exponent.
References
  • Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
  • A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl. (9) 62 (1983), no. 4, 441–491 (1984). MR 735932
    • —, Attracteurs maximaux dans les équations aux dérivées partielles, Séminaire du Collège de France, Pitman, Boston, Mass., 1985.
  • Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
  • Monique Dauge, Problèmes de Neumann et de Dirichlet sur un polyèdre dans $\textbf {R}^3$: régularité dans des espaces de Sobolev $L_p$, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 1, 27–32 (French, with English summary). MR 954085
  • Monique Dauge, Problèmes mixtes pour le laplacien dans des domaines polyédraux courbes, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 8, 553–558 (French, with English summary). MR 1053276
  • J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl. (9) 66 (1987), no. 3, 273–319. MR 913856
  • P. Grisvard, Alternative de Fredholm relative au problème de Dirichlet dans un polyèdre, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 3, 359–388 (English, with Italian summary). MR 397149
    • —, Elliptic problems in nonsmooth domains, Monographs and Studies in Math., vol. 24, Pitman, 1985.
  • J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, Nonlinear differential equations (Granada, 1984) Res. Notes in Math., vol. 132, Pitman, Boston, MA, 1985, pp. 1–42. MR 908897
  • Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
  • Jack K. Hale and Geneviève Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl. (9) 71 (1992), no. 1, 33–95. MR 1151557
  • Jack K. Hale and Geneviève Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations 73 (1988), no. 2, 197–214. MR 943939, DOI 10.1016/0022-0396(88)90104-0
  • A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983–1984) Res. Notes in Math., vol. 122, Pitman, Boston, MA, 1985, pp. 6, 161–179 (English, with French summary). MR 879461
  • Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • Tosio Kato, Fractional powers of dissipative operators. II, J. Math. Soc. Japan 14 (1962), 242–248. MR 151868, DOI 10.2969/jmsj/01420242
  • J.-L. Lions, Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs, J. Math. Soc. Japan 14 (1962), 233–241 (French). MR 152878, DOI 10.2969/jmsj/01420233
    • —, Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969. J. L. Lions and E. Magenes, Problèmes aux limites nonhomogènes et applications, tome 1, Dunod, Paris, 1968.
  • Jacques-Louis Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France 93 (1965), 43–96. MR 199519
    • O. Lopes and S. Ceron, Existence of forced periodic solutions of dissipative semilinear hyperbolic equations and systems, Preprint of the University of Campinas (UNICAMP), Campinas, Sao Paulo, Brazil, 1984.
  • Irving Segal, Non-linear semi-groups, Ann. of Math. (2) 78 (1963), 339–364. MR 152908, DOI 10.2307/1970347
    • R. Témam, Infinite dimensional dynamical systems in mechanics and physics, Springer-Verlag, 1988.
  • A. V. Babin and M. I. Vishik, Attraktory èvolyutsionnykh uravneniĭ, “Nauka”, Moscow, 1989 (Russian). MR 1007829
Similar Articles
Bibliographic Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 329 (1992), 185-219
  • MSC: Primary 58F12; Secondary 35K57, 35L70, 58D25
  • DOI: https://doi.org/10.1090/S0002-9947-1992-1040261-1
  • MathSciNet review: 1040261