Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-15T01:24:41.826Z Has data issue: false hasContentIssue false
  • English
  • Français

A reaction-diffusion equation on a thin L-shaped domain

Published online by Cambridge University Press:  14 November 2011

Jack K. Hale  and
Geneviève Raugel
Jack K. Hale
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.
Geneviève Raugel
Affiliation:
Laboratoire d'Analyse Numérique, Bâtiment 425, Université Paris-Sud, 91405 Orsay cédex, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a dissipative reaction–diffusion equation on a thin L-shaped domain (with the thinness measured by a parameter ε); we determine the limit equation for ε = 0 and prove the upper semicontinuity of the global attractors at ε = 0. We also state a lower semicontinuity result. When the limit equation is one-dimensional, we prove convergence of any orbit to a singleton.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 2 , 1995 , pp. 283 - 327
Copyright
Copyright © Royal Society of Edinburgh 1995

References

1Babin, A. V. and Vishik, M. I.. Regular attractors of semigroups of evolutionary equations. J. Math. Pures Appl. 62 (1983), 441–491.Google Scholar
2Bourquin, F. and Ciarlet, P. G.. Modeling and justification of eigenvalue problems for junctions between elastic structures. J. Fund. Anal. 87 (1989), 392–427.CrossRefGoogle Scholar
3Céa, J. and Geymonat, G.. Une méthode de linéarisation via l'optimisation. Symposia Matematica 10 (1972), 431–451.Google Scholar
4Ciarlet, P. G.. Plates and Junctions in Elastic Multi-structures: an Asymptotic Analysis, Collection RMA14 (Paris: Masson-Springer, 1990).Google Scholar
5Ciarlet, P. G., Dret, H. Le and Nzengwa, R.. Modelisation de la jonction entre un corps elastique tridimensionnel et une plaque. C.R. Acad. Sci. Paris, Sir. 1 305 (1987), 55–58.Google Scholar
6Ciarlet, P. G., Dret, H. Le and Nzengwa, R.. Junctions between three-dimensional and two-dimensional linearly elastic structures. J. Math. Pures Appl. 68 (1989), 261–295.Google Scholar
7Cioranescu, D. and Paulin, J. Saint Jean. Reinforced and honey-comb structures. J. Math. Pures Appl. 65(1986), 403–422.Google Scholar
8Crouzeix, M. and Rappaz, J.. On Numerical Approximation in Bifurcation Theory, Collection RMA 13 (Paris: Masson, 1990).Google Scholar
9Hale, J. K.. Asymptotic behavior and dynamics in infinite dimensions. In Research Notes in Mathematics 132, 1–41 (Boston: Pitman, 1985).Google Scholar
10Hale, J. K.. Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25 (Providence, R.I.: American Mathematical Society, 1988).Google Scholar
11Hale, J. K. and Raugel, G.. Lower semicontinuity of attractors of gradient systems and applications. Ann. Mat. Pura Appl. (IV) 154 (1989), 281–326.Google Scholar
12Hale, J. K. and Raugel, G.. Partial differential equations on thin domains, Proc. Int. Conf. in Alabama, 1990. In Differential Equations and Mathematical Physics, ed. Bennewitz, C., 63–97 (New York: Academic Press, 1991).Google Scholar
13Hale, J. K. and Raugel, G.. Reaction-diffusion equation on thin domains. J. Math. Pures Appl. 71 (1992), 33–95.Google Scholar
14Hale, j. K. and Raugel, G.. Convergence in gradient like systems and applications. Z. Angew. Math. Phys. 43(1992), 63–124.CrossRefGoogle Scholar
15Hale, J. K. and Raugel, G.. A damped hyperbolic equation on thin domains. Trans. Amer. Math. Soc. 329(1992), 185–219.CrossRefGoogle Scholar
16Hale, J. K. and Raugel, G.. Dynamics on thin domains (Preprint).Google Scholar
17Hale, J. K. and Raugel, G.. Attractors and convergence of PDE on thin L-shaped domains. In Progress in Partial Differential Equations: the Metz Surveys 2, ed. Chipot, M., 149–171 (Harlow: Longman, 1993).Google Scholar
18Hale, J. K. and Raugel, G.. Some additional remarks on PDE on thin L-shaped domains (in preparation).Google Scholar
19Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
20Iosif'yan, G. A., Oleinik, O. A. and Shamaev, A. S.. On the limiting behavior of the spectrum of a sequence of operators defined on different Hilbert spaces, Russian Math. Surv. 44 (1989), 195–196.CrossRefGoogle Scholar
21Kato, T.. Perturbation Theory for Linear Operators (Berlin and New York: Springer, 1966).Google Scholar
22Lagnese, J. E. and Lions, J. L.. Modelling, Analysis and Control of Thin Plates, Collection RMA 6 (Paris: Masson, 1988).Google Scholar
23Dret, H. Le. Modeling of the junction between two rods. J. Math. Pures Appl. 68 (1989), 365–397.Google Scholar
24Dret, H. Le. Folded plates revisited. Comput. Mech. 5 (1989), 345–365.CrossRefGoogle Scholar
25Dret, H. Le. Problemes Variationnels dans les Multi-domaines-Modelisation des Jonctions et Applications, Collection RMA 19 (Paris: Masson, 1991).Google Scholar
26Lions, J. L.. Perturbations singulieres dans les problemes aux limites et en controle optimal, Lecture Notes in Mathematics 323 (Berlin: Springer, 1973).Google Scholar
27Ramm, A. G.. Limit of the spectra of the interior Neumann problems when a solid domain shrinks to a plane one. J. Math. Anal. Appl. 108 (1985), 107–112.Google Scholar
28Raoult, A.. Asymptotic modeling of the elastodynamics of a multi-structure. Asymptotic Anal. 6 (1992), 73–108.CrossRefGoogle Scholar
29Raugel, G.. Persistence of Morse-Smale properties under some approximations and singular perturbations (Preprint).Google Scholar
30Raugel, G. and Sell, G.. Navier-Stokes equations on thin 3D domains I. Global attractors and regularity of solutions. J. Amer. Math. Soc. 6 (1993), 503–568.Google Scholar
31Raugel, G. and Sell, G.. Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic solutions, Seminaire du College de France, ed. Lions, J. L. (Boston: Longman, to appear).Google Scholar