×

Uniform attractors for impulsive reaction-diffusion equations. (English) Zbl 1204.49036

Summary: The goal of this paper is to consider the long time behavior of solutions of reaction-diffusion equations with impulsive effects at fixed moment of time. Under a new class of impulse functions, we prove the existence of uniform attractors in the spaces \(H^1_0(\Omega)\), \(L^p (\Omega)\) and \(L^{2p-2}(\Omega)\), respectively.

MSC:

49N25 Impulsive optimal control problems
35K20 Initial-boundary value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chepyzhov, V. V.; Vishik, M. I., Attractors for Equations of Mathematical Physics (2002), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0986.35001
[2] Temam, R., Infinite Dimension Dynamical System in Mechanics and Physics (1997), Springer: Springer New York
[3] Ciesielski, K., On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52, 81-91 (2004) · Zbl 1098.37017
[4] Kaul, S. K., Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stochastic Anal., 7, 509-523 (1994) · Zbl 0857.54039
[5] Lu, S. S.; Wu, H. Q.; Zhong, C. K., Attractors for nonautonomous 2D Navier-Stokes equations with normal external force, Disc. Cont. Dyn. Syst., 13, 701-719 (2005) · Zbl 1083.35094
[6] Bonotto, E. M.; Federson, M., Limit sets and the Poincare-Bendixson Theorem in impulsive semidynamical systems, J. Differ. Equat., 244, 2334-2349 (2008) · Zbl 1143.37014
[7] Haraux, A., Attractors of asymptotically compact process and application to nonlinear partial differential equations, Commun. Partial Differ. Equat., 13, 11, 1383-1414 (1988) · Zbl 0676.35008
[8] Haraux, A., Systems Dynamiques Dissipatifs et Applications (1991), Masson: Masson Paris · Zbl 0726.58001
[9] Chepyzhov, V. V.; Efendiev, M. A., Hausdorff dimension estimation for attractors of non-autonomous dynamical systems in unbounded domains: an example, Commun. Pure Appl. Math., LIII, 647-665 (2000) · Zbl 1022.37048
[10] Moise, I.; Rosa, R.; Wang, B. X., Attractors for noncompact non-autonomous systems via energy equations, Disc. Cont. Dyn. Syst., 10, 473-496 (2004) · Zbl 1060.35023
[11] Zhong, C. K.; Sun, C. Y.; Niu, M. F., On the existence of the global attractor for a class of infinite-dimensional nonlinear dissipative dynamical systems, Chin. Ann. Math. B, 26, 393-400 (2005) · Zbl 1079.35026
[12] Robinson, J. C., Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theorem of Global Attractors (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0980.35001
[13] Schmalfuss, B., Attractors for nonautonomous and random dynamical systems perturbed by impulses, Disc. Cont. Dyn. Syst., 9, 727-744 (2003) · Zbl 1029.37030
[14] Iovane, G.; Kapustyan, A. V., Global attractor for impulsive reaction-diffusion equation, Nonlinear Oscil., 8, 318-328 (2005) · Zbl 1108.35087
[15] Iovane, G.; Kapustyan, A. V.; Valero, J., Asymptotic behaviour of reaction-diffusion equations with non-damped impulsive effects, Nonlinear Anal., 68, 2516-2530 (2008) · Zbl 1228.35063
[16] Bainov, D. D.; Simeonov, P. S., Impulse Differential Equations: perodic solutions and applications, Pitman Monogr. Surv. Pure Appl. Math., 66 (1993) · Zbl 0793.34011
[17] Bainov, D. D.; Simeonov, P. S., Systems with impulse effect, stability, theory and applications. Systems with impulse effect, stability, theory and applications, Ellis Hirwood Ser. Math. (1989), Ellis Horwood: Ellis Horwood Chichister · Zbl 0676.34035
[18] Bainov, D. D.; Lakshmikantham, N.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), Word Scientific: Word Scientific Singapore · Zbl 0719.34002
[19] Zhong, C. K.; Yang, M. H.; Sun, C. Y., The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equat., 15, 367-399 (2006) · Zbl 1101.35022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.