×

Asymptotic regularity for some dissipative equations. (English) Zbl 1197.35072

Author’s abstract: This paper is devoted to proving some asymptotic regularity, for both reaction-diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity, which excel the sharp regularity allowed by the corresponding stationary equations (equilibrium points). Based on this regularity, the existence of the finite-dimensional global and exponential attractors can be obtained easily.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35B41 Attractors
35K57 Reaction-diffusion equations
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alikakos, A. D., An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33, 201-225 (1979) · Zbl 0386.34046
[2] Arrieta, J. M.; Carvalho, A. N.; Hale, J. K., A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17, 841-866 (1992) · Zbl 0815.35067
[3] Babin, A. V.; Vishik, M. I., Attractors of Evolution Equations (1992), North-Holland: North-Holland Amsterdam · Zbl 0778.58002
[4] Carvalho, A. N.; Cholewa, J. W., Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66, 443-463 (2002) · Zbl 1020.35059
[5] Carvalho, A. N.; Cholewa, J. W., Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207, 287-310 (2002) · Zbl 1060.35082
[6] Carvalho, A. N.; Cholewa, J. W., Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337, 932-948 (2008) · Zbl 1139.35026
[7] Carvalho, A. N.; Cholewa, J. W.; Dlotko, T., Strongly damped wave problems: bootstrapping and regularity of solutions, J. Differential Equations, 244, 2310-2333 (2008) · Zbl 1151.35056
[8] Cholewa, J. W.; Dlotko, T., Global Attractors in Abstract Parabolic Problems (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1024.35058
[9] Chepyzhov, V. V.; Vishik, M. I., Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., vol. 49 (2002), AMS: AMS Providence, RI · Zbl 0986.35001
[10] M. Conti, V. Pata, On the Regularity of Global Attractors, Discrete Contin. Dyn. Syst., in press; M. Conti, V. Pata, On the Regularity of Global Attractors, Discrete Contin. Dyn. Syst., in press · Zbl 1190.34066
[11] Di Plinio, F.; Pata, V., Robust exponential attractors for the strongly damped wave equation with memory. I, Russ. J. Math. Phys., 15, 301-315 (2008) · Zbl 1183.35051
[12] Di Plinio, F.; Pata, V.; Zelik, S. V., On the strongly damped wave equation with memory, Indiana Univ. Math. J., 57, 757-780 (2008) · Zbl 1149.35015
[13] Efendiev, M.; Miranville, A.; Zelik, S. V., Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135, 703-730 (2005) · Zbl 1088.37005
[14] Evans, L. C., Partial Differential Equations, Grad. Stud. Math., vol. 19 (1998), AMS: AMS Providence, RI
[15] Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions (1992), CRC Press · Zbl 0626.49007
[16] Fabrie, P.; Galusinski, C.; Miranville, A.; Zelik, S., Uniform exponential attractors for a singular perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10, 211-238 (2004) · Zbl 1060.35011
[17] Foias, C.; Sell, G. R.; Temam, R., Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73, 309-353 (1988) · Zbl 0643.58004
[18] Grasselli, M.; Pata, V., Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3, 849-881 (2004) · Zbl 1079.35022
[19] Hale, J. K., Asymptotic Behavior of Dissipative Systems (1988), AMS: AMS Providence, RI · Zbl 0642.58013
[20] Ladyzhenskaya, O. A., Attractors for Semigroups and Evolution Equations (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, New York, Leizioni Lincei · Zbl 0729.35066
[21] Marion, M., Attractors for reactions-diffusion equations: existence and estimate of their dimension, Appl. Anal., 25, 101-147 (1987) · Zbl 0609.35009
[22] Marion, M., Approximate inertial manifolds for reaction-diffusion equations in high space dimension, J. Dynam. Differential Equations, 1, 245-267 (1989) · Zbl 0702.35127
[23] Miranville, A.; Zelik, S. V., Attractors for dissipative partial differential equations in bounded and unbounded domains, (Dafermos, C. M.; Pokorny, M., Evolutionary Equations. Evolutionary Equations, Handb. Differ. Equ., vol. 4 (2008), Elsevier: Elsevier Amsterdam), 103 · Zbl 1221.37158
[24] Pata, V.; Squassina, M., On the strongly damped wave equation, Comm. Math. Phys., 253, 511-533 (2005) · Zbl 1068.35077
[25] Pata, V.; Zelik, S. V., A remark on the damped wave equation, Commun. Pure Appl. Anal., 5, 611-616 (2006) · Zbl 1140.35533
[26] Pata, V.; Zelik, S. V., Smooth attractors for strongly damped wave equations, Nonlinearity, 19, 1495-1506 (2006) · Zbl 1113.35023
[27] Robinson, J. C., Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors (2001), Cambridge University Press · Zbl 0980.35001
[28] Sell, G. R.; You, Y., Dynamics of Evolutionary Equations (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1254.37002
[29] Sun, C.; Cao, D.; Duan, J., Non-autonomous wave dynamics with memory — Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9, 743-761 (2008) · Zbl 1170.35026
[30] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1997), Springer: Springer New York · Zbl 0871.35001
[31] M. Yang, C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear Anal. Real World Appl. (2009), doi:10.1016/j.nonrwa.2009.01.022, in press; M. Yang, C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear Anal. Real World Appl. (2009), doi:10.1016/j.nonrwa.2009.01.022, in press
[32] Yang, M.; Sun, C.; Zhong, C., The existence of global attractors for the \(p\)-Laplacian equation, J. Math. Anal. Appl., 327, 1130-1142 (2007) · Zbl 1112.35031
[33] Zelik, S. V., Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3, 921-934 (2004) · Zbl 1197.35168
[34] Zhong, C.; Yang, M.; Sun, C., The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223, 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.