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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:
35B65Smoothness and regularity of solutions of PDE
35B41Attractors (PDE)
35K57Reaction-diffusion equations
35L05Wave equation (hyperbolic PDE)
35B40Asymptotic behavior of solutions of PDE
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 · doi:10.1016/0022-0396(79)90088-3
[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 · doi:10.1080/03605309208820866
[3]Babin, A. V.; Vishik, M. I.: Attractors of evolution equations, (1992)
[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 · doi:10.1017/S0004972700040296
[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 · doi:10.2140/pjm.2002.207.287 · doi:http://pjm.math.berkeley.edu/pjm/2002/207-2/p02.xhtml
[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 · doi:10.1016/j.jmaa.2007.04.051
[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 · doi:10.1016/j.jde.2008.02.011
[8]Cholewa, J. W.; Dlotko, T.: Global attractors in abstract parabolic problems, (2000)
[9]Chepyzhov, V. V.; Vishik, M. I.: Attractors for equations of mathematical physics, Amer. math. Soc. colloq. Publ. 49 (2002) · Zbl 0986.35001
[10]M. Conti, V. Pata, On the Regularity of Global Attractors, Discrete Contin. Dyn. Syst., in press
[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 · doi:10.1134/S1061920808030014
[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 · doi:10.1512/iumj.2008.57.3266
[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 · doi:10.1017/S030821050000408X
[14]Evans, L. C.: Partial differential equations, Grad stud. Math. 19 (1998) · Zbl 0902.35002
[15]Evans, L. C.; Gariepy, R. F.: Measure theory and fine properties of functions, (1992) · Zbl 0804.28001
[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 · doi:10.3934/dcds.2004.10.211
[17]Foias, C.; Sell, G. R.; Temam, R.: Inertial manifolds for nonlinear evolutionary equations, J. differential equations 73, 309-353 (1988) · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
[18]Grasselli, M.; Pata, V.: Asymptotic behavior of a parabolic – hyperbolic system, Commun. pure appl. Anal. 3, 849-881 (2004) · Zbl 1079.35022 · doi:10.3934/cpaa.2004.3.849
[19]Hale, J. K.: Asymptotic behavior of dissipative systems, (1988)
[20]Ladyzhenskaya, O. A.: Attractors for semigroups and evolution equations, (1991) · Zbl 0755.47049
[21]Marion, M.: Attractors for reactions – diffusion equations: existence and estimate of their dimension, Appl. anal. 25, 101-147 (1987) · Zbl 0609.35009 · doi:10.1080/00036818708839678
[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 · doi:10.1007/BF01053928
[23]Miranville, A.; Zelik, S. V.: Attractors for dissipative partial differential equations in bounded and unbounded domains, Handb. differ. Equ. 4, 103 (2008) · Zbl 1221.37158
[24]Pata, V.; Squassina, M.: On the strongly damped wave equation, Comm. math. Phys. 253, 511-533 (2005) · Zbl 1068.35077 · doi:10.1007/s00220-004-1233-1
[25]Pata, V.; Zelik, S. V.: A remark on the damped wave equation, Commun. pure appl. Anal. 5, 611-616 (2006) · Zbl 1140.35533 · doi:10.3934/cpaa.2006.5.611
[26]Pata, V.; Zelik, S. V.: Smooth attractors for strongly damped wave equations, Nonlinearity 19, 1495-1506 (2006) · Zbl 1113.35023 · doi:10.1088/0951-7715/19/7/001
[27]Robinson, J. C.: Infinite-dimensional dynamical systems: an introduction to dissipative parabolic pdes and the theory of global attractors, (2001)
[28]Sell, G. R.; You, Y.: Dynamics of evolutionary equations, (2002)
[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 · doi:10.3934/dcdsb.2008.9.743
[30]Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, (1997)
[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 · Zbl 1167.35319 · doi:10.1016/j.nonrwa.2007.12.001
[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 · doi:10.1016/j.jmaa.2006.04.085
[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 · doi:10.3934/cpaa.2004.3.921
[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 · doi:10.1016/j.jde.2005.06.008