Long-term analysis of degenerate parabolic equations in ℝ N

Springer Science and Business Media LLC - Tập 31 - Trang 383-410 - 2015
Gao Cheng Yue1, Cheng Kui Zhong2
1Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, P.R. China
2Department of Mathematics, Nanjing University, Nanjing, P. R. China

Tóm tắt

Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space ℝ N is considered. By using ℓ-trajectories methods, we proved that weak solutions generated by degenerate equations possess an (L 2 (ℝ N ), L loc 2 (ℝ N ))-global attractor. Moreover, the upper bounds of the Kolmogorov ɛ-entropy for such global attractor are also obtained.

Tài liệu tham khảo

Abergel, F.: Existence and finite dimensionality of the global attractor for evolution equationds on unbounded domains. J. Differential Equations, 83, 85–108 (1990) Arrieta, J., Cholewa, J. W., Dlotko, T., et al.: Linear parabolic equations in locally spaces. Math. Models Methods Appl. Sci., 14, 253–293 (2004) Arrieta, J., Cholewa, J. W., Dlotko, T., et al.: Asymptotic behavior and attractors for reaction diffusion equations in unbounded domain. Nonlinear Anal., 56, 515–554 (2004) Babin, A.V., Vishik, M. I.: Attractors of Evolutions, North-Holland, Amsterdam, 1992 Babin, A.V., Vishik, M. I.: Attractors of partial differential evolution equations in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A, 116, 221–243 (1990) Carvalho, A. N., Gentile, C. B.: Asymptotic behavior of non-linear parabolic equations with monotone principal part. J. Math. Anal. Appl., 280, 252–272 (2003) Carvalho, A. N., Dlotko, T.: Partly dissipative systems in locally uniform spaces. Colloq. Math., 100, 221–242 (2004) Chepyzhov, V. V., Vishik, M. I.: Kolmogorov’s ɛ-entropy for the attractor of reaction-diffusion equation. Math. Sbornik, 189, 81–110 (1998) Cholewa, J. W., Dlotko, T.: Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000 Cholewa, J. W., Dlotko, T.: Bi-spaces global attractors in abstract parabolic equations. Banach Center Pull. Evol. Equ., 60, 13–26 (2003) DiBenedetto, E.: Degenerate Parabolic Equations, Springer-Verlag, New York, 1993 Efendiev, M. A., Zelik, S. V.: The attractor for a nonlinear reaction-diffusion system in an bounded domain. Comm. Pure Appl. Math., 54, 625–688 (2001) Efendiev, M. A., Ôtani, M.: Infinite dimensional attractors for evolution equations with p-Laplacian and its Kolmogorov entropy. Differential and Integral Equations, 20, 1201–1209 (2007) Feireisl, E., Laurencot, P., Simondon, F.: Global attractors for degenerate parabolic equations on unbounded domain. J. Differential Equations, 129, 239–261 (1996) Grasselli, M., Pražák, D., Schimperna, G.: Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories. J. Differential Equations, 249, 2287–2315 (2010) Hale, J. K.: Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, RI, 1988 Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981 Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal., 58, 181–205 (1975) Khanmamedov, A.: Global attractors for one dimensional p-Laplacian equation. Nonlinear Anal., 71, 155–171 (2009) Khanmamedov, A.: Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain. J. Math. Anal. Appl., 318, 92–101 (2006) Kolmogorov, A. N., Tikhomirov, V. M.: ɛ-entropy and ɛ-Capacity of sets in functional spaces. In: Selected Works of A. N. Kolmogorov, Vol III, ed., Dordrecht, Kluver, 1993 Ladyzhenskaya, O. A.: Attractors for Semigroups and Evolution Equations, Leizioni Lincei/Canbridge Univ. Press, Cambridge/New York, 1991 Lions, J. L.: Quelques méthodes de résolution des problémes aux limites non linéaires, Gauthier-Villars, Paris, 1969 Lions, J. L., Magenes, E.: Non-homogeneous boundary value problems and applications, Spring-Verlag, Berlin, 1972 Málek, J., Pražák, D.: Large time behavior via the method of ℓ-trajectories. J. Differential Equations, 181, 243–279 (2002) Matsuura, K., Ôtani, M.: Exponential attractors for a quasilinear parabolic equation. Discrete Contin. Dyn. Syst., Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, Suppl., 713–720 (2007) Marion, M.: Attractors for reactions-diffusion equations: existence and estimate of their dimension. Appl. Anal., 25, 101–147 (1987) Nakao, M., Aris, N.: On global attractor for nonlinear parabolic equation of m-Laplacian type. J. Math. Anal. Appl., 331, 793–809 (2007) Prizzi, M.: A remark on reaction-diffusion equations in unbounded domains. Discrete Cont. Dyn. Syst., 9, 281–286 (2003) Prizzi, M., Rybakowski, K. P.: Attractors for reaction-diffusion equations on arbitrary unbounded domains. Topol. Methods Nonlinear Anal., 30, 251–277 (2007) Zelik, S.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain and Kolmogorov’s epsilon-entropy. Math. Nachr., 232, 129–179 (2001) Temam, R.: Infinite-Dimensional Dynamical Systems in Methanics and Physics, Second Edition, Springerverlag, Berlin, 1997 Wang, B. X.: Attractors for reaction-diffusion equation in unbounded domains. Physica D, 128, 41–52 (1999) Yang, M., Sun, C., Zhong, C.: Existence of a global attractor for a p-Laplacian equation in ℝn. Nonlinear Anal., 66, 1–13 (2007) Yang, M., Sun, C., Zhong, C.: Global attractor for a p-Laplacian equation. J. Math. Anal. Appl., 327, 1130–1142 (2007) Yin, J., Jin, C.: Travelling wavefronts for a non-divergent degenerate and singular parabolic equation with changing sign sources. Proc. Roy. Soc. Edinburgh Sect. A, 139, 1179–1207 (2009) Yue, G., Zhong, C.: Global attractors for plate equations with critical exponent in locally uniform spaces. Nonlinear Anal., 71, 4105–4114 (2009) 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)