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http://dx.doi.org/10.4134/CKMS.c190332

GLOBAL ATTRACTOR FOR A CLASS OF QUASILINEAR DEGENERATE PARABOLIC EQUATIONS WITH NONLINEARITY OF ARBITRARY ORDER  

Tran, Thi Quynh Chi (Department of Mathematics Electric Power University)
Le, Thi Thuy (Department of Mathematics Electric Power University)
Nguyen, Xuan Tu (Department of Mathematics Electric Power University)
Publication Information
Communications of the Korean Mathematical Society / v.36, no.3, 2021 , pp. 447-463 More about this Journal
Abstract
In this paper we study the existence and long-time behavior of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators with a new class of nonlinearities. First, we prove the existence and uniqueness of weak solutions by combining the compactness and monotone methods and the weak convergence techniques in Orlicz spaces. Then, we prove the existence of global attractors by using the asymptotic a priori estimates method.
Keywords
Quasilinear degenerate parabolic equation; weighted p-Laplacian operator; weak solution; global attractor; compactness method; monotonicity method; weak convergence techniques; Orlicz spaces; asymptotic a priori estimate method;
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1 H. Li and S. Ma, Asymptotic behavior of a class of degenerate parabolic equations, Abstr. Appl. Anal. 2012 (2012), Art. ID 673605, 15 pp. https://doi.org/10.1155/2012/673605   DOI
2 X. Li, C. Sun, and F. Zhou, Pullback attractors for a non-autonomous semilinear degenerate parabolic equation, Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 511-528.
3 W. Niu, Global attractors for degenerate semilinear parabolic equations, Nonlinear Anal. 77 (2013), 158-170. https://doi.org/10.1016/j.na.2012.09.010   DOI
4 W. Niu, Q. Meng, and X. Chai, Asymptotic behavior for nonlinear degenerate parabolic equations with irregular data, Appl. Anal. (2020). http://doi.org/10.1080/00036811.2020.1721470   DOI
5 W. Tan, Dynamics for a class of non-autonomous degenerate p-Laplacian equations, J. Math. Anal. Appl. 458 (2018), no. 2, 1546-1567. https://doi.org/10.1016/j.jmaa.2017.10.030   DOI
6 C. T. Anh, N. D. Binh, and L. T. Thuy, On the global attractors for a class of semilinear degenerate parabolic equations, Ann. Polon. Math. 98 (2010), no. 1, 71-89. https://doi.org/10.4064/ap98-1-5   DOI
7 C. T. Anh, N. D. Binh, and L. T. Thuy, Attractors for quasilinear parabolic equations involving weighted p-Laplacian operators, Vietnam J. Math. 38 (2010), no. 3, 261-280.
8 C. T. Anh and T. D. Ke, Long-time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Anal. 71 (2009), no. 10, 4415-4422. https://doi.org/10.1016/j.na.2009.02.125   DOI
9 A. V. Babin and M. I. Vishik, Attractors of evolution equations, translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.
10 C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 2, 195-212. https:// doi.org/10.1007/s00030-009-0048-3   DOI
11 P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl. 7 (2000), no. 2, 187-199. https://doi.org/10.1007/s000300050004   DOI
12 R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 1, translated from the French by Ian N. Sneddon, Springer-Verlag, Berlin, 1990.
13 R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4612-0645-3
14 P. G. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic p-Laplacian equation, Commun. Pure Appl. Anal. 12 (2013), no. 2, 735-754. https://doi.org/10.3934/cpaa.2013.12.735   DOI
15 P. G. Geredeli, On the existence of regular global attractor for p-Laplacian evolution equation, Appl. Math. Optim. 71 (2015), no. 3, 517-532. https://doi.org/10.1007/s00245-014-9268-y   DOI
16 X. Li, C. Sun, and N. Zhang, Dynamics for a non-autonomous degenerate parabolic equation in D10(Ω, σ), Discrete Contin. Dyn. Syst. 36 (2016), no. 12, 7063-7079. https://doi.org/10.3934/dcds.2016108   DOI
17 N. I. Karachalios and N. B. Zographopoulos, Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys. 56 (2005), no. 1, 11-30. https://doi.org/10.1007/s00033-004-2045-z   DOI
18 H. Li, S. Ma, and C. Zhong, Long-time behavior for a class of degenerate parabolic equations, Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2873-2892. https://doi.org/ 10.3934/dcds.2014.34.2873   DOI
19 J.-L. Lions, Quelques m'ethodes de r'esolution des probl'emes aux limites non lineaires, Dunod, 1969.
20 J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
21 N. I. Karachalios and N. B. Zographopoulos, On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations 25 (2006), no. 3, 361-393. https://doi.org/10.1007/s00526-005-0347-4   DOI