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Minimality and stable Bernoulliness in dimension 3
Long-time behavior for a class of weighted equations with degeneracy
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
In this paper we study the existence and some properties of the global attractors for a class of weighted equations when the weighted Sobolev space $ H_0^{1,a}(\Omega) $ (see Definition 1.1) cannot be bounded embedded into $ L^2(\Omega) $. We claim that the dimension of the global attractor is infinite by estimating its lower bound of $ Z_2 $-index. Moreover, we prove that there is an infinite sequence of stationary points in the global attractor which goes to 0 and the corresponding critical value sequence of the Lyapunov functional also goes to 0.
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Global attractors for a class of degenerate parabolic equations, Acta Mathematica Vietnamica, 34 (2009), 213-231.
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Global attractors for the m-semiflow generated by a quasilinear degenerate parabolic equations, J. Math. Anal. Appl., 363 (2010), 444-453.
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M. Efendiev and S. Zelik,
Finite- and infinite-dimensional attractors for porous media equations, Proc. London Math. Soc. (3), 96 (2008), 51-57.
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M. Efendiev, A. Miranville and S. Zelik,
Infinite-dimensional exponetial attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1107-1129.
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M. Efendiev,
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B. R. Hunt and V. Y. Kaloshin,
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N. I. Karachalios and N. B. Zographopoulos,
Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys., 56 (2005), 11-30.
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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), 361-393.
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F. Li, B. You and C. K. Zhong,
Multiple equilibrium points in global attractors for some $p$-Laplacian equations, Applicable Analysis, 97 (2018), 1591-1599.
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A. Miranville and S. Zelik,
Finite-dimensionality of attractors for degeneare equations of elliptic-parabolic type, Nonlinearity, 20 (2007), 1773-1797.
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B. You, F. Li and C. K. Zhong,
The existence of multiple equilibrium points in a global attractor for some $p$-Laplacian equation, J. Math. Anal. Appl., 418 (2014), 626-637.
doi: 10.1016/j.jmaa.2014.03.089. |
| [20] |
J. Zhang, C. K. Zhong and B. You,
The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems Ⅱ, Nonlinear Anal. Real World Appl., 36 (2017), 44-55.
doi: 10.1016/j.nonrwa.2017.01.002. |
| [21] |
C. K. Zhong and W. S. Niu,
On the $Z_2$ index of the global attractor for a class of $p$-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.
doi: 10.1016/j.na.2010.07.022. |
| [22] |
C. K. Zhong, B. You and R. Yang,
The existence of multiple equilibrium points in global attractor for some symmetric dynamical systems, Nonlinear Anal. Real World Appl., 19 (2014), 31-44.
doi: 10.1016/j.nonrwa.2014.02.008. |
show all references
References:
| [1] |
C. T. Anh and P. Q. Hung,
Global attractors for a class of degenerate parabolic equations, Acta Mathematica Vietnamica, 34 (2009), 213-231.
|
| [2] |
C. T. Anh, N. M. Chuong and T. D. Ke,
Global attractors for the m-semiflow generated by a quasilinear degenerate parabolic equations, J. Math. Anal. Appl., 363 (2010), 444-453.
doi: 10.1016/j.jmaa.2009.09.034. |
| [3] |
C. T. Anh and T. D. Ke,
Long-time behavior for quasilinear parabolic equations involving weighted $p$-Laplacian operators, Nonlinear Anal., 71 (2009), 4415-4422.
doi: 10.1016/j.na.2009.02.125. |
| [4] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. |
| [5] |
J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404. |
| [6] |
M. Efendiev and S. Zelik,
Finite- and infinite-dimensional attractors for porous media equations, Proc. London Math. Soc. (3), 96 (2008), 51-57.
doi: 10.1112/plms/pdm026. |
| [7] |
M. A. Efendiev and M. Ôtani,
Infinte-dimensional attractors for parabolic equations with $p$-Laplacian in heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 565-582.
doi: 10.1016/j.anihpc.2011.03.006. |
| [8] |
M. Efendiev, A. Miranville and S. Zelik,
Infinite-dimensional exponetial attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1107-1129.
doi: 10.1098/rspa.2003.1182. |
| [9] |
M. Efendiev,
Infinite-dimensional exponetial attractors for fourth-order nonlinear parabolic equations in unbounded domains, Math. Meth. Appl. Sci., 34 (2011), 939-949.
doi: 10.1002/mma.1412. |
| [10] |
J. K. Hale, L. T. Magalhães and W. M. Oliva, An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory, Applied Mathematical Sciences, 47. Springer-Verlag, New York, 1984.
doi: 10.1007/0-387-22896-9_9. |
| [11] |
B. R. Hunt and V. Y. Kaloshin,
Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275.
doi: 10.1088/0951-7715/12/5/303. |
| [12] |
N. I. Karachalios and N. B. Zographopoulos,
Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys., 56 (2005), 11-30.
doi: 10.1007/s00033-004-2045-z. |
| [13] |
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), 361-393.
doi: 10.1007/s00526-005-0347-4. |
| [14] |
N. I. Karachalios and N. B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations, Nonlinear Anal., 63 (2005), e1749–e1768.
doi: 10.1016/j.na.2005.03.022. |
| [15] |
F. Li, B. You and C. K. Zhong,
Multiple equilibrium points in global attractors for some $p$-Laplacian equations, Applicable Analysis, 97 (2018), 1591-1599.
doi: 10.1080/00036811.2017.1322199. |
| [16] |
A. Miranville and S. Zelik,
Finite-dimensionality of attractors for degeneare equations of elliptic-parabolic type, Nonlinearity, 20 (2007), 1773-1797.
doi: 10.1088/0951-7715/20/8/001. |
| [17] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
| [18] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
| [19] |
B. You, F. Li and C. K. Zhong,
The existence of multiple equilibrium points in a global attractor for some $p$-Laplacian equation, J. Math. Anal. Appl., 418 (2014), 626-637.
doi: 10.1016/j.jmaa.2014.03.089. |
| [20] |
J. Zhang, C. K. Zhong and B. You,
The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems Ⅱ, Nonlinear Anal. Real World Appl., 36 (2017), 44-55.
doi: 10.1016/j.nonrwa.2017.01.002. |
| [21] |
C. K. Zhong and W. S. Niu,
On the $Z_2$ index of the global attractor for a class of $p$-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.
doi: 10.1016/j.na.2010.07.022. |
| [22] |
C. K. Zhong, B. You and R. Yang,
The existence of multiple equilibrium points in global attractor for some symmetric dynamical systems, Nonlinear Anal. Real World Appl., 19 (2014), 31-44.
doi: 10.1016/j.nonrwa.2014.02.008. |
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