March  2020, 40(3): 1889-1902. doi: 10.3934/dcds.2020098

Long-time behavior for a class of weighted equations with degeneracy

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received  July 2019 Published  December 2019

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.

Citation: Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098
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.   Google Scholar

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C. T. AnhN. 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.  Google Scholar

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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.  doi: 10.1112/plms/pdm026.  Google Scholar

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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.  Google Scholar

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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.  doi: 10.1007/s00033-004-2045-z.  Google Scholar

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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.  doi: 10.1007/s00526-005-0347-4.  Google Scholar

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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.  Google Scholar

[15]

F. LiB. 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.  Google Scholar

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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.  doi: 10.1088/0951-7715/20/8/001.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[19]

B. YouF. 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.  Google Scholar

[20]

J. ZhangC. 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.  Google Scholar

[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.  Google Scholar

[22]

C. K. ZhongB. 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.  Google Scholar

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.   Google Scholar

[2]

C. T. AnhN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. EfendievA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. LiB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

B. YouF. 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.  Google Scholar

[20]

J. ZhangC. 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.  Google Scholar

[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.  Google Scholar

[22]

C. K. ZhongB. 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.  Google Scholar

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