American Institute of Mathematical Sciences

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

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

Shan Ma and  Chunyou Sun

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.

Keywords: Degenerate parabolic equations, global attractors, Z2-index, stationary points.
Mathematics Subject Classification: Primary: 35K65, 35P30, 35B41.
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

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[1]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139
[2]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226
[3]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[4]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
[5]

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067
[6]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105
[7]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021023
[8]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[9]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297
[10]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[11]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[12]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189
[13]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[14]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282
[15]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109
[16]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[17]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[18]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[19]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174
[20]

Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, , () : -. doi: 10.3934/era.2021001

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (179)
  • HTML views (118)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences