July  2014, 34(7): 2873-2892. doi: 10.3934/dcds.2014.34.2873

Long-time behavior for a class of degenerate parabolic equations

1. 

School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, 730070, China

2. 

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

3. 

Department of Mathematics, Nanjing University, Nanjing 210093

Received  April 2013 Revised  September 2013 Published  December 2013

The long-time behavior of a class of degenerate parabolic equations in a bounded domain will be considered in the sense that the nonnegative diffusion coefficient $a(x)$ is allowed to vanish on a nonempty closed subset with zero measure. For this purpose, some appropriate weighted Sobolev spaces are introduced and the corresponding embedding theorem is established. Then, we show the global existence and uniqueness of weak solutions. Finally, we distinguish two cases (subcritical and supcritical) to prove the existence of compact attractors for the semigroup associated with this class of equations.
Citation: Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873
References:
[1]

M. Aizenman, A sufficient condition for the avoidance of sets by measure preserving flows in $\mathbb R^n$, Duke Math. J., 45 (1978), 809-813. doi: 10.1215/S0012-7094-78-04538-6.  Google Scholar

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C. T. Anh and P. Q. Hung, Global attractors for a class of degenerate parabolic equations, Acta Math. Vietnam., 34 (2009), 213-231.  Google Scholar

[3]

C. T. Anh and P. Q. Hung, Global existence and long-time behavior of solutions to a class of degenerate parabolic equations, Ann. Polon. Math., 93 (2008), 217-230. doi: 10.4064/ap93-3-3.  Google Scholar

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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

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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

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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

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A. C. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Parana. Mat. (3), 26 (2008), 117-132. doi: 10.5269/bspm.v26i1-2.7415.  Google Scholar

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J. Chabrowski, Degenerate elliptic equation involving a subcritical Sobolev exponent, Portugal. Math., 53 (1996), 167-177.  Google Scholar

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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

[10]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004.  Google Scholar

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R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1. Physical Origins and Classical Methods, Reprinted from the 1984 edition, INSTN: Collection Enseignement, Masson, Paris, 1987.  Google Scholar

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E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

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C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010.  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

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V. K. Le and K. Schmitt, On boundary value problems for degenerate quasilinear elliptic equations and inequalities, J. Differential Equations, 144 (1998), 170-218. doi: 10.1006/jdeq.1997.3384.  Google Scholar

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M. Marion, Attractors for reactions-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.  Google Scholar

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M. Marion, Approximate inertial manifolds for reaction-diffusion equations in high space dimension, J. Dynam. Differential Equations, 1 (1989), 245-267. doi: 10.1007/BF01053928.  Google Scholar

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Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

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D. D. Monticelli and K. R. Payne, Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction, J. Differential Equations, 247 (2009), 1993-2026. doi: 10.1016/j.jde.2009.06.024.  Google Scholar

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F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal., 9 (2002), 31-54.  Google Scholar

[23]

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

[24]

E. Sánchez Palencia, Comportements local et macroscopique d'un type de milieux physiques hétérogènes, Intern. Jour. Engin. Sci., 12 (1974), 331-351. doi: 10.1016/0020-7225(74)90062-7.  Google Scholar

[25]

L. Tartar and F. Murat, H -convergence, in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997, 21-43.  Google Scholar

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[27]

J. Valero and A. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633. doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[28]

C. Wang and J. Yin, Evolutionary weighted p -Laplacian with boundary degeneracy, J. Differential Equations, 237 (2007), 421-445. doi: 10.1016/j.jde.2007.03.012.  Google Scholar

[29]

C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

show all references

References:
[1]

M. Aizenman, A sufficient condition for the avoidance of sets by measure preserving flows in $\mathbb R^n$, Duke Math. J., 45 (1978), 809-813. doi: 10.1215/S0012-7094-78-04538-6.  Google Scholar

[2]

C. T. Anh and P. Q. Hung, Global attractors for a class of degenerate parabolic equations, Acta Math. Vietnam., 34 (2009), 213-231.  Google Scholar

[3]

C. T. Anh and P. Q. Hung, Global existence and long-time behavior of solutions to a class of degenerate parabolic equations, Ann. Polon. Math., 93 (2008), 217-230. doi: 10.4064/ap93-3-3.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

A. C. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Parana. Mat. (3), 26 (2008), 117-132. doi: 10.5269/bspm.v26i1-2.7415.  Google Scholar

[8]

J. Chabrowski, Degenerate elliptic equation involving a subcritical Sobolev exponent, Portugal. Math., 53 (1996), 167-177.  Google Scholar

[9]

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

[10]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004.  Google Scholar

[11]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1. Physical Origins and Classical Methods, Reprinted from the 1984 edition, INSTN: Collection Enseignement, Masson, Paris, 1987.  Google Scholar

[12]

E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

V. K. Le and K. Schmitt, On boundary value problems for degenerate quasilinear elliptic equations and inequalities, J. Differential Equations, 144 (1998), 170-218. doi: 10.1006/jdeq.1997.3384.  Google Scholar

[18]

M. Marion, Attractors for reactions-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.  Google Scholar

[19]

M. Marion, Approximate inertial manifolds for reaction-diffusion equations in high space dimension, J. Dynam. Differential Equations, 1 (1989), 245-267. doi: 10.1007/BF01053928.  Google Scholar

[20]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[21]

D. D. Monticelli and K. R. Payne, Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction, J. Differential Equations, 247 (2009), 1993-2026. doi: 10.1016/j.jde.2009.06.024.  Google Scholar

[22]

F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal., 9 (2002), 31-54.  Google Scholar

[23]

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

[24]

E. Sánchez Palencia, Comportements local et macroscopique d'un type de milieux physiques hétérogènes, Intern. Jour. Engin. Sci., 12 (1974), 331-351. doi: 10.1016/0020-7225(74)90062-7.  Google Scholar

[25]

L. Tartar and F. Murat, H -convergence, in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997, 21-43.  Google Scholar

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[27]

J. Valero and A. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633. doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[28]

C. Wang and J. Yin, Evolutionary weighted p -Laplacian with boundary degeneracy, J. Differential Equations, 237 (2007), 421-445. doi: 10.1016/j.jde.2007.03.012.  Google Scholar

[29]

C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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