Long-time behavior for a class of degenerate parabolic equations

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

Long-time behavior for a class of degenerate parabolic equations

Hongtao Li ^1, , Shan Ma ^2, and Chengkui Zhong ^3,

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

Abstract
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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.

Keywords: global attractor, Global solution, degenerate parabolic equation..

Mathematics Subject Classification: Primary: 74H40, 35B41, 35K6.

Citation: Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete and 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.
[2]	C. T. Anh and P. Q. Hung, Global attractors for a class of degenerate parabolic equations, Acta Math. Vietnam., 34 (2009), 213-231.
[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.
[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.
[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.
[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.
[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.
[8]	J. Chabrowski, Degenerate elliptic equation involving a subcritical Sobolev exponent, Portugal. Math., 53 (1996), 167-177.
[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.
[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.
[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.
[12]	E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[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.
[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.
[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.
[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.
[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.
[18]	M. Marion, Attractors for reactions-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.
[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.
[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.
[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.
[22]	F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal., 9 (2002), 31-54.
[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.
[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.
[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.
[26]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
[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.
[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.
[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.

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.
[2]	C. T. Anh and P. Q. Hung, Global attractors for a class of degenerate parabolic equations, Acta Math. Vietnam., 34 (2009), 213-231.
[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.
[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.
[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.
[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.
[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.
[8]	J. Chabrowski, Degenerate elliptic equation involving a subcritical Sobolev exponent, Portugal. Math., 53 (1996), 167-177.
[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.
[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.
[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.
[12]	E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[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.
[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.
[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.
[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.
[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.
[18]	M. Marion, Attractors for reactions-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.
[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.
[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.
[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.
[22]	F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal., 9 (2002), 31-54.
[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.
[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.
[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.
[26]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
[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.
[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.
[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.

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