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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 |
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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