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On the symmetry of spatially periodic two-dimensional water waves
Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China |
2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000 |
References:
[1] |
C. T. Anh and T. Q. Bao, Pullback attractors for a non autonomous semilinear degenerate parabolic equation, Glasgow Math. J., 52 (2010), 537-554.
doi: 10.1017/S0017089510000418. |
[2] |
C. T. Anh, T. Q. Bao and L. T. Thuy, Regularity and fractal dimension of pullback attractors for a non-autonomous semilinear degenerate parabolic equation, Glasgow Math. J., 55 (2013), 431-448.
doi: 10.1017/S0017089512000663. |
[3] |
P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl., 7 (2000), 187-199.
doi: 10.1007/s000300050004. |
[4] |
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[5] |
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non autonomous 2D Navier Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.
doi: 10.1016/j.crma.2005.12.015. |
[6] |
A. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Parana. Mat., 26 (2008), 117-132.
doi: 10.5269/bspm.v26i1-2.7415. |
[7] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Physical Oringins and Classical Methods, Vol. I, Springer-Verlag, Berlin, 1990. |
[8] |
E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[9] |
D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830.
doi: 10.1090/S0002-9939-1994-1169025-2. |
[10] |
J. K. Hale and G. Raugel, {Reaction-diffusion equation on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. |
[11] |
N. Karachalios and N. 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. |
[12] |
N. Karachalios and N. 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. |
[13] |
H. Li, S. Ma and C. Zhong, Long-time behavior for a class of degenerate parabolic equations, Discrete Contin. Dyn. Syst., 34 (2014), 2873-2892.
doi: 10.3934/dcds.2014.34.2873. |
[14] |
X. Li, C. Sun and F. Zhou, Pullback attractors for a non-autonomous semilinear degenerate parabolic equation, Topol. Methods Nonlinear Anal., 47 (2016), 511-528.
doi: 10.12775/TMNA.2016.011. |
[15] |
D. Monticelli and K. 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. |
[16] |
F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal., 9 (2002), 31-54. |
[17] |
J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[18] |
C. Sun and Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.
doi: 10.1017/S0308210515000177. |
[19] |
C. Sun, Y. Xiao, Z. Tang and Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domain, Submitted. |
[20] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, Springer-Verlag, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[21] |
T. Trujillo and B. X. Wang, Continuity of strong solutions of the reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.
doi: 10.1016/j.na.2007.08.032. |
[22] |
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. |
[23] |
M. Yang and P. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Analysis: Real World Applications, 12 (2011), 2811-2821.
doi: 10.1016/j.nonrwa.2011.04.007. |
[24] |
W. Zhao, $H^{1}$-random attractors for stochastic reaction-diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72.
doi: 10.1016/j.na.2013.01.014. |
show all references
References:
[1] |
C. T. Anh and T. Q. Bao, Pullback attractors for a non autonomous semilinear degenerate parabolic equation, Glasgow Math. J., 52 (2010), 537-554.
doi: 10.1017/S0017089510000418. |
[2] |
C. T. Anh, T. Q. Bao and L. T. Thuy, Regularity and fractal dimension of pullback attractors for a non-autonomous semilinear degenerate parabolic equation, Glasgow Math. J., 55 (2013), 431-448.
doi: 10.1017/S0017089512000663. |
[3] |
P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl., 7 (2000), 187-199.
doi: 10.1007/s000300050004. |
[4] |
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[5] |
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non autonomous 2D Navier Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.
doi: 10.1016/j.crma.2005.12.015. |
[6] |
A. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Parana. Mat., 26 (2008), 117-132.
doi: 10.5269/bspm.v26i1-2.7415. |
[7] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Physical Oringins and Classical Methods, Vol. I, Springer-Verlag, Berlin, 1990. |
[8] |
E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[9] |
D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830.
doi: 10.1090/S0002-9939-1994-1169025-2. |
[10] |
J. K. Hale and G. Raugel, {Reaction-diffusion equation on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. |
[11] |
N. Karachalios and N. 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. |
[12] |
N. Karachalios and N. 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. |
[13] |
H. Li, S. Ma and C. Zhong, Long-time behavior for a class of degenerate parabolic equations, Discrete Contin. Dyn. Syst., 34 (2014), 2873-2892.
doi: 10.3934/dcds.2014.34.2873. |
[14] |
X. Li, C. Sun and F. Zhou, Pullback attractors for a non-autonomous semilinear degenerate parabolic equation, Topol. Methods Nonlinear Anal., 47 (2016), 511-528.
doi: 10.12775/TMNA.2016.011. |
[15] |
D. Monticelli and K. 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. |
[16] |
F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal., 9 (2002), 31-54. |
[17] |
J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[18] |
C. Sun and Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.
doi: 10.1017/S0308210515000177. |
[19] |
C. Sun, Y. Xiao, Z. Tang and Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domain, Submitted. |
[20] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, Springer-Verlag, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[21] |
T. Trujillo and B. X. Wang, Continuity of strong solutions of the reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.
doi: 10.1016/j.na.2007.08.032. |
[22] |
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. |
[23] |
M. Yang and P. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Analysis: Real World Applications, 12 (2011), 2811-2821.
doi: 10.1016/j.nonrwa.2011.04.007. |
[24] |
W. Zhao, $H^{1}$-random attractors for stochastic reaction-diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72.
doi: 10.1016/j.na.2013.01.014. |
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