Dynamics for a non-autonomous degenerate parabolic equation in \mathfrak{D}_{0}^{1}(\Omega, \sigma)

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December 2016, 36(12): 7063-7079. doi: 10.3934/dcds.2016108

Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$

Xin Li ^1, , Chunyou Sun ^2, and Na Zhang ^1,

1.	School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China
2.	School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000

Received May 2015 Revised July 2016 Published October 2016

Abstract
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In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation $u_{t}-div(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process $U(t,\tau)$ is continuous from $L^{2}(\Omega)$ to $\mathscr{D}_{0}^{1}(\Omega, \sigma)$ w.r.t. initial data; And finally show that the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract in $\mathscr{D}_{0}^{1}(\Omega, \sigma)$-norm. Any differentiability on the forcing term is not required.

Keywords: pullback attractor., asymptotic behavior, degenerate parabolic equation, Nash-Moser-Alikakos type a priori estimate, Non-autonomous.

Mathematics Subject Classification: Primary: 35K65, 35B40, 35B4.

Citation: Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108

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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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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