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

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

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

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.
Citation: Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108
References:
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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.  doi: 10.1017/S0017089512000663.  Google Scholar

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T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

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E. Dibenedetto, Degenerate Parabolic Equations,, Universitext, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

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N. Karachalios and N. Zographopoulos, Convergence towards attractors for a degenerate Ginzburg-Landau equation,, Z. Angew. Math. Phys., 56 (2005), 11.  doi: 10.1007/s00033-004-2045-z.  Google Scholar

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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.  doi: 10.1007/s00526-005-0347-4.  Google Scholar

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H. Li, S. Ma and C. Zhong, Long-time behavior for a class of degenerate parabolic equations,, Discrete Contin. Dyn. Syst., 34 (2014), 2873.  doi: 10.3934/dcds.2014.34.2873.  Google Scholar

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X. Li, C. Sun and F. Zhou, Pullback attractors for a non-autonomous semilinear degenerate parabolic equation,, Topol. Methods Nonlinear Anal., 47 (2016), 511.  doi: 10.12775/TMNA.2016.011.  Google Scholar

[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.  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.   Google Scholar

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J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge Univ. Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

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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.  doi: 10.1017/S0308210515000177.  Google Scholar

[19]

C. Sun, Y. Xiao, Z. Tang and Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domain,, Submitted., ().   Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, New York, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[21]

T. Trujillo and B. X. Wang, Continuity of strong solutions of the reaction-diffusion equation in initial data,, Nonlinear Anal., 69 (2008), 2525.  doi: 10.1016/j.na.2007.08.032.  Google Scholar

[22]

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

[23]

M. Yang and P. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations,, Nonlinear Analysis: Real World Applications, 12 (2011), 2811.  doi: 10.1016/j.nonrwa.2011.04.007.  Google Scholar

[24]

W. Zhao, $H^{1}$-random attractors for stochastic reaction-diffusion equations with additive noise,, Nonlinear Anal., 84 (2013), 61.  doi: 10.1016/j.na.2013.01.014.  Google Scholar

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.  doi: 10.1017/S0017089510000418.  Google Scholar

[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.  doi: 10.1017/S0017089512000663.  Google Scholar

[3]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem,, Nonlinear Diff. Equ. Appl., 7 (2000), 187.  doi: 10.1007/s000300050004.  Google Scholar

[4]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[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.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar

[6]

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

[7]

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, Physical Oringins and Classical Methods, (1990).   Google Scholar

[8]

E. Dibenedetto, Degenerate Parabolic Equations,, Universitext, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[9]

D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity,, Proc. Amer. Math. Soc., 120 (1994), 825.  doi: 10.1090/S0002-9939-1994-1169025-2.  Google Scholar

[10]

J. K. Hale and G. Raugel, {Reaction-diffusion equation on thin domains,, J. Math. Pures Appl., 71 (1992), 33.   Google Scholar

[11]

N. Karachalios and N. Zographopoulos, Convergence towards attractors for a degenerate Ginzburg-Landau equation,, Z. Angew. Math. Phys., 56 (2005), 11.  doi: 10.1007/s00033-004-2045-z.  Google Scholar

[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.  doi: 10.1007/s00526-005-0347-4.  Google Scholar

[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.  doi: 10.3934/dcds.2014.34.2873.  Google Scholar

[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.  doi: 10.12775/TMNA.2016.011.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.06.024.  Google Scholar

[16]

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

[17]

J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge Univ. Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[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.  doi: 10.1017/S0308210515000177.  Google Scholar

[19]

C. Sun, Y. Xiao, Z. Tang and Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domain,, Submitted., ().   Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, New York, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[21]

T. Trujillo and B. X. Wang, Continuity of strong solutions of the reaction-diffusion equation in initial data,, Nonlinear Anal., 69 (2008), 2525.  doi: 10.1016/j.na.2007.08.032.  Google Scholar

[22]

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

[23]

M. Yang and P. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations,, Nonlinear Analysis: Real World Applications, 12 (2011), 2811.  doi: 10.1016/j.nonrwa.2011.04.007.  Google Scholar

[24]

W. Zhao, $H^{1}$-random attractors for stochastic reaction-diffusion equations with additive noise,, Nonlinear Anal., 84 (2013), 61.  doi: 10.1016/j.na.2013.01.014.  Google Scholar

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