# American Institute of Mathematical Sciences

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, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108
##### References:
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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.  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), 431-448. doi: 10.1017/S0017089512000663.  Google Scholar [3] P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl., 7 (2000), 187-199. 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), 484-498. 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), 263-268. 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), 117-132. 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, Vol. I, Springer-Verlag, Berlin, 1990.  Google Scholar [8] E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 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), 825-830. 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), 33-95.  Google Scholar [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.  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), 361-393. 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), 2873-2892. 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), 511-528. 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), 1993-2026. 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), 31-54.  Google Scholar [17] J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 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), 1029-1052. 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, Springer-Verlag, 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), 2525-2532. 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), 421-445. 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), 2811-2821. 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), 61-72. doi: 10.1016/j.na.2013.01.014.  Google Scholar
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