Pullback attractors for a non-autonomous semilinear degenerate parabolic equation

Xin Li, Chunyou Sun, Feng Zhou

DOI: http://dx.doi.org/10.12775/TMNA.2016.011

Abstract


In this paper, we consider the pullback attractors for a non-autonomous semilinear degenerate parabolic equation $u_{t}-\rom{div}(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We provide that the usual $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract the $\mathscr{D}_{\lambda}$-class in $L^{2+\delta}(\Omega)$, where $\delta \in [0, \infty)$ can be arbitrary.

Keywords


Non-autonomous; degenerate parabolic equation; pullback attractor

Full Text:

PREVIEW FULL TEXT

References


C.T. Anh and T.Q. Bao, Pullback attractors for a non autonomous semilinear degenerate parabolic equation, Glasgow Math. J. 52 (2010), 537-554.

C.T. Anh, T.Q. Bao and L.T. Thuy, Regularity and fractal dimension of pullback attractors for a non autonomous semilnear degenerate parabolic equation, Glasgow Math. J. 55 (2013), 431-448.

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differential Equations Appl. 7 (2000), 1187-199.

A.C. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Parana. Mat. 26 (2008), 117-132.

A.N. Carvalho, J.A. Langa and J.C. Robinson, Attractors for infinite-dimensional non-autonomous dynamical systems, Applied Mathematical Sciences, Springer-Verlag New York Inc., 2012.

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non autonomous dynamical systems, Nonlinear Anal.64 (2006), 484-498.

E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.

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.

N.I. Karachalios and N.B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations, Nonlinear Anal. 63 (2005), 1749-1768.

P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011.

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.

G. Lukaszewicz, On pullback attractors in Lp for non-autonomous reaction-diffusion equations, Nonlinear Anal. 73 (2010), 350-357.

D. Monticelli and K. Payne, Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction, J. Differential Equations 274 (2009), 1993-2026.

F. Paronetto, Some new results on the convergence of degenerate elliptic and parabolic equations, J. Convex Anal. 9 (2002), 31-54.

C. Sun and Y. Yuan, Lp-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A 145A (2015), 1029-1052.

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, Springer-Verlag, 1977.

C. Wang and J. Yin, Evolutionary weighted p-Laplacian with boundary degeneracy, J. Differential Equations 237 (2007), 421-445.

M. Yang and P. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl. 12 (2011), 2811-2821.

W. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput. 239 (2014), 358-374.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism