Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains

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August 2009, 24(3): 855-882. doi: 10.3934/dcds.2009.24.855

Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains

1.	Department of Mathematics, National University of Defense Technology, Changsha, 410073, China
2.	Department of Mathematics, Auburn University, AL 36849-5310

Received December 2007 Revised April 2008 Published April 2009

Abstract
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The current paper is devoted to the study of pullback attractors for general nonautonomous and random parabolic equations on non-smooth domains $D$. Mild solutions are considered for such equations. We first extend various fundamental properties for solutions of smooth parabolic equations on smooth domains to solutions of general parabolic equations on non-smooth domains, including continuous dependence on parameters, monotonicity, and compactness, which are of great importance in their own. Under certain dissipative conditions on the nonlinear terms, we prove that mild solutions with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We then show that pullback attractors for nonautonomous and random parabolic equations on non-smooth domains exist in $L_q(D)$ for $1$ « $q$ < $\infty$.

Keywords: Non-smooth domains, nonautonomous dynamical systems, random dynamical systems, top Lyapunov exponents., pullback global attractors, absorbing sets, random parabolic equations, nonautonomous parabolic equations.

Mathematics Subject Classification: 35B41, 35K55, 35R60, 37H10, 37L30, 37L5.

Citation: Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855

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