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

Pages: 855 - 882, Volume 24, Issue 3, July 2009

doi:10.3934/dcds.2009.24.855       Abstract        Full Text (331.7K)       Related Articles

Jianhua Huang - Department of Mathematics, National University of Defense Technology, Changsha, 410073, China (email)
Wenxian Shen - Department of Mathematics, Auburn University, AL 36849-5310, United States (email)

Abstract: 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 parabolic equations, nonautonomous dynamical systems, random parabolic equations, random dynamical systems, absorbing sets, pullback global attractors, top Lyapunov exponents.
Mathematics Subject Classification:  35B41, 35K55, 35R60, 37H10, 37L30, 37L55.

Received: December 2007;      Revised: April 2008;      Published: April 2009.