Pullback attractors for nonautonomous and random parabolic
equations on non-smooth domains
Jianhua Huang - Department of Mathematics, National University of Defense Technology, Changsha, 410073, China (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
Received: December 2007; Revised: April 2008; Published: April 2009.
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