Pullback attractors for nonautonomous and random parabolicequations on non-smooth domains

Jianhua Huang; Wenxian Shen

doi:10.3934/dcds.2009.24.855

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2009, Volume 24, Issue 3: 855-882. Doi: 10.3934/dcds.2009.24.855

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

Jianhua Huang^1, and
Wenxian Shen^2,

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

Received: December 2007

Revised: April 2008

Published: August 2009

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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 $1$ « $q$ < $\infty$.

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Mathematics Subject Classification: 35B41, 35K55, 35R60, 37H10, 37L30, 37L55.

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