Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains

Wang, Yejuan; Zhou, Shengfan

doi:10.1090/S0033-569X-09-01150-0

Authors: Yejuan Wang and Shengfan Zhou
Journal: Quart. Appl. Math. 67 (2009), 343-378
MSC (2000): Primary 34B40, 35K57
DOI: https://doi.org/10.1090/S0033-569X-09-01150-0
Published electronically: March 25, 2009
MathSciNet review: 2514639
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Abstract: First, we introduce the concept of pullback $\omega$-limit compactness for multi-valued processes, as an extension of the similar concept in the autonomous and nonautonomous framework. Next, we present the necessary and sufficient conditions (pullback dissipativeness and pullback $\omega$-limit compactness) for the existence of a nonempty local bounded kernel (kernel sections are all compact, invariant and pullback attracting) of an infinite dimensional multi-valued process. In addition, we prove a result ensuring the existence of a uniform attractor and the uniform forward attraction of the inflated kernel sections of a family of multi-valued processes under the general assumptions of point dissipativeness and uniform $\omega$-limit compactness. Finally, we illustrate the abstract theory with a nonlinear reaction-diffusion model in an unbounded domain.

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