Kernel Sections of Multi-valued Processes with Application to the Nonlinear Reaction-diffusion Equations in Unbounded Domains

First, we introduce the concept of pullback ω-limit compactness for multivalued 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 ω-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 ω-limit compactness. Finally, we illustrate the abstract theory with a nonlinear reaction-diffusion model in an unbounded domain.