KERNEL SECTIONS OF MULTI-VALUED PROCESSES WITH APPLICATION TO THE NONLINEAR REACTION-DIFFUSION EQUATIONS IN UNBOUNDED DOMAINS
First, we introduce the concept of pullback omega-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 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.