We investigate additional regularity properties of all globally defined weak solutions, their global and trajectory attractors for classes of semi-linear parabolic differential inclusions with initial data from the natural phase space. The main contributions in this note are: (i) sufficient conditions for the existence of a Lyapunov function for a class of parabolic feedback control problems; (ii) convergence results for all weak solutions in the strongest topologies; and (iii) new structure and regularity properties for global and trajectory attractors. Results applied to the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; and (c) a climate energy balance model.
Introduction and Regularity of All Weak Solutions
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