Regularity of Weak Solutions and Their Attractors for a Parabolic Feedback Control Problem

We investigate additional regularity properties of all globally defined weak solutions, their global and trajectory attractors for a class of autonomous differential inclusion with upper semi-continuous interaction function, when initial data $u_{\tau}\in L^2(\Omega)$ . The main contributions of this paper are: (i) additional regularity and new topological properties of all weak solutions of parabolic feedback control problem with upper semi-continuous interaction function, (ii) a sufficient condition for regularity of global and trajectory attractors, and (iii) new a priory estimates for all weak solutions.