Autonomous and Non-autonomous Unbounded Attractors in Evolutionary Problems

Abstract If the semigroup is slowly non-dissipative, i.e., its solutions can diverge to infinity as time tends infinity, one still study dynamics via approach by unbounded attractors—the counterpart of classical notion global attractors. We continue development this theory started Chepyzhov and Goritskii (Unbounded attractors evolution equations. Advances in Soviet mathematics, American Mathematical Society, Providence, 1992). provide abstract results on attractor existence, we properties these attractors, well $$\omega $$ ω -limit sets non-dissipative setting. also develop pullback non-autonomous theory. The that illustrated analysis autonomous problem governed equation $$u_t = Au + f(u)$$ u t = A + f ( ) . In particular, using inertial manifold approach, criteria under which coincides with graph Lipschitz function, or becomes close function for large argument.