On maximal attractors for dynamical systems. Application to a semilinear parabolic problem with controlled growth

We introduce the concept of a maximal B-attractor for a semigroup, and find conditions under which a semigroup in a metric space possess a maximal B-attractor. We also establish a condition under which a maximal B-attractor is also a maximal attractor, in the sense of Chepyzhov and Goritskiĭ (in: Properties of global attractors of partial differential equations, volume 10 of Adv. Soviet Math., American Mathematical Society, Providence, 1992). Finally, we apply our result to a parabolic semilinear PDE, where the nonlinearity can be unbounded, as long as it grows linearly with a controlled growth constant, and find a maximal B-attractor for it. Furthermore, under a Lipschitz condition on the nonlinearity, with small Lipschitz constant, we prove that this maximal B-attractor is, in fact, a maximal attractor.