On the Boundedness of Global Solutions of Abstract Semilinear Parabolic Equations

In this paper we consider the question of the long time behavior of solutions of the initial value problem for evolution equations of the form durdtqAu t.s F u t.. in a Banach space where A is the infinitesimal generator of an analytic semigroup and F is a nonlinear function such that the initial value problem possesses nonglobal solutions for some initial data. We give sufficient conditions to ensure that global solutionswonw0, `.xare necessarily bounded in a suitable norm. We give several examples of problems to which the abstract result applies