ON THE BOUNDEDNESS OF GLOBAL-SOLUTIONS OF ABSTRACT SEMILINEAR PARABOLIC EQUATIONS

In this paper we consider the question of the long time behavior of so lutions of the initial value problem for evolution equations of the fo rm du/dt + Au(t) = F(u(t)) in a Banach space where A is the infinitesi mal generator of an analytic semigroup and F is a nonlinear function s uch that the initial value problem possesses nonglobal solutions for s ome initial data. We give sufficient conditions to ensure that global solutions [on [0,infinity)] are necessarily bounded in a suitable norm . We give several examples of problems to which the abstract result ap plies. (C) 1997 Academic Press.