A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on Bounded Domains

Abstract

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equationsunder the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case f (u) = u p . As a matter of fact, we prove: there is no global solution for any initial data if and only if the function f satisfiesfor every ǫ > 0 and nonnegative nontrivial initial data u 0 ∈ C 0 (Ω).Here, (S(t)) t≥0 is the heat semigroup with the Dirichlet boundary condition.