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

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 equations u t = Δ u + ψ ( t ) f ( u ) , in Ω × ( 0 , t ∗ ) , under 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 ∫ 0 ∞ ψ ( t ) f ( ‖ S ( t ) u 0 ‖ ∞ ) ‖ S ( t ) u 0 ‖ ∞ dt = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 ( Ω ) . Here, ( S ( t ) ) t ≥ 0 is the heat semigroup with the Dirichlet boundary condition.