A New 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 $$u_t=\mathrm{\Delta }u+\psi (t)f(u),in\mathrm{\Omega }\times (0,t^{\ast }),$$ 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: $$\begin{array}{rlrlr}{\displaystyle }& {\displaystyle thereisnoglobalsolutionforanyinitialdataifandonlyif}& {\displaystyle thefunctionfsatisfies}& {\displaystyle {\int }_0^{\mathrm{\infty }}\psi (t)\frac{f(\parallel S(t)u_0{\parallel }_{\mathrm{\infty }})}{\parallel S(t)u_0{\parallel }_{\mathrm{\infty }}}dt=\mathrm{\infty }}& {\displaystyle forevery\epsilon >0andnonnegativenontrivialinitialdata{u}_0\in C_0(\mathrm{\Omega })\mathrm{.}}\end{array}$$ Here, $(S(t){)}_{t\ge 0}$ is the heat semigroup with the Dirichlet boundary condition.