A necessary and sufficient condition for the existence of global solutions to reaction-diffusion 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 $$ u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), $$ under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case $f(u)=u^{p}$ . As a matter of fact, we prove: $$ \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} $$ Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the mixed boundary condition.