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 $u_t=\mathrm{\Delta }u+\psi (t)f(u),\text{in }\mathrm{\Omega }\times (0,\mathrm{\infty }),$ \( u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), \) 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 $f(u)=u^p$ $f(u)=u^{p}$ . As a matter of fact, we prove: $\begin{array}{rl}& \text{there is no global solution for any initial data if and only if }\\ & {\int }_0^{\mathrm{\infty }}\psi (t)\frac{f({\parallel S(t)u_0\parallel }_{\mathrm{\infty }})}{{\parallel S(t)u_0\parallel }_{\mathrm{\infty }}}dt=\mathrm{\infty }\\ & \text{for every nonnegative nontrivial initial data }{u}_0\in C_0(\mathrm{\Omega }).\end{array}$ \( \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} \) Here, ${(S(t))}_{t\ge 0}$ $(S(t))_{t\geq 0}$ is the heat semigroup with the mixed boundary condition.