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

TL;DR

This paper establishes a precise necessary and sufficient condition for the existence of global solutions to semilinear parabolic equations with Dirichlet boundary conditions on bounded domains, resolving a long-standing open problem.

Contribution

It provides the first exact criterion linking the non-existence of solutions to an integral condition involving the nonlinearity and the heat semigroup.

Findings

No global solution if and only if the integral diverges for all initial data.

The condition applies to general nonlinearities, including the power case.

Addresses an open problem in the theory of semilinear parabolic equations.

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 equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}), \] 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: \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if } &\mbox{the function } f \mbox{ satisfies} &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\lVert S(t)u_{0}\rVert_{\infty}\right)}{\lVert S(t)u_{0}\rVert_{\infty}}dt=\infty &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition.