A note on well-posedness of semilinear reaction-diffusion problem with singular initial data

TL;DR

This paper examines conditions under which semilinear reaction-diffusion equations with unbounded initial data are well-posed, highlighting differences between related ODE and PDE behaviors, especially regarding blow-up and global solutions.

Contribution

It provides new insights into the well-posedness of reaction-diffusion equations with singular initial data, contrasting standard growth conditions with the no-blow-up criterion.

Findings

The no-blow-up condition ensures global solutions for the related ODE.

An example shows PDE can be well-posed even if the toy ODE blows up.

Reaction-diffusion equations can be globally well-posed despite singular initial data.

Abstract

We discuss conditions for well-posedness of the scalar reaction-diffusion equation $u_{t}=\Delta u+f(u)$ equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition $\int_{1}^{\infty}1/f(s) \d s=\infty$ that guarantees global solutions for the related ODE $\dot u=f(u)$. We investigate well-posedness of the toy PDE $u_{t}=f(u)$ in $L^{p}$ under this no-blow-up condition. An example is given of a source term $f$ and an initial condition $\psi\in L^{2}(0,1)$ such that $\int_{1}^{\infty}1/f(s)\d s=\infty$ and the toy PDE blows-up instantaneously while the reaction-diffusion equation is globally well-posed in $L^{2}(0,1)$.