A note on well-posedness of semilinear reaction-diffusion problem with 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 $L^2(0,1)$.