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

We discuss conditions for well-posedness of the scalar reaction-diffusion equation $u_t=\mathrm{\Delta }u+f(u)$ equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition ${\int }_1^{\mathrm{\infty }}1\mathrm{/}f(s)\text{d}s=\mathrm{\infty }$ 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 ${\int }_1^{\mathrm{\infty }}1\mathrm{/}f(s)\text{d}s=\mathrm{\infty }$ and the toy PDE blows-up instantaneously while the reaction-diffusion equation is globally well-posed in $L^2(0,1)$.