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We discuss conditions for well-posedness of the scalar reaction-diffusionequation $u_{t}=\Delta u+f(u)$ equipped with Dirichlet boundary conditionswhere the initial data is unbounded. Standard growth conditions are juxtaposedwith the no-blow-up condition $\int_{1}^{\infty}1/f(s) \d s=\infty$ thatguarantees global solutions for the related ODE $\dot u=f(u)$. We investigatewell-posedness of the toy PDE $u_{t}=f(u)$ in $L^{p}$ under this no-blow-upcondition. 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 toyPDE blows-up instantaneously while the reaction-diffusion equation is globallywell-posed in $L^{2}(0,1)$.

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