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 ut=Δu+f(u) equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition that guarantees global solutions for the related ODE . We investigate well-posedness of the toy PDE ut=f(u) in Lp under this no-blow-up condition. An example is given of a source term f and an initial condition ψ∈L2(0,1) such that and the toy PDE blows-up instantaneously while the reaction–diffusion equation is globally well-posed in L2(0,1).