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 = Δ 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 ∫ 1 ∞ 1 / f ( s ) d s = ∞ that guarantees global solutions for the related ODE 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 ψ ∈ L 2 ( 0 , 1 ) such that ∫ 1 ∞ 1 / f ( s ) d s = ∞ and the toy PDE blows-up instantaneously while the reaction–diffusion equation is globally well-posed in L 2 ( 0 , 1 ) .