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) = 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 integral(infinity)(1) 1/integral (s) = infinity that guarantees global solutions for the related ODE (u) over dot = 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 epsilon L-2(0, 1) such that integral(infinity)(1) 1/f (s) ds = infinity and the toy PDE blows-up instantaneously while the reaction-diffusion equation is globally well-posed in L-2(0,1). (C) 2011 Elsevier Inc. All rights reserved.