On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equationequation(NLH)ut−Δu=α|u|u,ut−Δu=|u|αu, in the unit ball Ω   of RNRN with Dirichlet boundary conditions, in the subcritical case. More precisely, we study the set GG of initial values in C0(Ω)C0(Ω) for which the resulting solution of (NLH) is global. We obtain very precise information about a specific two-dimensional slice of GG, which (necessarily) contains sign-changing initial values. As a consequence of our study, we show that GG is not convex. This contrasts with the case of nonnegative initial values, where the analogous set G+G+ is known to be convex.