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

In this paper, we consider the nonlinear heat equation(NLH) u t − Δ u = | u | α u , in the unit ball Ω of R N with Dirichlet boundary conditions, in the subcritical case. More precisely, we study the set G of initial values in C 0 ( Ω ) for which the resulting solution of (NLH) is global. We obtain very precise information about a specific two-dimensional slice of G , which (necessarily) contains sign-changing initial values. As a consequence of our study, we show that G is not convex. This contrasts with the case of nonnegative initial values, where the analogous set G + is known to be convex.