A concrete realization of the slow-fast alternative for a semi linear heat equation with homogeneous Neumann boundary conditions

We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions.It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infnity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, andwe show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space