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

TL;DR

This paper characterizes the asymptotic behavior of solutions to a semilinear heat equation with Neumann boundary conditions, distinguishing between fast and slow decay solutions based on initial data and solution sign.

Contribution

It provides a detailed characterization of slow and fast solutions, showing the set of initial data leading to fast solutions forms a codimension-one graph in phase space.

Findings

Fast solutions decay exponentially to zero.

Slow solutions decay as negative powers of time.

The initial data set for fast solutions is a codimension-one graph.

Abstract

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 infinity), 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, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.