Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity(P){∂tu=Δu+eu,x∈RN,t>0,u(x,0)=u0(x),x∈RN, where u0 is a continuous initial function. In this paper, we consider the case where u0 decays to −∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for u0 is related to the decay rate of forward self-similar solutions of ∂tu=Δu+eu.