On the heat flow for the two-dimensional Gelfand equation

In this paper, we study the nonnegative solutions of semilinear heat equation ut=Δu+eu in Ω×(0,T)Ω×(0,T), where ΩΩ is a bounded domain in R2R2, and show that they are bounded whenever T=+∞T=+∞ and the energy functional along a solution tends to −∞−∞ as t→T−t→T− whenever the maximal existence time TT is finite.