Domain of existence and blowup for the exponential reaction-diffusion equation

Abstract

We investigate the existence, uniqueness, and blowup of solutions to the reaction-diffusion equation ut = Δu + λeu, λ > 0. The equation admits in any space dimension n > 2 the singular solution U (x) = -2log |X| + log(2(n - 2)/λ). In dimensions n ≥ 10 this solution plays an important role in defining a domain of existence and uniqueness of solutions of the equation. Thus, the Cauchy problem admits a unique solution for data 0 ≤ u0(x) ≤ U(x), while there exists no solution of the equation defined in a strip of the form Q = ℝn × (0, T) for any T > 0 if u0(x) ≥ U(x). We prove here that in the physical dimension n = 3 such borderline behaviour fails. Indeed, we show that for every dimension 3 ≤ n ≤ 9 the domain of existence expands in the following precise form: there exists a constant c# > 0, depending on n, such that the initial data u0(x) = U(x) + c# mark the borderline between global existence and instantaneous blow-up. In the same dimension range non-uniqueness occurs in a band around the solution U(x). The results extend to dimensions n = 1, 2, even if no singular solution like U exists.