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

We investigate the existence, uniqueness, and blowup of solutions to the reaction-diffusion equation ut = Delta u + lambda e(u), lambda > 0. The equation admits in any space dimension n > 2 the singular solution U(x) = -2log/x/ + log(2(n - 2)/lambda). In dimensions n greater than or equal to 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 less than or equal to u(0)(x) less than or equal to U(x), while there exists no solution of the equation defined in a strip of the form Q = R-n x (0, T) for any T > 0 if u(0)(x) > U(x). T;Ve prove here that in the physical dimension n = 3 such borderline behaviour fails. Indeed, we show that for every dimension 3 less than or equal to n less than or equal to 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 u(0)(x) = U(x) + c(#) mark the borderline between global existence and instantaneous blowup. 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.