Domain of existence and blowup for the exponential reaction-diffusion equation
Juan Luis Vazquez · Indiana University Mathematics Journal · 1999
We investigate the existence, uniqueness, and blowup of solutions to the reaction-diffusion equationThe equation admits in any space dimension n > 2 the singular solution U (x) = -2 log |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 ≤ u 0 (x) ≤ U (x), while there exists no solution of the equation defined in a strip of the form Q = R n × (0, T ) for any T > 0 if u 0 (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 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.