Uniqueness in the Cauchy problem for the heat equation

We relax the growth condition in time for uniqueness of solutions of the Cauchy problem for the heat equation as follows: Let u(x,t) be a continuous function on R-n x [0, T] satisfying the heat equation in R-n x (0, t) and the following: (i) There exist constants a > 0, 0 < alpha < 1, and C > 0 such that \u(x, t)\ less than or equal to C exp[(a/t)(z) + a\x\(2)] in R-n x (0, T). (ii) u(x, 0) = 0 for x is an element of R-n. Then u(x, t) = 0 on R-n x [0, T]. We also prove that the condition 0 < alpha < 1 is optimal.