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 ℝn × [0, T] satisfying the heat equation in ℝn × (0, t) and the following:
(i) There exist constants a > 0, 0 < α < 1, and C > 0 such that
(ii) u(x, 0) = 0 for x ∈ ℝn.
Then u(x, t)≡ 0 on ℝn × [0, T]
We also prove that the condition 0 < α < 1 is optimal.