recovered in (algebraic) form by the method of separation of variables and is also a solution to the heat equation, as can be seen from its explicit form. - Theorem A.1. Let u(x,t) be a solution to the heat equation on R^n for t > 0. Then (i) u(x,t) is smooth in x and t; (ii) u(x,t) -> f(x) as t -> 0+ where f is the initial data; (iii) max_x |u(x,t)| <= max_x |f(x)| for all t > 0. - Corollary A.2. If f is bounded and continuous, then there exists a unique solution u(x,t) to the heat equation with initial data f which satisfies the conditions of Theorem A.1. Proof. By the maximum principle, any two solutions must coincide if they have the same initial data. Thus uniqueness follows from the fact that zero is the only bounded continuous function satisfying the homogeneous problem. Existence can be shown by approximating f by smooth functions and using the explicit formula derived earlier. - Remark: This result generalizes to other parabolic equations under suitable assumptions.