I've heard the following quoted several times, but I can't seem to prove it or find a proof!
Let $u_0\in C^{\infty}(R;R)$ be of at most exponential growth, namely there exists a $C>0$ such that $|u_0(x)|\leq Ce^{Cx}$ for all $x$. Then there is a unique $u\in C^{1,2}(R_+\times R, R$) such that (1) $u$ and all of its derivatives are of at most exponential growth AND (2) $\partial_t u = \triangle u,\quad u(0,x)=u_0(x)$.
I know how to prove existence, and I know how to prove uniqueness given some integrability or boundedness assumptions. I also know how to prove the above theorem probabilistically so am seeking a nonprobabilistic proof -- any hints/ideas would be much appreciated!