Let $u(x, t)$ be the solution of the following Cauchy problem for the heat equation given by \begin{align*} u_{t}(x, t) - u_{xx}(x, t) &= 0 \quad (x, t) \in \mathbb{R} \times (0, T)\\ u(x, 0) &= 0. \end{align*} Furthermore, suppose there are positive absolute constants $C$ and $a$ such that $|u_{t}(x, t)|$, $|u_{x}(x, t)|$, $|u_{xx}(x, t)|$, and $|u(x, t)|$ are all $\leq Ce^{ax^{2}}$. The problem I am working on is to show that $u \equiv 0$ for $t \in (0, T)$.
This follows from the maximum principle for the Cauchy problem as mentioned on Page 57 of Evans, however, this result is much stronger than needed since it just requires that $u(x, t) \leq Ae^{ax^{2}}$ and the proof of the result is quite long.
Is there an alternative proof of the above fact that uses $|u_{t}(x, t)|$, $|u_{x}(x, t)|$, $|u_{xx}(x, t)|$, and $|u(x, t)|$ are all $\leq Ce^{ax^{2}}$?