I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation:

\begin{equation} u_{t} = \Delta u \quad \mathrm{in} ~\ (- \infty, 0] \times \mathbb{R}^{n} \end{equation}

In order to prove it is constant function, we can mimic the proof of Liouville's theorem in complex analysis by showing that $\nabla_{x} u(t,x)$ goes to zero. In usual heat equation defined in $(0, T] \times \mathbb{R}^{n}$, we have the following gradient estimation:

\begin{equation} \sup_{x \in \mathbb{R}^{n}}| \nabla_{x} u(T,x)| \leq \frac{C}{\sqrt{T}} \sup_{x \in \mathbb{R}^{n}} |u_{0}| \end{equation} where C is a constant relates to its dimension $n$ and $u_{0}$ is the initial value of $u(t,x)$. Originally, I want to change the variable $t \rightarrow -t$ such that I can apply the above gradient estimation to show that the solution of ancient heat equation $u(t,x)$ is constant function. However, there is a problem that $\sqrt{T}$ would be complex under this change of variable. Therefore, how can I combined these two concepts to show bounded solution for ancient heat equation is constant function, similar to Liouville's theorem in Complex analysis?