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Your approach works fine, just start with negative first momentmoments of time:
\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(t-T,x)| ,\quad t\le 0,\ T>0. \end{equation}
Your approach works fine, just start with negative first moment of time:
Your approach works fine, just start with negative moments of time:
