Heat equation and gradient estimation

Question

Assuming that $u_0 \in C^0(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ and considering the heat equation $$u_t - \Delta u = 0$$ in $\mathbb{R}^n \times (0,+\infty)$ and $u(x,0) = u_0(x)$ in $\mathbb{R}^n$, it is possible to show that some estimative in the following sense: $$||\nabla u(x,t)|| \leq C(t)||u_0||_{\infty}?$$