Let $(M,g(t))$ is a Ricci flow. And $$ \Box = \partial_t -\Delta_{g(t)} $$ is heat operator, which coupled to Ricci flow. If $$ \Box u =0 ~~~~~~\text{and }~~~~~~ \Box |\nabla u|\le 0 $$ Then how to show
If $u(\cdot, t_1)$ is L-Lipschitz, then so is $u(\cdot, t)$ for all $t\ge t_1$.
This problem is from the 7th page of Entropy and heat kernel bounds on a Ricci flow background. Namely the picture below.
By Arctic Char's hint, I have an answer. I use the follow maximum principle,which is from Topping's Lectures on the Ricci flow.
Besides, $u$ is smooth (I forgot it earlier). Since $|\nabla u(\cdot, t_1)|\le L$, in the theorem 3.1.1, letting $ X(t)=0, F(u,t)=0, \alpha=L$, then $\phi(t)=L$. Therefore, I have $$ |\nabla u(\cdot,t)|\le L ~~~\forall t\ge t_1 $$ namely, $u(\cdot,t)$ is L-Lipschitz.