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Let $M$ be a $C^3-$compact manifold and $v \in W^{2,1}_p(M\times [0,T])$ ($1\le p<\infty$) be the solution of:

$\begin{cases} \partial_t v-\Delta_{M} v=f, \quad M\times [0,T]\\ v(x,0)=v_0, \quad M \end{cases}$

for some $f\in L^{\infty}(M\times [0,T])$ and initial data $v_0$ such that ${\vert \vert v_0 \vert \vert}_{L^\infty}(M) \le C$

I would like to prove that ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^{\infty}(M)} \le Ct$

However applying the Hölder inequality, and using the regularity of the time derivative of $v$, I obtain: ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}(M) \le \hat C t^{1/q}$ without using at all the $L^{\infty}$ assumption on the initial data.

I think I will have to use a maximum principle technique but at this point I don't see from where to begin with.

I would appreciate any help or hints so I can fill in the details on my own.

Thanks a lot in advance!

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