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:

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

for some $f(v)\in L^{\infty}(M\times [0,T])$ 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.

EDIT: More precisely I find applying the Hölder inequality, ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p(M)} \le \int_0^t{\vert \vert \partial_sv(\cdot,s)\vert\vert}_{L^p}\;ds \le {\vert \vert \partial_sv(\cdot,s)\vert\vert}_{L^p(L^p)} t^{1-1/p}$ I noticed that letting $p\to \infty$ then it seems reasonable to expect $(*)$. However this could be a proof if I could show that ${\vert \vert \partial_sv(\cdot,s)\vert\vert}_{L^p(L^p)}$ is bounded also for $p=\infty$ Yet I don't know how to prove this...

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

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

DISCLAIMER: I already asked this in MSE but since I got no answer, I thought it 'd be better to post it also here.

Thanks a lot in advance!