# Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}\le \hat C t^{1/q}$

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!