I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here.

Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial_tu-\Delta u$. Let $f\in C^{0,0,\alpha}((0,T)\times M,\mathbb{R})$, $u_0\in C^{2,\alpha}(M,\mathbb{R})$ and let $u\in C^{1,2,\alpha}((0,T)\times M,\mathbb{R})$ be the unique solution of

$$ \begin{cases} Lu=f \\ u(t,.)\to u_0(.) & \text{as $t\to 0$} \end{cases} $$

Here $C^{k,l,\alpha}$, $C^{k,\alpha}$ are the appropriate (parabolic) Hölder spaces.

Then we have

$$||u||_{C^{1,2,\alpha}((0,T)\times M,\mathbb{R})}\le C(||f||_{C^{0,0,\alpha}((0,T)\times M,\mathbb{R})}+||u_0||_{C^{2,\alpha}(M,\mathbb{R})})$$

For some $C>0$ independent of $f,$ $u$ and $u_0$.

Question: Is it possible to choose $C$ independent of $T$? If C depends on $T$, i.e. $C=C(T)$, is it possible to choose $C(T)$ in a way that $C(T)$ is bounded for $T\to 0$?

If one replaces $M$ by a suitable domain in $\mathbb{R}^n$ the answer of my question seems to be "Yes". However, I need the result for compact manifolds and was wondering if there occur any problems.