Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$.

Let $$u\in C^{2,1}(M\times[0,T))\cap C^\infty(M\times(0,T))$$ solve $$Lu=f$$ where $$f\in C^0(M\times[0,T))\cap C^\infty(M\times(0,T))$$ is such that $f(\cdot,t)\in C^\alpha(M)$ for any $t\in[0,T)$ and $$\sup_{t\in[0,T)}\|f(\cdot,t)\|_{C^\alpha(M)}<\infty.$$

QUESTION: Do we have the follow

a prioriestimate? $$\sup_{t\in[t_0,T)}\|u(\cdot,t)\|_{C^{2+\alpha}(M)}\leq C\bigg(\sup_{s\in[t_0,T]}\|f(\cdot,s)\|_{C^\alpha(M)}+\|u\|_{L^\infty(M\times[0,T])}\bigg)$$

This estimate was needed for proving the global existence of a solution for the harmonic map heat flow, which is a quasilinear parabolic equation: $$(\Delta_g-\partial_t)u=\Pi(u)(du,du).$$ Basically it says that $u(\cdot,t)$ does not blow up as $t\nearrow T$.

This is stated on page 182 of the book *Variational Problems in Geometry* by Seiki Nishikawa, where the author calls it Schauder estimate. However, the standard Schauder estimate, which involves the Hölder seminorm of $f$ in $t$ as well. The author lists Gilbarg–Trudinger (which obviously do not talk about parabolic equations) and a Japanese book (I don't speak Japanese) for such Schauder estimates.

So is the above estimate correct? If so, where can I find a proof?

Thanks in advance!

Edit: A possible approach might be to use the heat kernel, so that $u$ has an inegral expression which we can analyze. However I'm totally unfamiliar with estimates on the heat kernel. For example, is the estimate true for $M=\mathbb{R}^n$?

Edit: On page 316 of *Partial Differential Equations III: Nonlinear Equations* (Second Edition) by Michael E. Taylor, $(1.13)$ is the following $$\|e^{t\Delta}\|_{\mathcal{L}(C^r,C^{r+s})}\leq C_st^{-s/2},\quad0<t\leq1,$$ where $\Delta$ is the Laplacian on a complete Riemannian manifold. This implies my wanted estimate, but unfortunately no proof was given. I skimmed the earlier parts of the book and think maybe this follows from the estimates of pseudo-differential operators acting on Hölder–Zygmund spaces that is established earlier. Am I right?