Reference request: Schauder estimate in the space variable for parabolic equations

Question

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

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?

Answer 1

For the heat equation in $\mathbb R^n$ the estimate holds. The main thing is the smoothness of the volume potential $g=\Gamma*f$, where $\Gamma$ is the fundamental solution. It is obtained in O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, ch.4, $\S2$. Though in the beginning of the paragraph the assumption on $f$ is Holder continuity wrt $x$ and $t$ both, in the process it is proven that $$ |D^2_xg(x,t)-D^2_xg(x',t)|\le C|x-x'|\sup_{s\in[0,T]}\|f(\cdot,s)\|_{C^\alpha(\mathbb R^n)},\quad x,x'\in \mathbb R^n,\ t>0. $$