I saw a type of Hölder estimate in Friedman's book: Partial Differential Equations of Parabolic Type (page 200, 3.24) which goes as follows:
Suppose we have a uniformly parabolic equation with Hölder coefficients,$$\begin{cases}\partial_tu-Lu=f(x)\quad &\text{in }\Omega\times(0,T]\\ u=0\quad &\text{on }\partial\Omega\\u(x,0)=0\end{cases}$$where $f(x)=0$ on $\partial \Omega$, then for any $0<\delta<1$, there exist constant $K$ and $\sigma$ such that $$|u|_{C^{1+\delta}(\bar{Q}_T)}\leq KT^\sigma|f|_\infty$$
Right now I am concerned with the version with Neumann boundary condition. Suppose we have a uniformly parabolic equation with Hölder continuous (or as smooth as you want)$$\begin{cases}\partial_tu-Lu=f(x)\quad &\text{in }\Omega\times(0,T]\\\frac{\partial u}{\partial v}=0\quad &\text{on }\partial\Omega\\u(x,0)=0\end{cases}$$$\Omega$ has smooth boundary and $f$ satisfies the compatibility condition and as smooth as you want, do we have the following Hölder estimate:$$|u|_{C^{1+\delta}(Q_T)}\leq CT^\sigma|f|_{\infty}$$
I believe this is correct, but I don't know where to find the reference. Friedman's book ceased to talk about the Neumann boundary condition case.
Thank you very much.
