Reference for Hölder estimate on parabolic equation with Neumann boundary condition

Question

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.

Answer 1

At least when $\Omega$ is bounded and $T<T_0<\infty$, it can be done with the Sobolev spaces theory for parabolic equations. For $Q_T=\Omega\times[0,T]$ taking $p$ close enough to infinity we have (using the embedding theorem) $$ |u|_{C^{1+\delta}(Q_T)}\le C |u|_{W_{p}^{2,1}(Q_T)}\le C |f|_{L_p(Q_T)}= C\left(\int_0^T\int_\Omega|f|^p\, dxdt\right)^{1/p}\le CT^{1/p}|f|_{L_\infty(Q_T)}. $$