$\textbf{The problem}$: Let $[-r_0,r_0]$ be a segment in $\mathbb{R}$, let $T>0$ and $u$ be a smooth function satisfying: \begin{align} \begin{cases} u_t-\Delta u = 0 & \qquad \text{on $[-r_0,r_0] \times [0,T]$;} \\ |u(x,t)| \leq \epsilon & \qquad \text{on $[-r_0,r_0] \times [0,T]$;}\\ \end{cases} \end{align}
I would like to obtain a bound for $|u_{tt}(x,t)|$ on $[-r_0,r_0] \times [0,T]$, but i have difficulty finding it.
$\textbf{My idea}$: To make use of Prop. 12.1, Chapter 5 of DiBenedetto's 'PDE's'. Before stating the proposition, we first define the cilinder $Q_{\rho}$ with vertex at $(x_0,t_0)$ as: \begin{align} (x_0,t_0)+Q_{\rho} = \left[|x-x_0|<\rho \right] \times \left[t_0 - \rho^2,t_0 \right] \end{align}
$\textbf{Proposition:} $Let $E\subset \mathbb{R}^N$ and let $u \in C^{1,2}(E \times (0,T) )$ be a solution to the heat equation in $E \times (0,T)$. There exist constants $\gamma$ and $C$ depending only on $N$ such that for every box $(x_0,t_0)+Q_{4\rho} \subset E \times (0,T)$ $$ \sup_{(x_0,t_0)+Q_{\rho}} \left|D^{\alpha}u \right| \leq \gamma \frac{C^{|\alpha|}|\alpha|!}{\rho^{\alpha}} ⨍_{(x_0,t_0)+Q_{4\rho}} \left|u \right|\,dy \, ds $$ for all multi-indices $\alpha$. Moreover $$\sup_{(x_0,t_0)+Q_{\rho}} \left|\frac{\partial^k u}{\partial t^k} \right| \leq \gamma \frac{C^{2k}(2k)!}{\rho^{2k}} ⨍_{(x_0,t_0)+Q_{4\rho}} \left|u \right|\,dy \, ds $$ for all integers $k$.
I don't see how to apply this proposition such that we get an explicit bound for $|u_{tt}(x,t)|$ on the whole $[-r_0,r_0] \times [0,T]$.
Or am i wrong in general to consider this proposition to help me out?
Thank you in advance.