Let $S = [0,1]\times [0,T]$ for $0 < T \leq 1$. Let $X([0,T]) = \tilde{C}^{2, \alpha}(S\times [0,T])$ and $Y([0,T])=\tilde{C}^{0, \alpha}(S\times [0,T])$ be functions in the respective parabolic Hölder spaces.
Consider the PDE $u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u + f$ $u(\cdot, 0) \equiv 0$ If $a, b, c, f \in Y([0,T])$, then we have the a-priori estimate $\lVert u \rVert_{X([0,T])} \leq C(T)\lVert f \rVert_{Y([0,T])}$ where $C(T)$ is a constant depending on $T$. This is a standard result.
Now define coefficients $a'(x,t) = \begin{cases}a(x,t) &\text{if $0 \leq t \leq T$}\\ a(x,T) &\text{if $T \leq t \leq 1$} \end{cases}$ and likewise for the other coefficients.
Then it holds that $\lVert a'(x,t) \rVert_{Y([0,1])} \leq \lVert a(x,t) \rVert_{Y([0,T])}$ and similarly for the others.
From this, $\lVert u \rVert_{X([0,T])} \leq C\lVert f \rVert_{Y([0,T])}$ where $C$ does not depend on $T$ anymore.
Is this a valid argument? Can someone elaborate a bit on this? This seems a very easy to get rid of dependence on $T$, so why are people not using it more often?
Thanks