# A-priori bound on parabolic PDE that doesn't depend on end time

I have a PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_{x} + c(x,t)u + f$$ where the coefficients are in parabolic Holder space $\widetilde{C}^{0, \alpha}(I \times [0,T])$ where $I=[0,2\pi]$. The a-priori bound (eg. from Krylov's book) is $$\lVert u \rVert_{\widetilde{C}^{2, \alpha}(I \times [0,T])} \leq C\left(\lVert f \rVert_{\widetilde{C}^{0, \alpha}(I \times [0,T])} + \lVert u_0 \rVert_{\widetilde{C}^{2, \alpha}(I \times [0,T])}\right)$$ where the constant $C$ depends on the endtime $T.$

Does anyone know how to remove this dependence on $T$? I think there might be a way to do it by considering the PDE obtained when we change the coefficients somehow but am not sure.

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What is $I$? An interval in the real line? Bounded or not? – YangMills Jul 30 '12 at 18:26
@YangMills yes, $I=[0,2\pi]$. – poe Jul 30 '12 at 18:34
What is the norm of $f$ and $u_0$ in the estimate? – timur Jul 30 '12 at 19:45
Sorry for the omissions. @timur I edited it. – poe Jul 30 '12 at 20:50
And what happens if you look at simple cases where for example $a$ and $b$ are identically zero and $f$ is equal to $1$? – Deane Yang Jul 30 '12 at 20:58