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.