Let us say $\Omega$ is a smooth bounded domain.
I have a baby quesiton: when we can say a solution $u$ of a heat equation is in $C^{2, 1}(\overline\Omega\times (0, T])\cap C(\overline\Omega\times [0, T])$, rather than $C^{2, 1}(\Omega\times (0, T])\cap C(\overline\Omega\times [0, T])$? I can understand that if the initial condition satisfies a certain compatibility condition, then the solution belongs to $C^{2+\alpha, 1+\alpha/2}(\overline\Omega\times [0, T])$, which is classical result. I was told that the compatibility condition only has impact on $t=0$? So if we remove the compatibility condition, then we have $u\in C^{2+\alpha, 1+\alpha/2}(\overline\Omega\times [\epsilon, T])$ for any small $\epsilon$? ..Thanks...
To be more specific, let me say the problem I was considering is
$u_t-\Delta u= f(x, t), \ \ (x, t)\in \Omega\times (0, T]$
$u(x, 0)= u_0(x), \ \ \ x\in\Omega$
$u(x, t)=0, \ \ (x, t)\in \partial\Omega\times (0, T]$
with $f$ and $u_0$ sufficiently smooth. And the compatibility condition is
$u_0(x)=0, \ \ x\in\partial\Omega$, and $-\Delta u_0=f(x, 0)$.