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I'm trying to understand the compatibility conditions for regularity of second order parabolic equations. Let's consider the equation $u_t - Lu = f$ with $u(0)=g$ on $\Omega \times [0,T)$ with $u = 0$ on $\Gamma_T$, the parabolic boundary of $\Omega \times [0,T)$.

We need the assumptions that $\frac{d^{m-1}f}{d^{m-1}}(0) - Lg_{m-1} \in H_0^1(\Omega)$ for the $L^2$ theory to work to conclude smoothness up to the boundary (I'm presuming $f$ and $\Omega$ are smooth). As far as I can tell this is required so that we don't have the following situation:

$u_t - u_{xx} = 0$ on $[0,1] \times [0,1]$ with $u(0,x)=1$ and $u(t,0)=u(t,1)=0$. Here we clearly can't have differentiability up to the boundary since $1 \neq 0$. However after time $t = 0$ it would appear to me that this problem goes away, ie. that the solutions are smooth up to the boundary except at $t=0$. Is this the case? Can people perhaps share insightful examples?

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I don't know what $f$, $g_{m-1}$, and $L$ are in the second sentence. Also, you appear to have a typo in the formula that starts the second paragraph. –  Deane Yang Oct 7 '10 at 14:43
    
I also continue to be baffled why you aren't consulting your classmates and teachers at Courant before posting on MathOverflow. Aren't you able to get helpful answers from them? –  Deane Yang Oct 7 '10 at 14:50
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More often than not I don't get helpful answers. I perhaps would if I bothered professors more but I like to do that in extreme moderation. At least here if people don't care to answer they don't need to. If they have something helpful to say they can offer it. –  Dorian Oct 7 '10 at 15:15
    
I should also add that I have my qualification exams next week and so this is why I am constantly asking questions. I have a million questions which come up and I do indeed ask people here. However some of them I appreciate multiple perspectives which this website offers. –  Dorian Oct 7 '10 at 15:16
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What about more advanced students who have already passed the exam? You have been asking very good, interesting, but often difficult questions that many of us already with phds and published papers can't answer. You should really try to get a better sense of what is expected of you so you don't expend too much effort things that you don't need to know. Also, could you clarify the question above? –  Deane Yang Oct 7 '10 at 17:45
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