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Let $U\subseteq \mathbb{R}^n$ be a bounded Lipschitz domain, $T>0$, and let $u$ solve, classically: $$\partial_tu-\Delta u = 0 \text{ on } U\times (0,T) \\ u=0 \text{ at }t=0$$

Can I conclude that $u$ is real analytic in space, for $t>0$? How about the case $t\geq 0$?

I am sorry for the probably easy question for the experts among you but by browsing around I couldn't find, up to now, a reasonable result in this direction. Would you be so kind to point me to a reference? Thank you in advance.

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  • $\begingroup$ Do you mean $u=0$ on $\partial U$? Otherwise the solution is just zero. $\endgroup$ Jun 8, 2022 at 15:44
  • $\begingroup$ Why so? I might impose a non-homogeneous Dirichlet condition at $\partial U \times (0,T)$, say, $u=g$ and $g$ vanishing at $t=0$, and the solution would not be zero @TrevorNorton $\endgroup$
    – Lilla
    Jun 8, 2022 at 15:52
  • $\begingroup$ That's true. I was supposing homogenous BCs. Do you know what the BCs are in this case? Are you supposing smooth Dirichlet BCS? $\endgroup$ Jun 8, 2022 at 15:56
  • $\begingroup$ @TrevorNorton I am assuming $u=g$ with $g$ be a smooth function in a neighbourhood of $\partial U \times (0,T)$ (because I can't define spatial smoothness by just requiring $g$ to live on $\partial U$) $\endgroup$
    – Lilla
    Jun 8, 2022 at 16:00

1 Answer 1

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On page 62 of "Partial Differential Equations" by Evans it says Evans

The book [M] is "Partial Differential Equations" by Mikhailov.

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