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Let $\Omega$ be an open bounded subset of $\mathbb R^N$.

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$

Consider the following boundary value problem for the heat equation: $$ \begin{cases} u_t - \Delta u = f \\ u|_{\partial\Omega} = 0\\ u(0) = u_0 \end{cases} $$

Questions:

  • Let $k \ge 2$. Assume $u_0 \in C^k(\bar \Omega)$, $f \in C^k([0,T) \times\bar \Omega)$ such that $u_0 = \Delta u_0 = 0$ on $\partial \Omega$ and assume that $\Omega$ is of class $C^k$. Is it true that there exists a unique solution $u \in L^\infty((0,T)\times\bar\Omega) \cap C^k([0,T)\times \bar\Omega)$? How can one prove it? Do we need some additional assumptions on $f$?

  • Fix $U$ a neighborhood of $x_0 \in \Omega$, and assume that $u_0 \in C^k(U)$, and $f \in C^k([0,T) \times U)$. Is it true that there exists a unique (weak) solution of the heat equation that is regular in $U$, that is $u \in C^k([0,T)\times U) \cap L^\infty$?

  • Are the results in the first two questions true even if we assume $\Omega$ Lipschitz? An are they true with less regularity assumptions on $f$?

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    $\begingroup$ Welcome to MathOverflow! Do I understand correctly that you are merely interested in the regularity of $u$ on $(0,T) \times \Omega$, and not on $(0,T) \times \overline{\Omega}$? In this case I don't see why the boundary regularity of $\Omega$ (e.g. being Lipschitz or of class $C^k$) should be relevant. $\endgroup$
    – Jochen Glueck
    May 6, 2019 at 20:26
  • $\begingroup$ @JochenGlueck I had a few typos. I'm interested in the boundary regularity as well. $\endgroup$
    – user139845
    May 6, 2019 at 21:06
  • $\begingroup$ What are the values of $k$? $\endgroup$
    – Andrew
    May 7, 2019 at 8:52
  • $\begingroup$ @Andrew Natural numbers. $\endgroup$
    – user139845
    May 7, 2019 at 12:20
  • $\begingroup$ Only a small remark on the special case $k=0$: it is shown in the paper "Arendt and Bénilan: Wiener Regularity and Heat Semigroups on Spaces of Continuous Functions (1999)" that the distributional Laplace operator generates a $C_0$-semigroup on the space $C_0(\Omega)$ iff $\Omega$ is Wiener regular (which is for instance fulfilled if $\Omega$ is Lipschitz). Now, one can use the convolution formula for inhomogenious evolution equations to prove continuity results for $u$. $\endgroup$
    – Jochen Glueck
    May 7, 2019 at 16:09

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