Regularity result for the boundary value problem for the heat equation

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 $$f$$?

• 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. – Jochen Glueck May 6 at 20:26
• @JochenGlueck I had a few typos. I'm interested in the boundary regularity as well. – user139845 May 6 at 21:06
• What are the values of $k$? – Andrew May 7 at 8:52
• @Andrew Natural numbers. – user139845 May 7 at 12:20
• 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 $u$. – Jochen Glueck May 7 at 16:09