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 $f$?