Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation $$ \begin{cases} \frac{\partial u}{\partial t} - \Delta u &= 0 \qquad \text{on } \Omega \times (0,T]\\ u &= g \qquad \text{on } \partial \Omega \times [0,T]\\ u &= h \qquad \text{on } \Omega \times \{t = 0\} \end{cases} $$ The existence of a weak solution to this equation has been proven Theorem 6.1 in Non-Homogeneous Boundary Value Problems and Applications II (Lions and Magenes, 1972). Throughout this book it is assumed that $\Omega$ is a $C^{\infty}$-domain.

I was wondering if this condition can be relaxed for certain domains, in particular for a hypercube $\Omega = [0,1]^n \subset \mathbb{R}^n$? Texts other than Lions and Magenes only ever seem to the consider homogeneous Dirichlet condition $g \equiv 0$. Furthermore, a book considering parabolic equations that is analogous to Elliptic Problems in Nonsmooth Domains (Grisvard, 2011) doesn't seem to exist.

Is there anything known about the existence of solutions to such PDEs?