Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good theory for existence and uniqueness of solutions to \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \Delta_g u=0\,\quad &\text{on $\mathbb R^+\times \Omega$}, \\ u(t,x)=f(t,x)\,\quad &\text{on $\Sigma=\mathbb R^+\times \partial \Omega$,}\\ u(0,x)=0 \,\quad &\text{on $\Omega$.} \end{cases} \end{aligned} \end{equation} subject to some decay properties at infinity? For example, ideally I would like to know if there are suitable assumptions on the metric $g$ and the boundary data $f$ that guarantees existence and uniqueness of a solution $u \in L^2(\mathbb R\times \Omega)$ or perhaps some other Sobolev space with some weight representing growth or decay of solutions when $t$ approaches infinity.
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