It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation \begin{equation}\label{pf2} \begin{aligned} \begin{cases} \partial^2_{t}u- \Delta u=0\,\quad &\text{on $(0,T)\times \Omega$}, \\ u=f\,\quad &\text{on $(0,T)\times \partial \Omega$,}\\ u(0,x)=\partial_t u(0,x)=0\,\quad &\text{on $\Omega$,} \end{cases} \end{aligned} \end{equation} admits a unique solution $u\in C(0,T;L^2(\Omega))\cap C^1(0,T;H^{-1}(\Omega))$ with $\partial_\nu u \in H^{-1}((0,T)\times \partial \Omega)$.
I want to derive these estimates via a slightly unorthodox method, namely by considering spectral decomposition in the spatial variables. Indeed, let us write $\{\phi_k\}_{k=1}^{\infty}$ and $\{\lambda_k\}_{k=1}^{\infty}$ for the Dirichlet eigenfunctions and eigenvalues of $-\Delta$ respectively.
Writing $u(t,x)= \sum_{k=1}^{\infty} u_k(t)\phi_k(x)$, it is trivial to see that $$\partial^2_t u_k(t) + \lambda_k u_k(t) = -\int_{(0,T)\times \partial \Omega} f(t,x)\,\partial_\nu \phi_k(x)\,dx$$
How can I conclude the classical properties of $u$ above from this representation.
