Let $\Omega$ be open and bounded in $\mathbb{R}^n$ and $I$ an interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the vector associates to the eigenvalues.
We denote $$u=\sum_{n=1}^{+\infty} [(f,e_n) \cos[(t-t_0) \sqrt{\lambda_n}] + \dfrac{(g,e_n)}{\sqrt{\lambda_n}}\sin[(t-t_0)\sqrt{\lambda_n}]]e_n$$
such that $u \in C^0(I,H^1_0(\Omega))$ and $\dfrac{\partial u}{\partial t} \in C^0(I,L^2(\Omega))$.
How we can prove that $$\Box u = 0 \quad\text{in}\quad \mathcal{D}'(I^{0} \times \Omega)$$ where $\Box$ is wave operator, ($\Box u = \dfrac{\partial^2 u}{\partial t^2} - \Delta u$, and $I^{0}$ is the interior of the interval $I$?
Thanks for the help.