Uniqueness result for Almost Periodic Distributions depending on time and space and an application to the unique continuation for the wave equation

Abstract: Let $\Omega\subset \mathbb{R}^N$, $N=1,2,3$, be an open bounded and connected set with continuous piecewise $\mathrm{C}^{\infty}$ boundary. Here we deal with almost periodic distributions of the form $u(t,x)=\sum_{n=0}^{+\infty} c_n S_n(x) \mathrm{e}^{i \lambda_n t}$ where $(c_n){n\in \mathbb{N}}\subset \mathbb{C}$ belong to the space of slowing growing sequences $s^\prime$, and $(\lambda_n^2){n\in\mathbb{N}}\subset \mathbb{R}$ and $(S_n){n\in\mathbb{N}}\subset \mathrm{H}_0^{1}(\Omega)$ are respectively the eigenvalues and eigenvectors of the Laplacian. Given $\omega\subset\Omega$, we prove that there exists $T{max}(\Omega,\omega)>0$ depending only on $\Omega$ and $\omega$ such that if $T>T_{max}(\Omega,\omega)$ and $u|_{\omega\times ]-T,T[}=0$, then $u\equiv 0$. Using this result we prove a unique continuation property for the wave equation.