On the inverse source problem for the wave operator (Q1373043)

We study the following inverse source problem for the wave operator \[ P_n:={\partial^2\over\partial t^2}- a^2\Delta_n\quad a>0, \] where \(\Delta_n\) is the Laplace operator, on the closed strip \(\overline G_T:= \{(x,t)\in\mathbb{R}^{n+1}: x\in\mathbb{R}^n\), \(0\leq t\leq T\}\), \(T>0\). Given any distribution \(v\in {\mathcal D}'(\mathbb{R}^{n+1})\) with \(\text{supp }v\subseteq \mathbb{R}^n\times(t>T)\), satisfying the wave equation \[ P_nv(x,t)= 0,\quad t>0,\quad\text{in }{\mathcal D}'(\mathbb{R}^n\times (t>T)), \] find a (source) distribution \(\nu\) with \(\text{supp }\nu\subseteq\overline G_T\) such that the wave potential \(E_n*\nu\), the convolution of the fundamental solution \(E_n\) of \(P_n\) and \(\nu\), satisfies the condition \(E_n*\nu(x,t)= v(x,t)\), \(t>T\). We deal with the solvability, the structure of the set of solutions, and the stability of the proposed problem.