Determination of source terms in diffusion and wave equations by observations after incidents: uniqueness and stability

We consider a diffusion and a wave equations: $$\partial_t^ku(x,t) = \Delta u(x,t) + \mu(t)f(x), \quad x\in \Omega, \, t>0, \quad k=1,2$$ with the zero initial and boundary conditions, where $\Omega \subset \mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of 1. determining $\mu(t)$, $0<t<T$ with given $f(x)$; 2. determining $f(x)$, $x\in \Omega$ with given $\mu(t)$ \end{itemize} by data of $u$: $u(x_0,\cdot)$ with fixed point $x_0\in \Omega$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_1$, by assuming that $T<T_1<T_2$ and $\mu(t)=0$ for $t\ge T$, which means that the source stops to be active after the time $T$ and the observations are started only after $T$. This assumption is practical by such a posteriori data after incidents, although inverse problems had been well studied in the case of $T=0$. We establish the non-uniqueness, the uniqueness and conditional stability for a diffusion and a wave equations. The proofs are based on eigenfunction expansions of the solutions $u(x,t)$, and we rely on various knowledge of the generalized Weierstrass theorem on polynomial approximation, almost periodic functions, Carleman estimate, non-harmonic Fourier series.