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

TL;DR

This paper investigates inverse problems for diffusion and wave equations, focusing on determining source terms or coefficients from post-incident observations, establishing conditions for uniqueness and stability with practical assumptions on data collection timing.

Contribution

It introduces new uniqueness and stability results for inverse source problems in diffusion and wave equations using observations after incidents, with practical timing assumptions.

Findings

Proves uniqueness of source term identification from limited data.

Establishes stability conditions under specific assumptions.

Provides methods based on eigenfunction expansions and Carleman estimates.

Abstract

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.