Uniqueness for inverse heat equation (Q1768037)

Let \(s:[0,T]\to [0,1]\) be a function such that i) \(s(0)=b\in(0,1)\); ii) \(s(t)<1\) for all \(t\in (0,T]\). The author considers the parabolic problem \[ \begin{aligned} D_tu=D_x[a(x,t)D_xu],&\qquad (x,t)\in Q_T, \\ u(x,0)=f(x),&\qquad x\in [0,1],\\ u(0,t)=0,\quad t\in (0,T], &\qquad u(s(t),t)=g(t),\quad t\in (0,T), \end{aligned} \] where \(Q_T=\{(x,t)\in \mathbb R\times (0,T): x\in (0,1)\backslash \{s(t)\}\), and \(a\in C^1([0,1]\times [0,T])\), \(0<a_1\leq a(x,t)\leq a_2\), for all \((x,t)\in [0,1]\times [0,T]\), \(a_1\) and \(a_2\) being two positive constants. The aim of the paper is to recover the boundary datum \(u(1,t)=\psi(t)\), when \(f\), \(g\) and \(s\) are given data. Taking advantage of a uniqueness result by \textit{J.-C. Saut} and \textit{B. Scheurer}, the author can show that \(\psi\) can be uniquely determined in terms of the data.