Abstract

The problem of finding the pair of functions \(\{u(x,t),f(x)\}\) from the system \(-\partial u/\partial t + u_{xx} = \varphi(t)f(x),\) \((x,t) \in (0,1)\times(0,1),\quad u(1,t) =0, u_x(0,t)=u_x(1,t) =0, u(x,0) =0, u(x,1) = g(x)\) with \(\varphi\) and \(g\) given, is considered. It is proved that for \(\varphi \not\equiv 0\) the solution is unique. Under various assumptions on \(\varphi\) and \(g\) two regularizations based on the Fourier transform associated with a Lebesgue measure for this ill-posed problem are suggested.