Some inverse problems for parabolic equations (Q1363987)

The unique solvability of the following inverse problems for parabolic equations is studied: In the domain \(Q_T=\{t,x\mid 0<t<T, 0<x<x_0\}\) two functions \(u(t,x)\), \(F(t,x)= f(t)+ g(x)\) are to be found, satisfying the relationships \[ u_t= u_{xx}+ (f(t)+ g(x))u,\quad (t,x)\in Q_T, \] \[ u(0,x)= u^0(x),\quad u(T,x)= u^1(x),\quad x\in[0,x_0], \] \[ u_x(t,0)= u_x(t,x_0)= 0,\quad u(t,x_1)= b(t),\quad t\in[0,T],\quad 0<x_1<x_0. \] The input data are supposed to be consistent.