Determination of the source function and the parabolic equation coefficient (Q1363287)

This paper examines the uniqueness of the solution to the following problem: In the domain \(Q_T=\{t,x\mid 0<t<T, 0<x<x_0\}\), consider the problem of finding two functions \(u(t,x)\) and \(F(t,x)= f(t)g(x)\) satisfying the conditions: \[ u_t(t,x)= u_{xx}(t,x)+ f(t)g(x),\quad (t,x)\in Q_T, \] \[ u(0,x)= u^0(x),\;u(T,x)= u^1(x),\;x\in[0,x_0],\quad u_x(t,0)= u_x(t,x_0)= 0,\;t\in[0,T], \] and \(\int^{x_0}_0 u(t,x)dx= b(t)\), \(t\in[0,T]\). We assume that these conditions are consistent. The problem of finding a source of the form \(f(t)g(x)\) arises, e.g., in studying the influence of radioactive decay on crust temperature.