Inverse problem of determining the kernel in an integro-differential equation of parabolic type (Q2251838)

Consider the problem of finding the pair of functions \(u(x,t)\) and \(K(x',t)\), from the equations \[ u_{t}-\Delta u = \int^{t}_0 K(x',t)u(x,t-\tau )d\tau,\,\, u(x,0) = \phi (x),\,\, u(x',0,t) = f(x',t). \] A local existence and uniqueness theorem for the inverse problem is proved.