Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition

summary:We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value $u_0\in H^1(\Omega )$ is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe's method is constructed for the problem when $u_0\in L^2(\Omega )$ and the integral kernel in the nonlocal boundary condition is symmetric