An inverse source problem with single Dirichlet type measured output data for a linear parabolic equation

The problem of determining an unknown source term in a linear parabolic equation ut=(k(x)ux)x+F(x,t)ut=(k(x)ux)x+F(x,t), (x,t)∈ΩT(x,t)∈ΩT, from the Dirichlet type measured output data h(t):=u(0,t)h(t):=u(0,t) is studied. A formula for the Fréchet gradient of the cost functional J(F)=‖u(0,t;F)−h(t)‖2J(F)=‖u(0,t;F)−h(t)‖2 is derived via the solution of the corresponding adjoint problem, within the weak solution theory for PDEs and the quasi-solution approach. The Lipschitz continuity of the gradient is proved. Based on the obtained results the convergence theorem for the gradient method is proposed.