Regularization of an inverse nonlinear parabolic problem with time-dependent coefficient and locally Lipschitz source term

We consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form ut+a(t)Au(t)=f(t,u(t)) subject to the final condition u(T)=φ. Here A is a positive self-adjoint unbounded operator in a Hilbert space H and f satisfies a locally Lipschitz condition. This problem is ill-posed. Using quasi-reversibility method, we shall construct a regularized solution uε from the measured data aε and φε. We show that the regularized problems are well-posed and that their solutions converge to the exact solutions. Error estimates of logarithmic type are given and a simple numerical example is presented to illustrate the method as well as verify the error estimates given in the theoretical parts.