LAUSR

Abstract We consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form u t + a ( t ) A u ( 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 u e from the measured data a e and φ e . 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.