On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t + A u ( t ) = f ( u ( t ) , t ) , u ( 1 ) = φ , where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β > 0 ) is well-posed and that its solution U β ( t ) converges on [ 0 , 1 ] to the exact solution u ( t ) as β → 0 + . These results extend some earlier works on the nonlinear backward problem.