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)+Au(t) -= f(u(t), t), u(1) = xi, 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 beta > 0) is well-posed and that its solution U-beta(t) converges on [0,1] to the exact solution u(t) as beta -> 0(+). These results extend some earlier works on the nonlinear backward problem. (c) 2014 Elsevier Inc. All rights reserved.