An approximate solution for nonlinear backward parabolic equations

We consider the backward parabolic equation { u(t) + Au = f(t, u(t)). 0 < t < T. u(T) = g. where A is a positive unbounded operator and f is a nonlinear function satisfying a Lipschitz condition, with an approximate datum g. The problem is severely ill-posed. Using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces. According to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of Holder type for all t is an element of inverted right perpendicular0, Tinverted left perpendicular. An example on heat equations and numerical experiments are given. (C) 2010 Elsevier Inc. All rights reserved.