An approximate solution for nonlinear backward parabolic equations

Abstract

We consider the backward parabolic equation { u t + A u = 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 Hölder type for all t ∈ [ 0 , T ] . An example on heat equations and numerical experiments are given.