Approximation of a parabolic non-linear evolution equation backwards in time

Abstract

We consider the inverse time problem for the non-linear heat equation in the form ut+Au=f(u), u(1)=X, where A is any non-negative, self adjoint operator. Using the strongly continuous contraction semi-group generated by Abeta =-A(I+ beta A)-1, beta >0, we derive an estimation of the error on the whole interval (0,1) between ut(t), solution of the initial problem, and upsilon beta (t), solution of the regularized problem: upsilon beta (t)+Abeta upsilon beta (t)=e-(1-t beta AA( beta )).f( upsilon beta ), upsilon beta (1)=X.