An approximate solution for nonlinear backward parabolic equations

We consider the backward parabolic equation { ut +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 Hölder type for all t ∈ [0, T ]. An example on heat equations and numerical experiments are given. Mathematics Subject Classification 2000: 35R30, 35K05, 65J22.