Stability results for the heat equation backward in time

For the heat equation backward in time ut = u(xx), x is an element of R, t is an element of(0,T), parallel to u(.T)-phi(.)parallel to(Lp(R)) <= is an element of Subject to the constraint parallel to u(.,0)parallel to(Lp(R)) <= E with T > 0, phi is an element of L-p(R), 0 < is an element of < E, 1 < p < infinity being given, we prove that if u(1) and u(2) are two solutions of the problem, then there is a constant c > 0 such that parallel to u(1)(.,t) - u(2)(.,t)parallel to(Lp(R)) <= ce(t/T) E1-t/T, for all t vertical bar 0,T vertical bar. In case p = 2 we establish stability estimates of Holder type for all derivatives with respect to x and t of the solutions. We suggest a useful strategy of choosing mollification parameters which provides a continuity at t = 0 when an additional condition on the smoothness of u(x,0) is given. Furthermore, we propose a stable marching difference scheme for this ill-posed problem and test several related numerical methods for it. (C) 2008 Elsevier Inc. All rights reserved.