Stability results for the heat equation backward in time

Abstract

For the heat equation backward in time u t = u x x , x ∈ R , t ∈ ( 0 , T ) , ‖ u ( ⋅ , T ) − φ ( ⋅ ) ‖ L p ( R ) ⩽ ϵ subject to the constraint ‖ u ( ⋅ , 0 ) ‖ L p ( R ) ⩽ E with T > 0 , φ ∈ L p ( R ) , 0 < ϵ < E , 1 < p < ∞ 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 ‖ u 1 ( ⋅ , t ) − u 2 ( ⋅ , t ) ‖ L p ( R ) ⩽ c ϵ t / T E 1 − t / T , ∀ t ∈ [ 0 , T ] . In case p = 2 we establish stability estimates of Hölder 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.