Stability results for the heat equation backward in time

For the heat equation backward in timeut=uxx,x∈R,t∈(0,T),‖u(⋅,T)−φ(⋅)‖Lp(R)⩽ϵ subject to the constraint‖u(⋅,0)‖Lp(R)⩽E‖u(⋅,0)‖Lp(R)⩽E with T>0T>0, φ∈Lp(R)φ∈Lp(R), 0<ϵ0c>0 such that‖u1(⋅,t)−u2(⋅,t)‖Lp(R)⩽cϵt/TE1−t/T,∀t∈[0,T]. In case p=2p=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=0t=0 when an additional condition on the smoothness of u(x,0)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.