F John's stability conditions versus A Carasso's SECB constraint for backward parabolic problems

Abstract

In order to solve backward parabolic problems John (1960 Commun. Pure. Appl. Math.13 551–85) introduced the two constraints ‖u(T)‖ ⩽ M and ‖u(0) − g‖ ⩽ δ where u(t) satisfies the backward heat equation for t ∊ (0, T) with the initial data u(0). The slow evolution from the continuation boundary (SECB) constraint was introduced by Carasso (1994 SIAM J. Numer. Anal. 31 1535–57) to attain continuous dependence on data for backward parabolic problems even at the continuation boundary t = T. The additional ‘SECB constraint’ guarantees a significant improvement in stability up to t = T. In this paper, we prove that the same type of stability can be obtained by using only two constraints among the three. More precisely, we show that the a priori boundedness condition ‖u(T)‖ ⩽ M is redundant. This implies that Carasso's SECB condition can be used to replace the a priori boundedness condition of John with an improved stability estimate. Also, a new class of regularized solutions is introduced for backward parabolic problems with an SECB constraint. The new regularized solutions are optimally stable and we also provide a constructive scheme to compute. Finally, numerical examples are provided.