Ja n 20 09 F . John ’ s stability conditions vs . A . Carasso ’ s SECB constraint for backward parabolic problems ∗

In order to solve backward parabolic problems F. John [Comm. Pure. Appl. Math. (1960)] 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 has been introduced by A. Carasso in [SIAM J. Numer. Anal. (1994)] 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 the Carasso’s SECB condition can be used to replace the a priori boundedness condition of F. 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.