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In order to solve backward parabolic problems F. John [{\it Comm. Pure. Appl.Math.} (1960)] introduced the two constraints "$\|u(T)\|\le M$" and $\|u(0) - g\| \le \delta$ where $u(t)$ satisfies the backward heat equation for$t\in(0,T)$ with the initial data $u(0).$  The {\it slow-evolution-from-the-continuation-boundary} (SECB) constraint hasbeen introduced by A. Carasso in [{\it SIAM J. Numer. Anal.} (1994)] to attaincontinuous dependence on data for backward parabolic problems even at thecontinuation boundary $t=T$. The additional "SECB constraint" guarantees asignificant improvement in stability up to $t=T.$ In this paper we prove thatthe same type of stability can be obtained by using only two constraints amongthe three. More precisely, we show that the a priori boundedness condition$\|u(T)\|\le M$ is redundant. This implies that the Carasso's SECB conditioncan be used to replace the a priori boundedness condition of F. John with animproved stability estimate. Also a new class of regularized solutions isintroduced for backward parabolic problems with an SECB constraint. The newregularized solutions are optimally stable and we also provide a constructivescheme to compute. Finally numerical examples are provided.

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