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)
Nearby in the stack
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.