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

Jinwoo Lee; Dongwoo Sheen

Paper

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

Abstract

In order to solve backward parabolic problems F. John [{\it Comm. Pure. Appl. Math.} (1960)] introduced the two constraints "

" and

where

satisfies the backward heat equation for

with the initial data

The {\it slow-evolution-from-the-continuation-boundary} (SECB) constraint has been introduced by A. Carasso in [{\it SIAM J. Numer. Anal.} (1994)] to attain continuous dependence on data for backward parabolic problems even at the continuation boundary

. The additional "SECB constraint" guarantees a significant improvement in stability up to

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

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.