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

TL;DR

This paper compares two stability conditions for backward parabolic problems, showing that Carasso's SECB constraint can replace John's boundedness condition, leading to improved stability and a new class of regularized solutions.

Contribution

The paper demonstrates that the SECB constraint alone suffices for stability, replacing John's boundedness condition, and introduces a new class of optimally stable regularized solutions with a constructive scheme.

Findings

SECB constraint can replace John's boundedness condition

New regularized solutions are optimally stable

Constructive scheme for computing regularized solutions

Abstract

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 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 $t=T$. 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 $\|u(T)\|\le M$ 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.