Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: F. John's stability conditions vs. A. Carasso's SECB constraint for
  backward parabolic problems   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.

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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: eess