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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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Online
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1569-3945
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Volume 20, Issue 5-6

Issues

Regularization for ill-posed parabolic evolution problems

Matthew A. Fury
Published Online: 2012-12-14 | DOI: https://doi.org/10.1515/jip-2012-0018

Abstract.

We prove regularization for a certain ill-posed parabolic evolution problem in a Banach space X by obtaining Hölder-continuous dependence of its solution on modeling. In particular, we consider the generally ill-posed problem , , with initial data , in X where is the infinitesimal generator of a bounded holomorphic semigroup of angle on X, and for each with for . Assuming there exists a solution of the problem adhering to certain stabilizing conditions, we approximate by the solution of an approximate well-posed problem. We then use this estimate to prove the existence of a family of regularizing operators for the given ill-posed problem. The theory has applications to the backwards heat equation and other ill-posed partial differential equations in , , with time-dependent coefficients.

Keywords: Ill-posed problems; regularizing families of operators; continuous dependence on modeling; parabolic evolution problems

About the article

Received: 2012-03-22

Published Online: 2012-12-14

Published in Print: 2012-12-01


Citation Information: , Volume 20, Issue 5-6, Pages 667–699, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jip-2012-0018.

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© 2012 by Walter de Gruyter Berlin Boston.Get Permission

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