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International mathematical journal of analysis and its applications

CiteScore 2018: 0.72

SCImago Journal Rank (SJR) 2018: 0.363
Source Normalized Impact per Paper (SNIP) 2018: 0.530

Mathematical Citation Quotient (MCQ) 2018: 0.36

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Volume 35, Issue 1


Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

Hajime Koba / Hideki Matsuoka
  • Department of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan
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Published Online: 2015-02-06 | DOI: https://doi.org/10.1515/anly-2014-1262


This paper considers a backward problem on a heat equation with a fractional Laplacian. It is not easy to solve a backward heat equation directly. This problem is a well-known ill-posed problem. In order to consider a backward heat equation with a fractional Laplacian, we apply the N-th power of the Dirichlet-Laplacian and small parameters to regularize the equation. This method is called a quasi-reversibility method. We use the generalized quasi-reversibility method to change the backward heat system into another system. This paper shows the existence of a strong solution of the modified backward heat system, and derives L2-estimates of the difference between a solution of the heat equation with the fractional Laplacian and a solution of our system.

Keywords: Primary 35R30; secondary 35R25; 35K05

MSC: Backward heat equation; quasi-reversibility method; fractional Laplacian; fractional diffusion

About the article

Received: 2014-04-07

Revised: 2014-10-20

Accepted: 2015-01-30

Published Online: 2015-02-06

Published in Print: 2015-03-01

Citation Information: Analysis, Volume 35, Issue 1, Pages 47–57, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747, DOI: https://doi.org/10.1515/anly-2014-1262.

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