### Abstract

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 L^{2}-estimates of the difference between a solution of the heat equation with the fractional Laplacian and a solution of our system.

Original language | English |
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Pages (from-to) | 47-57 |

Number of pages | 11 |

Journal | Analysis |

Volume | 35 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 Mar 1 |

### Fingerprint

### Keywords

- 35K05
- Primary 35R30
- secondary 35R25

### ASJC Scopus subject areas

- Applied Mathematics
- Analysis
- Numerical Analysis

### Cite this

*Analysis*,

*35*(1), 47-57. https://doi.org/10.1515/anly-2014-1262

**Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian.** / Koba, Hajime; Matsuoka, Hideki.

Research output: Contribution to journal › Article

*Analysis*, vol. 35, no. 1, pp. 47-57. https://doi.org/10.1515/anly-2014-1262

}

TY - JOUR

T1 - Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

AU - Koba, Hajime

AU - Matsuoka, Hideki

PY - 2015/3/1

Y1 - 2015/3/1

N2 - 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.

AB - 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.

KW - 35K05

KW - Primary 35R30

KW - secondary 35R25

UR - http://www.scopus.com/inward/record.url?scp=84925450481&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925450481&partnerID=8YFLogxK

U2 - 10.1515/anly-2014-1262

DO - 10.1515/anly-2014-1262

M3 - Article

VL - 35

SP - 47

EP - 57

JO - Analysis (Germany)

JF - Analysis (Germany)

SN - 0174-4747

IS - 1

ER -