### Abstract

For reaction-diffusion equations, equilibrium solutions or traveling wave solutions with thin transition layers are constructed by singular perturbation methods. It is usually difficult to study their stability. This isbecause the linearized eigenvalue problem has a critical eigenvalue in a small neighborhood of zero, and its location is difficult to determine. The SLEP method is known as one of the most powerful tools to study this critical eigenvalue. To apply this method rigorously, a uniform convergence theorem for the inverse of a differential operator, for instance the inverse Allen-Cahn operator, in some function space plays a crucial role. However, there has been a significant difficulty in the cases of unbounded intervals including those of traveling waves, and no rigorous result was available previously. This paper presents a uniform convergence theorem in a general framework. Our new uniform convergence theorem makes the SLEP method applicable to various kinds of problems including stability of traveling waves.

Original language | English |
---|---|

Pages (from-to) | 29-54 |

Number of pages | 26 |

Journal | Advances in Differential Equations |

Volume | 8 |

Issue number | 1 |

Publication status | Published - 2003 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**A uniform convergence theorem for singular limit eigenvalue problems.** / Taniguchi, Masaharu.

Research output: Contribution to journal › Article

*Advances in Differential Equations*, vol. 8, no. 1, pp. 29-54.

}

TY - JOUR

T1 - A uniform convergence theorem for singular limit eigenvalue problems

AU - Taniguchi, Masaharu

PY - 2003

Y1 - 2003

N2 - For reaction-diffusion equations, equilibrium solutions or traveling wave solutions with thin transition layers are constructed by singular perturbation methods. It is usually difficult to study their stability. This isbecause the linearized eigenvalue problem has a critical eigenvalue in a small neighborhood of zero, and its location is difficult to determine. The SLEP method is known as one of the most powerful tools to study this critical eigenvalue. To apply this method rigorously, a uniform convergence theorem for the inverse of a differential operator, for instance the inverse Allen-Cahn operator, in some function space plays a crucial role. However, there has been a significant difficulty in the cases of unbounded intervals including those of traveling waves, and no rigorous result was available previously. This paper presents a uniform convergence theorem in a general framework. Our new uniform convergence theorem makes the SLEP method applicable to various kinds of problems including stability of traveling waves.

AB - For reaction-diffusion equations, equilibrium solutions or traveling wave solutions with thin transition layers are constructed by singular perturbation methods. It is usually difficult to study their stability. This isbecause the linearized eigenvalue problem has a critical eigenvalue in a small neighborhood of zero, and its location is difficult to determine. The SLEP method is known as one of the most powerful tools to study this critical eigenvalue. To apply this method rigorously, a uniform convergence theorem for the inverse of a differential operator, for instance the inverse Allen-Cahn operator, in some function space plays a crucial role. However, there has been a significant difficulty in the cases of unbounded intervals including those of traveling waves, and no rigorous result was available previously. This paper presents a uniform convergence theorem in a general framework. Our new uniform convergence theorem makes the SLEP method applicable to various kinds of problems including stability of traveling waves.

UR - http://www.scopus.com/inward/record.url?scp=33747190035&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747190035&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33747190035

VL - 8

SP - 29

EP - 54

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 1

ER -