A uniform convergence theorem for singular limit eigenvalue problems

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)29-54
Number of pages26
JournalAdvances in Differential Equations
Volume8
Issue number1
Publication statusPublished - 2003
Externally publishedYes

Fingerprint

Singular Limit
Uniform convergence
Convergence Theorem
Eigenvalue Problem
Traveling Wave
Eigenvalue
Singular Perturbation Method
Transition Layer
Equilibrium Solution
Thin Layer
Traveling Wave Solutions
Reaction-diffusion Equations
Function Space
Mathematical operators
Differential operator
Interval
Zero
Operator

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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

In: Advances in Differential Equations, Vol. 8, No. 1, 2003, p. 29-54.

Research output: Contribution to journalArticle

@article{02b8a794c7b948c59152f5769df6950e,
title = "A uniform convergence theorem for singular limit eigenvalue problems",
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.",
author = "Masaharu Taniguchi",
year = "2003",
language = "English",
volume = "8",
pages = "29--54",
journal = "Advances in Differential Equations",
issn = "1079-9389",
publisher = "Khayyam Publishing, Inc.",
number = "1",

}

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 -