Advances in Differential Equations

A uniform convergence theorem for singular limit eigenvalue problems

Masaharu Taniguchi

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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 is because 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.

Article information

Adv. Differential Equations Volume 8, Number 1 (2003), 29-54.

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 34E15: Singular perturbations, general theory 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 35B25: Singular perturbations 35B35: Stability


Taniguchi, Masaharu. A uniform convergence theorem for singular limit eigenvalue problems. Adv. Differential Equations 8 (2003), no. 1, 29--54.

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