2003 A uniform convergence theorem for singular limit eigenvalue problems
Masaharu Taniguchi
Adv. Differential Equations 8(1): 29-54 (2003). DOI: 10.57262/ade/1355926867


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.


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Masaharu Taniguchi. "A uniform convergence theorem for singular limit eigenvalue problems." Adv. Differential Equations 8 (1) 29 - 54, 2003. https://doi.org/10.57262/ade/1355926867


Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1030.35103
MathSciNet: MR1946557
Digital Object Identifier: 10.57262/ade/1355926867

Primary: 35K57
Secondary: 34E15 , 34L15 , 35B25 , 35B35

Rights: Copyright © 2003 Khayyam Publishing, Inc.


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Vol.8 • No. 1 • 2003
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