Spectral analysis for transition front solutions in Cahn-Hilliard systems

Peter Howard; Bongsuk Kwon

doi:10.3934/dcds.2012.32.125

Article Contents

2012, Volume 32, Issue 1: 125-166. Doi: 10.3934/dcds.2012.32.125

This issue Previous Article Traveling wave solution for a lattice dynamical system with convolution type nonlinearity Next Article Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements

Spectral analysis for transition front solutions in Cahn-Hilliard systems

Peter Howard^1, and
Bongsuk Kwon^1,

1.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Received: June 30, 2010

Revised: February 28, 2011

Published: December 2011

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Abstract

We consider the spectrum associated with the linear operator obtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized about a transition wave solution. In many cases it's possible to show that the only non-negative eigenvalue is $\lambda = 0$, and so stability depends entirely on the nature of this neutral eigenvalue. In such cases, we identify a stability condition based on an appropriate Evans function, and we verify this condition under strong structural conditions on our equations. More generally, we discuss and implement a straightforward numerical check of our condition, valid under mild structural conditions.

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Mathematics Subject Classification: Primary: 35B35, 35P05; Secondary: 35Q99.

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