On the spectral stability of standing waves of the one-dimensional $M^5$-model

Salvador  Cruz-García; Catherine  García-Reimbert

doi:10.3934/dcdsb.2016.21.1079

Article Contents

2016, Volume 21, Issue 4: 1079-1099. Doi: 10.3934/dcdsb.2016.21.1079

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On the spectral stability of standing waves of the one-dimensional $M^5$-model

Salvador Cruz-García^1, and
Catherine García-Reimbert^1,

1.
Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apdo. Postal 20-726, C.P. 01000 México D.F.

Received: March 31, 2015

Revised: October 31, 2015

Published: May 2016

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Abstract

We consider the spectral stability problem for a family of standing pulse and wave front solutions to the one-dimensional version of the $M^5$-model formulated by Hillen [T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585--616], to describe the mesenchymal cell motion inside tissue. The stability analysis requires the definition of spectrum, which is divided into two disjoint sets: the point spectrum and the essential spectrum. Under this partition the eigenvalue zero belongs to the essential spectrum and not to the point spectrum. By excluding the eigenvalue zero we can bring the spectral problem into an equivalent scalar quadratic eigenvalue problem. This leads, naturally, to deduce the existence of a negative eigenvalue which also turns out to belong to the essential spectrum. Beyond this result, the scalar formulation enables us to use the integrated equation technique to establish, via energy methods, that the point spectrum is empty. Our main result is that the family of standing waves is spectrally stable. To prove it, we go back to the original scalar problem and show that the rest of the essential spectrum is a subset of the open left-half complex plane.

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Mathematics Subject Classification: Primary: 34L05, 35B35, 35A18; Secondary: 92C17, 35L40, 35B40.

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