| Issue |
Math. Model. Nat. Phenom.
Volume 15, 2020
|
|
|---|---|---|
| Article Number | 66 | |
| Number of page(s) | 24 | |
| DOI | https://doi.org/10.1051/mmnp/2020039 | |
| Published online | 03 December 2020 | |
Spectrum of the M5-traveling waves*
Escuela Superior de Apan, Universidad Autónoma del Estado de Hidalgo, Carretera Apan-Calpulalpan Km 8, Col. Chimalpa,
C.P 43920,
Apan,
Hidalgo, Mexico.
** Corresponding author: s.c.garcia.fr@gmail.com
Received:
9
January
2020
Accepted:
24
October
2020
In this paper, we study the essential spectrum of the operator obtained by linearizing at traveling waves that occur in the one-dimensional version of the M5-model for mesenchymal cell movement inside a directed tissue made up of highly aligned fibers. We show that traveling waves are spectrally unstable in L2(ℝ; ℂ3) as the essential spectrum includes the imaginary axis. Tools in the proof include exponential dichotomies and Fredholm properties. We prove that a weighted space Lw2(ℝ; ℂ3) with the same function for the tree variables of the linearized operator is no suitable to shift the essential spectrum to the left of the imaginary axis. We find a pair of appropriate weight functions whereby on the weighted space Lwα2(ℝ; ℂ2) × Lwε2(ℝ; ℂ) the essential spectrum lies on {Reλ<0}, outside the imaginary axis.
Mathematics Subject Classification: 34L05 / 35B35 / 35A18 / 92C17 / 35L40
Key words: Mesenchymal motion / traveling waves / spectral stability / essential spectrum / two-weighted space
© The authors. Published by EDP Sciences, 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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