Geodesic
Spectrum of the M5-traveling waves
Salvador Cruz-García1
1 Escuela Superior de Apan, Universidad Autónoma del Estado de Hidalgo, Carretera Apan-Calpulalpan Km 8, Col. Chimalpa, C.P 43920, Apan, Hidalgo, Mexico.
Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 66

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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 , outside the imaginary axis.
DOI : 10.1051/mmnp/2020039

Salvador Cruz-García 1

1 Escuela Superior de Apan, Universidad Autónoma del Estado de Hidalgo, Carretera Apan-Calpulalpan Km 8, Col. Chimalpa, C.P 43920, Apan, Hidalgo, Mexico. 
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Salvador Cruz-García. Spectrum of the M5-traveling waves. Mathematical modelling of natural phenomena, Tome 15 (2020), article  no. 66. doi: 10.1051/mmnp/2020039

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