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. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2020039/

[1] J. Alexander, R. Gardner, C.K.R.T. Jones A topological invariant arising in the stability analysis of travelling waves J. Reine Angew. Math 1990 167 212

[2] S. Cruz-García, C. García-Reimbert On the spectral stability of standing waves of the one-dimensional M5-model Discrete Contin. Dyn. Syst. B 2016 1079 1099

[3] S. Cruz-García, C. García-Reimbert Approximations and error bounds for traveling and standing wave solutions of the one-dimensional M5-model for mesenchymal motion Bol. Soc. Mat. Mex 2020 147 169

[4] M. Egeblad, Z. Werb New functions for the matrix metalloproteinases in cancer progression Nat. Rev. Cancer 2002 161 174

[5] G. Flores, R.G. Plaza Stability of post-fertilization traveling waves J. Differ. Equ 2009 1529 1590

[6] P. Friedl, K. Wolf Tumor cell invasion and migration: diversity and escape mechanisms Nat. Rev. Cancer 2003 362 374

[7] J. Goodman Nonlinear asymptotic stability of viscous shock profiles for conservation laws Arch. Rational Mech. Anal 1986 325 344

[8] J. Goodman, Remarks on the stability of viscous shock waves. In Viscous Profiles and Numerical Methods for Shock Waves, edited by M. Shearer. SIAM, Philadelphia, PA (1991) 66–72.

[9] T. Hillen M5 mesoscopic and macroscopic models for mesenchymal motion J. Math. Biol 2006 585 616

[10] J. Humpherys, K. Zumbrun An efficient shooting algorithm for Evans function calculations in large systems Phys. D 2006 116 126

[11] J. Humpherys On the shock wave spectrum for isentropic gas dynamics with capillarity J. Differ. Equ 2009 2938 2957

[12] H.-Y. Jin, J. Li, Z.-A. Wang Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity J. Differ. Equ 2013 193 219

[13] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves. Springer, New York (2013).

[14] J.A. McDonald and R.P. Mecham, Receptors for Extracellular Matrix. Academic Press, San Diego, California (1991).

[15] K.J. Palmer Exponential dichotomies and transversal homoclinic points J. Differ. Equ 1984 225 256

[16] K.J. Palmer Exponential dichotomies and Fredholm operators Proc. Amer. Math. Soc 1988 149 156

[17] J. Rottmann-Matthes Linear stability of traveling waves in first-order hyperbolic PDEs J. Dyn. Differ. Equ 2011 365 393

[18] J. Rottmann-Matthes Stability and freezing of nonlinear waves in first-order hyperbolic PDEs J. Dyn. Differ. Equ 2012 341 367

[19] B. Sandstede, Stability of travelling waves, In Handbook of Dynamical Systems. North-, Amsterdam (2002) 983–1055.

[20] D.H. Sattinger On the stability of waves of nonlinear parabolic systems Adv. Math 1976 312 355

[21] Z.-A. Wang, T. Hillen, M. Li Mesenchymal motion models in one dimension SIAM J. Appl. Math 2008 375 397

[22] K. Wolf, I. Mazo, H. Leung, K. Engelke, U.H. Von Andrian, E.I. Deryugina, A.Y. Strongin, E.-B. Bröcker, P. Friedl Compensation mechanism in tumor cell migration: mesenchymal-amoeboid transition after blocking of pericellular proteolysis J. Cell Biol 2003 267 277

[23] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, in Vol 3 of Handbook of Fluid Mechanics (2005) 311–533.

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