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June  2016, 21(4): 1079-1099. doi: 10.3934/dcdsb.2016.21.1079

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

 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., Mexico, Mexico

Received  April 2015 Revised  November 2015 Published  March 2016

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.
Citation: Salvador Cruz-García, Catherine García-Reimbert. On the spectral stability of standing waves of the one-dimensional $M^5$-model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1079-1099. doi: 10.3934/dcdsb.2016.21.1079
##### References:
 [1] J. Alexander, R. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., 410 (1990), 167-212. [2] G. Flores and R. G. Plaza, Stability of post-fertilization traveling waves, J. Differential Equations, 247 (2009), 1529-1590. doi: 10.1016/j.jde.2009.05.007. [3] P. Friedl and K. Wolf, Tumor cell invasion and migration: Diversity and escape mechanisms, Nat. Rev. Cancer, 3 (2003), 362-374. [4] A. Ghazaryan and C. K. R. T. Jones, On the stability of high lewis number combustion fronts, Discrete Contin. Dyn. Sys. A., 24 (2009), 809-826. doi: 10.3934/dcds.2009.24.809. [5] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840. [6] T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math Biol., 53 (2006), 585-616. doi: 10.1007/s00285-006-0017-y. [7] T. Hillen, P. Hinow and Z. A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, Discrete Contin. Dyn. Sys. B., 14 (2010), 1055-1080. doi: 10.3934/dcdsb.2010.14.1055. [8] J. Humpherys, On the shock wave spectrum for isentropic gas dynamics with capillarity, J. Differential Equations, 246 (2009), 2938-2957. doi: 10.1016/j.jde.2008.07.028. [9] J. Humpherys, Spectral Energy Methods and the Stability of Shock Waves, Ph.D thesis, Indiana University, 2002. [10] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, New York, 2013. doi: 10.1007/978-1-4614-6995-7. [11] K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543. doi: 10.1007/s00285-008-0217-8. [12] K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [13] K. J. Palmer, Exponential dichotomies and Freholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. [14] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Commun. Math. Phys., 164 (1994), 305-349. doi: 10.1007/BF02101705. [15] R. G. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discr. and Cont. Dynam. Syst., 10 (2004), 885-924. doi: 10.3934/dcds.2004.10.885. [16] J. Rottmann-Matthes, Linear stability of traveling waves in first-order hyperbolic PDEs, J. Dyn. Diff. Equat., 23 (2011), 365-393. doi: 10.1007/s10884-011-9216-3. [17] J. Rottmann-Matthes, Stability and freezing of nonlinear waves in first-order hyperbolic PDEs, J. Dyn. Diff. Equat., 24 (2012), 341-367. doi: 10.1007/s10884-012-9241-x. [18] B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems (ed. B. Fiedler), North-Holland, Amsterdam, 2 (2002), 983-1055. doi: 10.1016/S1874-575X(02)80039-X. [19] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7. [20] Z. A. Wang, T. Hillen and M. Li, Mesenchymal motion models in one dimension, SIAM J. Appl. Math., 69 (2008), 375-397. doi: 10.1137/080714178. [21] K. Wolf, I. Mazo, H. Leung, K. Engelke, U. H. von Andrian, E. I. Deryugina, A. Y. Strongin, E.-B. Bröcker and P. Friedl, Compensation mechanism in tumor cell migration: Mesenchymal-amoeboid transition after blocking of pericellular proteolysis, J. Cell Biol., 160 (2003), 267-277. doi: 10.1083/jcb.200209006. [22] Y. Wu and X. Xing, Stability of traveling waves with critical speeds for $p$-degree Fisher-type equations, Discrete Contin. Dyn. Sys., 20 (2008), 1123-1139. doi: 10.3934/dcds.2008.20.1123. [23] K. Zumbrun, Stability and dynamics of viscous shock waves, in Nonlinear Conservation Laws and Applications (eds. A. Bressan et al.), Springer, 153 (2011), 123-167. doi: 10.1007/978-1-4419-9554-4_5. [24] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics (eds. S. Friedlander and D. Serre), North-Holland, Amsterdam, 3 (2004), 311-533.

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##### References:
 [1] J. Alexander, R. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., 410 (1990), 167-212. [2] G. Flores and R. G. Plaza, Stability of post-fertilization traveling waves, J. Differential Equations, 247 (2009), 1529-1590. doi: 10.1016/j.jde.2009.05.007. [3] P. Friedl and K. Wolf, Tumor cell invasion and migration: Diversity and escape mechanisms, Nat. Rev. Cancer, 3 (2003), 362-374. [4] A. Ghazaryan and C. K. R. T. Jones, On the stability of high lewis number combustion fronts, Discrete Contin. Dyn. Sys. A., 24 (2009), 809-826. doi: 10.3934/dcds.2009.24.809. [5] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840. [6] T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math Biol., 53 (2006), 585-616. doi: 10.1007/s00285-006-0017-y. [7] T. Hillen, P. Hinow and Z. A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, Discrete Contin. Dyn. Sys. B., 14 (2010), 1055-1080. doi: 10.3934/dcdsb.2010.14.1055. [8] J. Humpherys, On the shock wave spectrum for isentropic gas dynamics with capillarity, J. Differential Equations, 246 (2009), 2938-2957. doi: 10.1016/j.jde.2008.07.028. [9] J. Humpherys, Spectral Energy Methods and the Stability of Shock Waves, Ph.D thesis, Indiana University, 2002. [10] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, New York, 2013. doi: 10.1007/978-1-4614-6995-7. [11] K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543. doi: 10.1007/s00285-008-0217-8. [12] K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [13] K. J. Palmer, Exponential dichotomies and Freholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. [14] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Commun. Math. Phys., 164 (1994), 305-349. doi: 10.1007/BF02101705. [15] R. G. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discr. and Cont. Dynam. Syst., 10 (2004), 885-924. doi: 10.3934/dcds.2004.10.885. [16] J. Rottmann-Matthes, Linear stability of traveling waves in first-order hyperbolic PDEs, J. Dyn. Diff. Equat., 23 (2011), 365-393. doi: 10.1007/s10884-011-9216-3. [17] J. Rottmann-Matthes, Stability and freezing of nonlinear waves in first-order hyperbolic PDEs, J. Dyn. Diff. Equat., 24 (2012), 341-367. doi: 10.1007/s10884-012-9241-x. [18] B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems (ed. B. Fiedler), North-Holland, Amsterdam, 2 (2002), 983-1055. doi: 10.1016/S1874-575X(02)80039-X. [19] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7. [20] Z. A. Wang, T. Hillen and M. Li, Mesenchymal motion models in one dimension, SIAM J. Appl. Math., 69 (2008), 375-397. doi: 10.1137/080714178. [21] K. Wolf, I. Mazo, H. Leung, K. Engelke, U. H. von Andrian, E. I. Deryugina, A. Y. Strongin, E.-B. Bröcker and P. Friedl, Compensation mechanism in tumor cell migration: Mesenchymal-amoeboid transition after blocking of pericellular proteolysis, J. Cell Biol., 160 (2003), 267-277. doi: 10.1083/jcb.200209006. [22] Y. Wu and X. Xing, Stability of traveling waves with critical speeds for $p$-degree Fisher-type equations, Discrete Contin. Dyn. Sys., 20 (2008), 1123-1139. doi: 10.3934/dcds.2008.20.1123. [23] K. Zumbrun, Stability and dynamics of viscous shock waves, in Nonlinear Conservation Laws and Applications (eds. A. Bressan et al.), Springer, 153 (2011), 123-167. doi: 10.1007/978-1-4419-9554-4_5. [24] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics (eds. S. Friedlander and D. Serre), North-Holland, Amsterdam, 3 (2004), 311-533.
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