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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 |
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. |
show all references
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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