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Spreading in kinetic reaction–transport equations in higher velocity dimensions

Published online by Cambridge University Press:  07 February 2018

EMERIC BOUIN  and
NILS CAILLERIE
EMERIC BOUIN
Affiliation:
CEREMADE – Université Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France email: bouin@ceremade.dauphine.fr
NILS CAILLERIE
Affiliation:
Institut Camille Jordan (ICJ), Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France email: caillerie@math.univ-lyon1.fr
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Abstract

In this paper, we extend and complement previous works about propagation in kinetic reaction–transport equations. The model we study describes particles moving according to a velocity-jump process, and proliferating according to a reaction term of monostable type. We focus on the case of bounded velocities, having dimension higher than one. We extend previous results obtained by the first author with Calvez and Nadin in dimension one. We study the large time/large-scale hyperbolic limit via an Hamilton–Jacobi framework together with the half-relaxed limits method. We deduce spreading results and the existence of travelling wave solutions. A crucial difference with the mono-dimensional case is the resolution of the spectral problem at the edge of the front, that yields potential singular velocity distributions. As a consequence, the minimal speed of propagation may not be determined by a first-order condition.

Keywords

Kinetic equationstravelling wavesdispersion relation35Q9245K0535C07
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 219 - 247
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

EB and NC acknowledge the support of the ERC Grant MATKIT (ERC-2011-StG). NC has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement no. 639638).

References

[1] Aronson, D. G. & Weinberger, H. F. (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 (1), 33–76.Google Scholar
[2] Barles, G. (1994) Solutions de Viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications, Vol. 17. Springer–Verlag, New York.Google Scholar
[3] Barles, G., Evans, L. C. & Souganidis, P. E. (1990) Wavefront propagation for reaction-diffusion systems of PDE. Duke Math. J. 61 (3), 835–858.Google Scholar
[4] Barles, G., Mirrahimi, S. & Perthame, B. (2009) Concentration in Lotka-Volterra parabolic or integral equations: A general convergence result. Methods Appl. Anal. 16 (3), 321–340.Google Scholar
[5] Barles, G. & Perthame, B. (1998) Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Opt. 26 (5), 1133–1148.Google Scholar
[6] Berestycki, H. & Nadin, G. (2012) Spreading speeds for one-dimensional monostable reaction-diffusion equations. J. Math. Phys. 53 (11), 115619.Google Scholar
[7] Bouin, E. (2015) A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic Relat. Models 8 (2), 255–280.Google Scholar
[8] Bouin, E. & Calvez, V. (2012) A kinetic eikonal equation. Comptes Rendus Math. 350 (5), 243–248.Google Scholar
[9] Bouin, E., Calvez, V., Grenier, E. & Nadin, G. (2016) Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations. arXiv:1607.03676.Google Scholar
[10] Bouin, E., Calvez, V., Meunier, N., Mirrahimi, S., Perthame, B., Raoul, G. & Voituriez, R. (2012) Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration. Comptes Rendus Math. 350 (15), 761–766.Google Scholar
[11] Bouin, E., Calvez, V. & Nadin, G. (2015) Propagation in a kinetic reaction-transport equation: Travelling waves and accelerating fronts. Arch. Ration. Mech. Anal. 217 (2), 571–617.Google Scholar
[12] Bouin, E. & Mirrahimi, S. (2015) A Hamilton-Jacobi approach for a model of population structured by space and trait. Commun. Math. Sci. 13 (6), 1431–1452.Google Scholar
[13] Bressloff, P. C. & Faugeras, O. (2017) On the Hamiltonian structure of large deviations in stochastic hybrid systems. J. Stat. Mech.: Theory Exp. 2017 (3), 033206.Google Scholar
[14] Bressloff, P. C. & Newby, J. M. (2014) Path integrals and large deviations in stochastic hybrid systems. Phys. Rev. E 89 (4), pages: 042701.Google Scholar
[15] Caillerie, N. (2017) Large deviations of a velocity jump process with a Hamilton-Jacobi approach. Comptes Rendus Math. 355 (2), 170–175.Google Scholar
[16] Calvez, V., Gabriel, P. & Mateos González, Á. (2016) Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation. Preprint arXiv:1609.06933.Google Scholar
[17] Coville, J. (2013) Singular measure as principal eigenfunction of some nonlocal operators. Appl. Math. Lett. 26 (8), 831–835.Google Scholar
[18] Crandall, M. G., Ishii, H. & Lions, P.-L. (1992) User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27 (1), 1–68.Google Scholar
[19] Cuesta, C. M., Hittmeir, S. & Schmeiser, C. (2012) Traveling waves of a kinetic transport model for the KPP-Fisher equation. SIAM J. Math. Anal. 44 (6), 4128–4146.Google Scholar
[20] Diekmann, O., Capasso, V. (1999) Mathematics inspired by biology. In: Lectures Given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Martina Franca, Italy, June 13–20, 1997. Springer, Berlin, New York. OCLC: 731926204.Google Scholar
[21] Diekmann, O., Jabin, P. E., Mischler, S. & Perthame, B. (2005) The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach. Theor. Popul. Biol. 67 (4), 257–271.Google Scholar
[22] Evans, L. C. & Souganidis, P. E. (1989) A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J. 38 (1), 141–172.Google Scholar
[23] Faggionato, A. Gabrielli, D. & Ribezzi Crivellari, M. (2010) Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors. Markov Process. Relat. Fields 16 (3), 497–548.Google Scholar
[24] Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7 (4), 355–369.Google Scholar
[25] Freidlin, M. (1985) Functional Integration and Partial Differential Equations. (AM-109). Princeton University Press, Princeton, N.J.Google Scholar
[26] Freidlin, M. I. & Gärtner, J. (1979) On the propagation of concentration waves in periodic and random media. Sov. Math. Dokl. 20, 1282–1286.Google Scholar
[27] Hadeler, K. P. (1988) Hyperbolic travelling fronts. Proc. Edinb. Math. Soc., 31 (01), 89.Google Scholar
[28] Hivert, H. (2017) An asymptotic preserving scheme for front propagation in a kinetic reaction-transport equation. HAL ID: hal-01522278.Google Scholar
[29] Kifer, Y. (2009) Large Deviations and Adiabatic Transitions for Dynamical Systems and Markov Processes in Fully Coupled Averaging. Memoirs of the American Mathematical Society, no. 944. American Mathematical Society, Providence, R.I.Google Scholar
[30] Kolmogorov, A. N., Petrovsky, I. G. & Piskunov, N. S. (1937) Etude de lquation de la diffusion avec croissance de la quantit de matire et son application un problme biologique. Mosc. Univ. Math. Bull., 1, 1–25.Google Scholar
[31] Lorz, A., Mirrahimi, S. & Perthame, B. (2011) Dirac mass dynamics in multidimensional nonlocal parabolic equations. Commun. Partial Differential Equ. 36 (6), 1071–1098.Google Scholar
[32] Perthame, B. & Souganidis, P. E. (2009) Asymmetric potentials and motor effect: A homogenization approach. Annales de l'Institut Henri Poincare (C) Non Linear Anal. 26 (6), 2055–2071.Google Scholar
[33] Souganidis, P. E. (1997) Front propagation: Theory and applications. In: Viscosity Solutions and Applications, Lecture notes in mathematics, Vol. 1660, Springer, Berlin, Heidelberg, pp. 186–242.Google Scholar