Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics
[Propagation de fronts dirigée par une ligne de diffusion rapide : limite en temps long et grande diffusion]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017) , pp. 141-176.

Nous étudions une équation de réaction-diffusion posée dans une bande horizontale, couplée à une équation de diffusion sur son bord supérieur à travers une condition de Robin. Cette classe de modèles a été proposée par H. Berestycki, L. Rossi et le deuxième auteur afin d’étudier l’influence d’une ligne de diffusion rapide (par exemple une route) sur les invasions biologiques. Ils prouvent que la vitesse d’invasion est augmentée par une forte diffusivité sur la ligne, et plus précisément asymptotiquement proportionnelle à la racine carrée de cette dernière. Dans le cas d’une croissance logistique, ces résultats peuvent être réduits à des calculs algébriques. Le but de cet article est de généraliser ce résultat à des non-linéarités différentes et pour lesquelles ces calculs ne peuvent être accomplis. Nous mettons aussi en lumière un nouveau phénomène de transition entre deux ondes progressives différentes, qu’on explique en détail.

The system under study is a reaction-diffusion equation in a horizontal strip, coupled to a diffusion equation on its upper boundary via an exchange condition of the Robin type. This class of models was introduced by H. Berestycki, L. Rossi and the second author in order to model biological invasions directed by lines of fast diffusion. They proved, in particular, that the speed of invasion was enhanced by a fast diffusion on the line, the spreading velocity being asymptotically proportional to the square root of the fast diffusion coefficient. These results could be reduced, in the logistic case, to explicit algebraic computations. The goal of this paper is to prove that the same phenomenon holds, with a different type of nonlinearity, which precludes explicit computations. We discover a new transition phenomenon, that we explain in detail.

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DOI : https://doi.org/10.5802/jep.40
Classification : 35K57,  35B40,  35C07
Mots clés : Réaction-diffusion, fronts, ondes progressives, asymptotique, accélération, propagation, couplage, invasion, extinction
@article{JEP_2017__4__141_0,
     author = {Laurent Dietrich and Jean-Michel Roquejoffre},
     title = {Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {141--176},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.40},
     mrnumber = {3611101},
     zbl = {06754325},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.40/}
}
Laurent Dietrich; Jean-Michel Roquejoffre. Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017) , pp. 141-176. doi : 10.5802/jep.40. https://jep.centre-mersenne.org/articles/10.5802/jep.40/

[1] D. G. Aronson - “Bounds for the fundamental solution of a parabolic equation”, Bull. Amer. Math. Soc. 73 (1967), p. 890-896 | Article | MR 217444 | Zbl 0153.42002

[2] D. G. Aronson & H. F. Weinberger - “Multidimensional nonlinear diffusion arising in population genetics”, Adv. in Math. 30 (1978) no. 1, p. 33-76 | Article | MR 511740 | Zbl 0407.92014

[3] B. Audoly, H. Berestycki & Y. Pomeau - “Réaction diffusion en écoulement stationnaire rapide”, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 328 (2000) no. 3, p. 255-262 | Zbl 0992.76097

[4] H. Berestycki, A.-C. Coulon, J.-M. Roquejoffre & L. Rossi - “The effect of a line with nonlocal diffusion on Fisher-KPP propagation”, Math. Models Methods Appl. Sci. 25 (2015) no. 13, p. 2519-2562 | Article | MR 3397542 | Zbl 1327.35175

[5] H. Berestycki & L. Nirenberg - “Travelling fronts in cylinders”, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) no. 5, p. 497-572 | Article | Numdam | MR 1191008 | Zbl 0799.35073

[6] H. Berestycki, J.-M. Roquejoffre & L. Rossi - “The influence of a line with fast diffusion on Fisher-KPP propagation”, J. Math. Biol. 66 (2013) no. 4-5, p. 743-766 | Article | MR 3020920 | Zbl 1270.35264

[7] P. Constantin, A. Kiselev, A. Oberman & L. Ryzhik - “Bulk burning rate in passive-reactive diffusion”, Arch. Rational Mech. Anal. 154 (2000) no. 1, p. 53-91 | Article | MR 1778121 | Zbl 0979.76093

[8] P. Constantin, A. Kiselev & L. Ryzhik - “Quenching of flames by fluid advection”, Comm. Pure Appl. Math. 54 (2001) no. 11, p. 1320-1342 | Article | MR 1846800 | Zbl 1032.35087

[9] P. Constantin, J.-M. Roquejoffre, L. Ryzhik & N. Vladimirova - “Propagation and quenching in a reactive Burgers-Boussinesq system”, Nonlinearity 21 (2008) no. 2, p. 221-271 | Article | MR 2384547 | Zbl 1155.35085

[10] A.-C. Coulon Chalmin - Fast propagation in reaction-diffusion equations with fractional diffusion, Université de Toulouse, 2014, Ph. D. Thesis

[11] L. Dietrich - “Existence of travelling waves for a reaction-diffusion system with a line of fast diffusion”, Appl. Math. Res. Express. AMRX (2015) no. 2, p. 204-252 | Article | MR 3394265 | Zbl 1328.35093

[12] L. Dietrich - “Velocity enhancement of reaction-diffusion fronts by a line of fast diffusion”, Trans. Amer. Math. Soc. (2016), online | Article | MR 3605970

[13] Y. Du & H. Matano - “Convergence and sharp thresholds for propagation in nonlinear diffusion problems”, J. Eur. Math. Soc. (JEMS) 12 (2010) no. 2, p. 279-312 | Article | MR 2608941 | Zbl 1207.35061

[14] A. Fannjiang, A. Kiselev & L. Ryzhik - “Quenching of reaction by cellular flows”, Geom. Funct. Anal. 16 (2006) no. 1, p. 40-69 | Article | MR 2221252 | Zbl 1097.35077

[15] P. C. Fife & J. B. McLeod - “The approach of solutions of nonlinear diffusion equations to travelling front solutions”, Arch. Rational Mech. Anal. 65 (1977) no. 4, p. 335-361 | Article | MR 442480 | Zbl 0361.35035

[16] D. Gilbarg & N. S. Trudinger - Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 | Zbl 1042.35002

[17] F. Hamel & A. Zlatoš - “Speed-up of combustion fronts in shear flows”, Math. Ann. 356 (2013) no. 3, p. 845-867 | Article | MR 3063899 | Zbl 1283.35081

[18] Ja. I. Kanel’ - “Stabilization of the solutions of the equations of combustion theory with finite initial functions”, Mat. Sb. (N.S.) 65 (107) (1964), p. 398-413 | MR 177209

[19] A. Kiselev & A. Zlatoš - “Quenching of combustion by shear flows”, Duke Math. J. 132 (2006) no. 1, p. 49-72 | Article | MR 2219254 | Zbl 1103.35048

[20] M. A. Lewis, S. V. Petrovskii & J. R. Potts - The mathematics behind biological invasions, Interdisciplinary Applied Mathematics, vol. 44, Springer, 2016 | MR 3496147 | Zbl 1338.92001

[21] A. Mellet, J. Nolen, J.-M. Roquejoffre & L. Ryzhik - “Stability of generalized transition fronts”, Comm. Partial Differential Equations 34 (2009) no. 4-6, p. 521-552 | Article | MR 2530708 | Zbl 1173.35021

[22] A. Novikov & L. Ryzhik - “Boundary layers and KPP fronts in a cellular flow”, Arch. Rational Mech. Anal. 184 (2007) no. 1, p. 23-48 | Article | MR 2289862 | Zbl 1109.76064

[23] L. Roques, J.-P. Rossi, H. Berestycki, J. Rousselet, J. Garnier, J.-M. Roquejoffre, L. Rossi, S. Soubeyrand & C. Robinet - “Modeling the spatio-temporal dynamics of the pine processionary moth”, in Processionary Moths and Climate Change: An Update (A. Roques, ed.), Springer Netherlands, 2015, p. 227-263

[24] , Vespa velutina Lepeletier, 1836, Inventaire National du Patrimoine Naturel, http://inpn.mnhn.fr/espece/cd_nom/433589

[25] N. Vladimirova, P. Constantin, A. Kiselev, O. Ruchayskiy & L. Ryzhik - “Flame enhancement and quenching in fluid flows”, Combust. Theory Model. 7 (2003) no. 3, p. 487-508 | Article | MR 2007570 | Zbl 1068.76570

[26] A. Zlatoš - “Sharp transition between extinction and propagation of reaction”, J. Amer. Math. Soc. 19 (2006) no. 1, p. 251-263 | Article | MR 2169048 | Zbl 1081.35011

[27] A. Zlatoš - “Reaction-diffusion front speed enhancement by flows”, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011) no. 5, p. 711-726 | Article | Numdam | MR 2838397 | Zbl 1328.35105