Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media

Hamel, François; Nadin, Grégoire

doi:10.1090/proc/15997

Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media
HTML articles powered by AMS MathViewer

by François Hamel and Grégoire Nadin PDF

Proc. Amer. Math. Soc. 150 (2022), 3549-3564 Request permission

Abstract:

We consider in this article reaction-diffusion equations of the Fisher-KPP type with a nonlinearity depending on the space variable $x$, oscillating slowly and non-periodically. We are interested in the width of the interface between the unstable steady state $0$ and the stable steady state $1$ of the solutions of the Cauchy problem. We prove that, if the heterogeneity has large enough oscillations, then the width of this interface, that is, the diameter of some level sets, diverges linearly as $t\rightarrow +\infty$ along some sequences of times, while it is sublinear along other sequences. As a corollary, under these conditions, generalized transition fronts do not exist for this equation.

References

Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
Henri Berestycki and François Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math. 65 (2012), no. 5, 592–648. MR 2898886, DOI 10.1002/cpa.21389
Henri Berestycki, François Hamel, and Luca Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 469–507. MR 2317650, DOI 10.1007/s10231-006-0015-0
H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous equations. Mem. Amer. Math. Soc., to appear.
Anton Bovier, Gaussian processes on trees, Cambridge Studies in Advanced Mathematics, vol. 163, Cambridge University Press, Cambridge, 2017. From spin glasses to branching Brownian motion. MR 3618123, DOI 10.1017/9781316675779
Maury Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983), no. 285, iv+190. MR 705746, DOI 10.1090/memo/0285
J. Černý, A. Drewitz, and L. Schmitz. (Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation. arXiv:2102.01049, 2021.
A. Drewitz and L. Schmitz. Invariance principles and Log-distance of F-KPP fronts in a random medium. arXiv:2102.01047, 2021.
Arnaud Ducrot, Thomas Giletti, and Hiroshi Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5541–5566. MR 3240934, DOI 10.1090/S0002-9947-2014-06105-9
R.A. Fisher. The advance of advantageous genes. Ann. Eugenics 7 (1937), 335–369.
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Jimmy Garnier, Thomas Giletti, and Gregoire Nadin, Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media, J. Dynam. Differential Equations 24 (2012), no. 3, 521–538. MR 2964791, DOI 10.1007/s10884-012-9254-5
François Hamel and Grégoire Nadin, Spreading properties and complex dynamics for monostable reaction-diffusion equations, Comm. Partial Differential Equations 37 (2012), no. 3, 511–537. MR 2889561, DOI 10.1080/03605302.2011.647198
François Hamel and Lionel Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations 249 (2010), no. 7, 1726–1745. MR 2677813, DOI 10.1016/j.jde.2010.06.025
Christopher Henderson, Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data, Nonlinearity 29 (2016), no. 11, 3215–3240. MR 3567089, DOI 10.1088/0951-7715/29/11/3215
A.N. Kolmogorov, I.G. Petrovsky, and N.S. Piskunov, Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bjul. Moskowskogo Gos. Univ.1 (1937), 1–26.
Grégoire Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 4, 841–873. MR 3390087, DOI 10.1016/j.anihpc.2014.03.007
James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik, and Andrej Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 217–246. MR 2864411, DOI 10.1007/s00205-011-0449-4
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Wenxian Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations 16 (2004), no. 4, 1011–1060. MR 2110054, DOI 10.1007/s10884-004-7832-x
Eiji Yanagida, Irregular behavior of solutions for Fisher’s equation, J. Dynam. Differential Equations 19 (2007), no. 4, 895–914. MR 2357530, DOI 10.1007/s10884-007-9096-8
Andrej Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. (9) 98 (2012), no. 1, 89–102. MR 2935371, DOI 10.1016/j.matpur.2011.11.007