Universal selection of pulled fronts

Avery, Montie; Scheel, Arnd

doi:10.1090/cams/8

Universal selection of pulled fronts
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by Montie Avery and Arnd Scheel

Comm. Amer. Math. Soc. 2 (2022), 172-231

DOI: https://doi.org/10.1090/cams/8

Published electronically: July 14, 2022

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Abstract:

We establish selection of critical pulled fronts in invasion processes as predicted by the marginal stability conjecture. Our result shows convergence to a pulled front with a logarithmic shift for open sets of steep initial data, including one-sided compactly supported initial conditions. We rely on robust, conceptual assumptions, namely existence and marginal spectral stability of a front traveling at the linear spreading speed and demonstrate that the assumptions hold for open classes of spatially extended systems. Previous results relied on comparison principles or probabilistic tools with implied nonopen conditions on initial data and structure of the equation. Technically, we describe the invasion process through the interaction of a Gaussian leading edge with the pulled front in the wake. Key ingredients are sharp linear decay estimates to control errors in the nonlinear matching and corrections from initial data.

References

Igor S. Aranson and Lorenz Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys. 74 (2002), no. 1, 99–143. MR 1895097, DOI 10.1103/RevModPhys.74.99
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
M. Avery and L. Garénaux, Spectral stability of the critical front in the extended Fisher-KPP equation, Preprint, 2020, arXiv:2009.01506
Montie Avery and Arnd Scheel, Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum, SIAM J. Math. Anal. 53 (2021), no. 2, 2206–2242. MR 4244535, DOI 10.1137/20M1343476
M. Avery and A. Scheel, Sharp decay rates for localized perturbations to the critical front in the Ginzburg-Landau equation, J. Dyn. Diff. Equat. (2021), DOI: https://doi.org/10.1007/s10884-021-10093-3.
Margaret Beck and Jonathan Jaquette, Validated spectral stability via conjugate points, SIAM J. Appl. Dyn. Syst. 21 (2022), no. 1, 366–404. MR 4369568, DOI 10.1137/21M1420095
Henri Berestycki and Louis Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré C Anal. Non Linéaire 9 (1992), no. 5, 497–572 (English, with English and French summaries). MR 1191008, DOI 10.1016/S0294-1449(16)30229-3
A. Bers, M. Rosenbluth, and R. Sagdeev, Handbook of plasma physics, vol. 1, 2983, Chapter 3.2.
Emeric Bouin, Christopher Henderson, and Lenya Ryzhik, The Bramson logarithmic delay in the cane toads equations, Quart. Appl. Math. 75 (2017), no. 4, 599–634. MR 3686514, DOI 10.1090/qam/1470
Emeric Bouin, Christopher Henderson, and Lenya Ryzhik, The Bramson delay in the non-local Fisher-KPP equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), no. 1, 51–77 (English, with English and French summaries). MR 4049916, DOI 10.1016/j.anihpc.2019.07.001
Maury D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978), no. 5, 531–581. MR 494541, DOI 10.1002/cpa.3160310502
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
Leonid Brevdo, A dynamical system approach to the absolute instability of spatially developing localized open flows and media, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2022, 1375–1397. MR 1910331, DOI 10.1098/rspa.2001.0912
Paul Carter and Arnd Scheel, Wave train selection by invasion fronts in the FitzHugh-Nagumo equation, Nonlinearity 31 (2018), no. 12, 5536–5572. MR 3876553, DOI 10.1088/1361-6544/aae1db
Pierre Collet and Jean-Pierre Eckmann, Instabilities and fronts in extended systems, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1990. MR 1109707, DOI 10.1515/9781400861026
Gerhard Dangelmayr, Bernold Fiedler, Klaus Kirchgässner, and Alexander Mielke, Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability, Pitman Research Notes in Mathematics Series, vol. 352, Longman, Harlow, 1996. With a contribution by G. Raugel. MR 1458064
Jonathan H. P. Dawes, After 1952: the later development of Alan Turing’s ideas on the mathematics of pattern formation, Historia Math. 43 (2016), no. 1, 49–64 (English, with English and French summaries). MR 3450591, DOI 10.1016/j.hm.2015.03.003
G. Dee and J. S. Langer, Propagating pattern selection, Phys. Rev. Lett. 50 (1983), 383–386.
G. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 60 (1988), no. 25, 2641–2644.
Arjen Doelman, Björn Sandstede, Arnd Scheel, and Guido Schneider, Propagation of hexagonal patterns near onset, European J. Appl. Math. 14 (2003), no. 1, 85–110. MR 1970238, DOI 10.1017/S095679250200503X
Arjen Doelman, Björn Sandstede, Arnd Scheel, and Guido Schneider, The dynamics of modulated wave trains, Mem. Amer. Math. Soc. 199 (2009), no. 934, viii+105. MR 2507940, DOI 10.1090/memo/0934
Ute Ebert and Wim van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000), no. 1-4, 1–99. MR 1787406, DOI 10.1016/S0167-2789(00)00068-3
J.-P. Eckmann and G. Schneider, Non-linear stability of modulated fronts for the Swift-Hohenberg equation, Comm. Math. Phys. 225 (2002), no. 2, 361–397. MR 1889229, DOI 10.1007/s002200100577
Jean-Pierre Eckmann and C. Eugene Wayne, The nonlinear stability of front solutions for parabolic partial differential equations, Comm. Math. Phys. 161 (1994), no. 2, 323–334. MR 1266487, DOI 10.1007/BF02099781
K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing, Phys. Rev. B 75 (2007), 064107.
Grégory Faye and Matt Holzer, Asymptotic stability of the critical Fisher-KPP front using pointwise estimates, Z. Angew. Math. Phys. 70 (2019), no. 1, Paper No. 13, 21. MR 3881827, DOI 10.1007/s00033-018-1048-0
Grégory Faye and Matt Holzer, Asymptotic stability of the critical pulled front in a Lotka-Volterra competition model, J. Differential Equations 269 (2020), no. 9, 6559–6601. MR 4107062, DOI 10.1016/j.jde.2020.05.012
Grégory Faye, Matt Holzer, and Arnd Scheel, Linear spreading speeds from nonlinear resonant interaction, Nonlinearity 30 (2017), no. 6, 2403–2442. MR 3655106, DOI 10.1088/1361-6544/aa6c74
G. Faye, M. Holzer, A. Scheel, and L. Siemer, Invasion into remnant instability: a case study of front dynamics, Indiana Univ. Math. J., To appear.
Bernold Fiedler and Arnd Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, Trends in nonlinear analysis, Springer, Berlin, 2003, pp. 23–152. MR 1999097
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), no. 4, 355–369.
Mitsuo Funaki, Masayasu Mimura, and Tohru Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound. 8 (2006), no. 2, 223–245. MR 2256842, DOI 10.4171/IFB/141
Th. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity 7 (1994), no. 3, 741–764. MR 1275528, DOI 10.1088/0951-7715/7/3/003
Thierry Gallay and Arnd Scheel, Diffusive stability of oscillations in reaction-diffusion systems, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2571–2598. MR 2763727, DOI 10.1090/S0002-9947-2010-05148-7
Thierry Gallay and C. Eugene Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbf R^2$, Arch. Ration. Mech. Anal. 163 (2002), no. 3, 209–258. MR 1912106, DOI 10.1007/s002050200200
Robert A. Gardner and Kevin Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998), no. 7, 797–855. MR 1617251, DOI 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1
Ryan N. Goh, Samantha Mesuro, and Arnd Scheel, Spatial wavenumber selection in recurrent precipitation, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 1, 360–402. MR 2788928, DOI 10.1137/100793086
Cole Graham, Precise asymptotics for Fisher-KPP fronts, Nonlinearity 32 (2019), no. 6, 1967–1998. MR 3947215, DOI 10.1088/1361-6544/aaffe8
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol. 2 (1975), no. 3, 251–263. MR 411693, DOI 10.1007/BF00277154
F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52 (1999), no. 10, 1255–1276. MR 1699968, DOI 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.3.CO;2-N
François Hamel, James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media 8 (2013), no. 1, 275–289. MR 3043938, DOI 10.3934/nhm.2013.8.275
S. Heinze, Travelling waves for semilinear parabolic partial differential equations in cylindrical domains, Ph.D. Thesis, Ruprecht-Karls Universitaet Heidelberg, 1989.
Bernard Helffer, Spectral theory and its applications, Cambridge Studies in Advanced Mathematics, vol. 139, Cambridge University Press, Cambridge, 2013. MR 3027462, DOI 10.1017/CBO9781139505727
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
Bastian Hilder, Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law, J. Differential Equations 269 (2020), no. 5, 4353–4380. MR 4097934, DOI 10.1016/j.jde.2020.03.033
Matt Holzer, Anomalous spreading in a system of coupled Fisher-KPP equations, Phys. D 270 (2014), 1–10. MR 3151313, DOI 10.1016/j.physd.2013.12.003
Matt Holzer, A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 2069–2084. MR 3411553, DOI 10.3934/dcds.2016.36.2069
Matt Holzer and Arnd Scheel, Accelerated fronts in a two-stage invasion process, SIAM J. Math. Anal. 46 (2014), no. 1, 397–427. MR 3154144, DOI 10.1137/120887746
Matt Holzer and Arnd Scheel, Criteria for pointwise growth and their role in invasion processes, J. Nonlinear Sci. 24 (2014), no. 4, 661–709. MR 3228472, DOI 10.1007/s00332-014-9202-0
Peter Howard, Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form, Comm. Math. Phys. 269 (2007), no. 3, 765–808. MR 2276360, DOI 10.1007/s00220-006-0102-5
Kevin Zumbrun and Peter Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741–871. MR 1665788, DOI 10.1512/iumj.1998.47.1604
Sameer Iyer and Björn Sandstede, Mixing in reaction-diffusion systems: large phase offsets, Arch. Ration. Mech. Anal. 233 (2019), no. 1, 323–384. MR 3974642, DOI 10.1007/s00205-019-01358-9
Gabriela Jaramillo, Arnd Scheel, and Qiliang Wu, The effect of impurities on striped phases, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 1, 131–168. MR 3922810, DOI 10.1017/S0308210518000197
Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun, Nonlocalized modulation of periodic reaction diffusion waves: nonlinear stability, Arch. Ration. Mech. Anal. 207 (2013), no. 2, 693–715. MR 3005327, DOI 10.1007/s00205-012-0573-9
Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun, Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation, Arch. Ration. Mech. Anal. 207 (2013), no. 2, 669–692. MR 3005326, DOI 10.1007/s00205-012-0572-x
Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun, Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations, Invent. Math. 197 (2014), no. 1, 115–213. MR 3219516, DOI 10.1007/s00222-013-0481-0
Todd Kapitula and Keith Promislow, Spectral and dynamical stability of nonlinear waves, Applied Mathematical Sciences, vol. 185, Springer, New York, 2013. With a foreword by Christopher K. R. T. Jones. MR 3100266, DOI 10.1007/978-1-4614-6995-7
Todd Kapitula and Björn Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Phys. D 124 (1998), no. 1-3, 58–103. MR 1662530, DOI 10.1016/S0167-2789(98)00172-9
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
E. Knobloch, Spatial localization in dissipative systems, Annu. Rev. Condensed Matter Phys. 6 (2015), no. 1,325–359.
A. Kolmogorov, I. Petrovskii, and N. Piskunov, Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bjul. Moskowskogo Gos. Univ. Ser. Internat. Sec. A 1 (1937), 1–26.
Ka-Sing Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differential Equations 59 (1985), no. 1, 44–70. MR 803086, DOI 10.1016/0022-0396(85)90137-8
Tong Li and Zhi-An Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math. 70 (2009/10), no. 5, 1522–1541. MR 2578681, DOI 10.1137/09075161X
Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547]. MR 3012216
Alan C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), no. 2, 279–303. MR 3363403, DOI 10.1017/S0022112069000176
James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, Convergence to a single wave in the Fisher-KPP equation, Chinese Ann. Math. Ser. B 38 (2017), no. 2, 629–646. MR 3615508, DOI 10.1007/s11401-017-1087-4
James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, Refined long-time asymptotics for Fisher-KPP fronts, Commun. Contemp. Math. 21 (2019), no. 7, 1850072, 25. MR 4017787, DOI 10.1142/S0219199718500724
Kenneth J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), no. 2, 225–256. MR 764125, DOI 10.1016/0022-0396(84)90082-2
Kenneth J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 149–156. MR 958058, DOI 10.1090/S0002-9939-1988-0958058-1
Matthew F. Pennybacker, Patrick D. Shipman, and Alan C. Newell, Phyllotaxis: some progress, but a story far from over, Phys. D 306 (2015), 48–81. MR 3367573, DOI 10.1016/j.physd.2015.05.003
Alin Pogan and Arnd Scheel, Instability of spikes in the presence of conservation laws, Z. Angew. Math. Phys. 61 (2010), no. 6, 979–998. MR 2738299, DOI 10.1007/s00033-010-0058-3
G. Raugel and K. Kirchgässner, Stability of fronts for a KPP-system. II. The critical case, J. Differential Equations 146 (1998), no. 2, 399–456. MR 1631295, DOI 10.1006/jdeq.1997.3391
Jean-Michel Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), no. 4, 499–552 (English, with English and French summaries). MR 1464532, DOI 10.1016/S0294-1449(97)80137-0
Jean-Michel Roquejoffre, Luca Rossi, and Violaine Roussier-Michon, Sharp large time behaviour in $N$-dimensional Fisher-KPP equations, Discrete Contin. Dyn. Syst. 39 (2019), no. 12, 7265–7290. MR 4026189, DOI 10.3934/dcds.2019303
V. Rottschäfer and C. E. Wayne, Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation, J. Differential Equations 176 (2001), no. 2, 532–560. MR 1866286, DOI 10.1006/jdeq.2000.3984
Vivi Rottschäfer and Arjen Doelman, On the transition from the Ginzburg-Landau equation to the extended Fisher-Kolmogorov equation, Phys. D 118 (1998), no. 3-4, 261–292. MR 1633034, DOI 10.1016/S0167-2789(98)00035-9
Björn Sandstede, Arnd Scheel, Guido Schneider, and Hannes Uecker, Diffusive mixing of periodic wave trains in reaction-diffusion systems, J. Differential Equations 252 (2012), no. 5, 3541–3574. MR 2876664, DOI 10.1016/j.jde.2011.10.014
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
Arnd Scheel, Coarsening fronts, Arch. Ration. Mech. Anal. 181 (2006), no. 3, 505–534. MR 2231782, DOI 10.1007/s00205-006-0422-9
Arnd Scheel, Spinodal decomposition and coarsening fronts in the Cahn-Hilliard equation, J. Dynam. Differential Equations 29 (2017), no. 2, 431–464. MR 3651596, DOI 10.1007/s10884-015-9491-5
J. A. Sherratt, M. J. Smith, and J. D. M. Rademacher, Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion, Proc. Nati. Acad. Sci. 106 (2009), no. 27, 10890–10895.
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15 (1997), 319–328.
Wim van Saarloos, Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence, Phys. Rev. A (3) 39 (1989), no. 12, 6367–6390. MR 1003567, DOI 10.1103/PhysRevA.39.6367
W. van Saarloos, Front propagation into unstable states, Phys. Rep. 386 (2003), 29–222.
Wim van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Phys. D 56 (1992), no. 4, 303–367. MR 1169610, DOI 10.1016/0167-2789(92)90175-M
S. van Teeffelen, R. Backofen, A. Voigt, and H. Löwen, Derivation of the phase-field-crystal model for colloidal solidification, Phys. Rev. E 79 (2009), 051404.
Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224, DOI 10.1007/s00285-002-0169-3