Fast propagation for KPP equations with slowly decaying initial conditions

Abstract This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher–KPP reaction–diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions.

[1]  Richard Haberman,et al.  The Accommodation of Traveling Waves of Fisher's Type to the Dynamics of the Leading Tail , 1993, SIAM J. Appl. Math..

[2]  François Hamel,et al.  The speed of propagation for KPP type problems. I: Periodic framework , 2005 .

[3]  K. Lau.  On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov , 1985 .

[4]  Jean-Michel Roquejoffre,et al.  A parabolic equation of the KPP type in higher dimensions , 1995 .

[5]  Koichi Uchiyama,et al.  The behavior of solutions of some non-linear diffusion equations for large time , 1977 .

[6]  L. Roques,et al.  Recolonisation by diffusion can generate increasing rates of spread. , 2010, Theoretical population biology.

[7]  J. Roquejoffre,et al.  Stability of Generalized Transition Fronts , 2009 .

[8]  Andrej Zlatos,et al.  Sharp transition between extinction and propagation of reaction , 2005, math/0504333.

[9]  Adam M. Oberman,et al.  Bulk Burning Rate in¶Passive–Reactive Diffusion , 1999, math/9907132.

[10]  J. McLeod,et al.  The approach of solutions of nonlinear diffusion equations to travelling front solutions , 1977 .

[11]  Henri Berestycki,et al.  Asymptotic spreading in heterogeneous diffusive excitable media , 2008 .

[12]  Jack Xin,et al.  Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds , 2005 .

[13]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[14]  P. Martínez,et al.  Dynamique en grand temps pour une classe d'équations de type KPP en milieu périodique , 2008 .

[15]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[16]  Henri Berestycki,et al.  Generalized travelling waves for reaction-diffusion equations , 2006 .

[17]  François Hamel,et al.  Spreading Speeds for Some Reaction-Diffusion Equations with General Initial Conditions , 2009, SIAM J. Math. Anal..

[18]  J. Roquejoffre.  Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders , 1997 .

[19]  Jianhua Huang,et al.  Speeds of Spread and Propagation for KPP Models in Time Almost and Space Periodic Media , 2009, SIAM J. Appl. Dyn. Syst..

[20]  L. Roques,et al.  Uniqueness and stability properties of monostable pulsating fronts , 2011 .

[21]  H. McKean.  Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov , 1975 .

[22]  D. A. Larson.  Transient Bounds and Time-Asymptotic Behavior of Solutions to Nonlinear Equations of Fisher Type , 1978 .

[23]  W. Saarloos,et al.  Front propagation into unstable states : universal algebraic convergence towards uniformly translating pulled fronts , 2000, cond-mat/0003181.

[24]  J. Xin.  Multidimensional Stability of Traveling Waves in a Bistable Reaction–Diffusion Equation, I , 1992 .

[25]  Xiao-Qiang Zhao,et al.  Spreading speeds and traveling waves for periodic evolution systems , 2006 .

[26]  D. J. Needham,et al.  Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations , 1999 .

[27]  J. Roquejoffre.  Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability , 1992 .

[28]  J. Roquejoffre,et al.  Stability of travelling fronts in a model for flame propagation part I: Linear analysis , 1992 .

[29]  Hans F. Weinberger,et al.  On spreading speeds and traveling waves for growth and migration models in a periodic habitat , 2002, Journal of mathematical biology.

[30]  Henri Berestycki,et al.  Generalized Transition Waves and Their Properties , 2010, 1012.0794.

[31]  M. Bramson.  Convergence of solutions of the Kolmogorov equation to travelling waves , 1983 .

[32]  A. Kiselev,et al.  Enhancement of the traveling front speeds in reaction-diffusion equations with advection , 2000, math/0002175.

[33]  X. Xin,et al.  Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity , 1991 .

[34]  F. Rothe.  Convergence to travelling fronts in semilinear parabolic equations , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[35]  Yoshinori Kametaka,et al.  On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type , 1975 .

[36]  Henri Berestycki,et al.  The speed of propagation for KPP type problems. II , 2010 .

[37]  M. Taniguchi,et al.  Stability of Planar Waves in the Allen–Cahn Equation , 2009 .

[38]  David H. Sattinger,et al.  On the stability of waves of nonlinear parabolic systems , 1976 .

[39]  J. Sherratt,et al.  Comparison theorems and variable speed waves for a scalar reaction–diffusion equation , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[40]  Jack Xin,et al.  Front Propagation in Heterogeneous Media , 2000, SIAM Rev..

[41]  Jonathan A. Sherratt,et al.  Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation , 1996 .

[42]  Andrej Zlatoš.  Quenching and propagation of combustion without ignition temperature cutoff , 2004, math/0409172.