How travelling waves attract the solutions of KPP-type equations
type: Item
instance of: scholarly article
title: How travelling waves attract the solutions of KPP-type equations
zbMATH Open document ID: 1284.35108
DOI: 10.1090/S0002-9947-2012-05554-1
author: Michaël Bages
author: Patrick Martinez
author: J. M. Roquejoffre
published in: Transactions of the American Mathematical Society
publication date: 2013-07-25T00:00:00Z
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 6191474
zbMATH Keywords: reaction diffusion equations
zbMATH Keywords: KPP equations
zbMATH Keywords: traveling waves
zbMATH Keywords: attractive properties
zbMATH Keywords: stability
zbMATH Keywords: Kolmogorov-Petrovskii-Piskunov
zbMATH Keywords: Cole-Hopf transformation
MaRDI profile type: Publication
cites work: Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion
cites work: Large-time dynamics for a class of KPP-type equations in periodic media
cites work: Front propagation in periodic excitable media
cites work: Q3057559
cites work: Q5201633
cites work: Travelling fronts in cylinders
cites work: Convergence of solutions of the Kolmogorov equation to travelling waves
cites work: Space-time behaviour in problems of hydrodynamic type: a case study
cites work: Existence of pulsating waves in a model of flames in sprays
cites work: Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
cites work: On the structure of the spectra of periodic travelling waves
cites work: Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity
cites work: Travelling fronts and entire solutions of the Fisher-KPP equation in \({\mathbb R}^N\)
cites work: Uniqueness and stability properties of monostable pulsating fronts
cites work: Non-adiabatic KPP fronts with an arbitrary Lewis number
cites work: Geometric theory of semilinear parabolic equations
cites work: A Parabolic Equation of the KPP Type in Higher Dimensions
cites work: The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow
cites work: Traveling fronts in space-time periodic media
cites work: Long-time behaviour of heat flow: Global estimates and exact asymptotics
cites work: Q4951570
cites work: On bounded and unbounded global solutions of a supercritical semilinear heat equation
cites work: Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders
cites work: Nontrivial large-time behaviour in bistable reaction-diffusion equations
cites work: On the stability of waves of nonlinear parabolic systems
cites work: Generation of Analytic Semigroups by Strongly Elliptic Operators Under General Boundary Conditions
cites work: The behavior of solutions of some non-linear diffusion equations for large time
cites work: Complexity of large time behaviour of evolution equations with bounded data
cites work: Irregular behavior of solutions for Fisher's equation
review text: The Kolmogorov-Petrovskii-Piskunov (KPP) reaction diffusion equations NEWLINE\[NEWLINEu_t -\text{div}(A(x,y)Du) + B(x,y).Du =f(x,y,u), \quad (x,y)\in \mathbb{R}\times \mathbb{T}^{N-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{t\to -\infty} u(t,x,y) = 0, \lim_{t\to \infty} u(t,x,y) = 1,NEWLINE\]NEWLINE with \(f\) smooth, concave and 1-periodic in \(u\), with \(f(x,y,0)=f(x,y,1)=0\) and NEWLINE\[NEWLINE\sup_{(x,y)}f_u(x,y,1)<0<\sup_{(x,y)}f_u(x,y,0),NEWLINE\]NEWLINE and with \( \mathbb{T}^n\) being the \(n\)-dimensional torus, is known to have pulsating waves \(u(t,x,y)=\phi_c(t,x-ct,y)\) if \(c>c_*\) for some critical values \(c_*>0\). The authors study the large time asymptotics of solutions with initial data \(u_0\) lying between two translates of one pulsating wave: NEWLINE\[NEWLINE\phi_c(-M,x,y) \leq u_0(x,y) \leq \phi_c(M,x,y) \quad \text{for some} \;M>0, c>c_*.NEWLINE\]NEWLINE They show that the solution \(u\) converges to the pulsating wave with a shift \(m(t,x,y)\) determined as a solution of a nonlinear parabolic equation with periodic coefficients. In detail NEWLINE\[NEWLINE\sup_{(x,y)\in \mathbb{R}\times \mathbb{T}^{N-1}}|u(t,x,y)-\phi_c(t+m(t,x,y),x,y)|\to 0 \quad \text{as} \;t\to \infty.NEWLINE\]NEWLINE In the 1D case \(u_t-u_{xx}=f(u)\) with \(\phi_c(x-M)\leq u_0(x)\leq \phi_c(x+M)\) an explicit approximation of \(m\) is provided employing the Cole-Hopf transformation of the \(m\)-equation to a linear one and using the heat kernel. Moreover, in the 1D case a convergence rate for the asymptotics of \(u\) is obtained.NEWLINENEWLINEThe main techniques of the proof are the derivation of a simplified model for the shift \(m\), which can be analyzed, and linearizing it via the Cole-Hopf transform. For the convergence at \(t\to \infty\) the authors use mainly parabolic estimates and the weak maximum principle.
reviewed by: Tomáš Dohnal
description: scientific article; zbMATH DE number 6191474