Convergence to travelling waves for quasilinear Fisher–KPP type equations

• Published in: Journal of mathematical analysis and applications Vol. 390; no. 1; pp. 74 - 85
• Main Authors: Díaz, J.I., Kamin, S.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.06.2012
• Subjects: Asymptotic convergence; Kolmogorov, Petrovsky and Piscunov equation; Travelling waves; Kolmogorov, Petrovsky and Piscunov equation; Asymptotic convergence; Travelling waves
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2012.01.018

We consider the Cauchy problem{ut=φ(u)xx+ψ(u),(t,x)∈R+×R,u(0,x)=u0(x),x∈R, when the increasing function φ satisfies that φ(0)=0 and the equation may degenerate at u=0 (in the case of φ′(0)=0). We consider the case of u0∈L∞(R), 0⩽u0(x)⩽1 a.e. x∈R and the special case of ψ(u)=u−φ(u). We prove that the solution approaches the travelling wave solution (with speed c=1), spreading either to the right or to the left, or to the two travelling waves moving in opposite directions.