TY - JOUR

T1 - Convergence to travelling waves for quasilinear Fisher-KPP type equations

AU - Díaz, J. I.

AU - Kamin, S.

N1 - Funding Information:
Both authors thank Phillip Rosenau for many useful conversations on the model during the long preparation of this work. The research of the first author was partially supported by the project ref. MTM200806208 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) supported by UCM. The research of the author has received funding from the ITN FIRST of the Seventh Framework Programme of the European Community’s (grant agreement number 238702).

PY - 2012/6/1

Y1 - 2012/6/1

N2 - 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≤u 0(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.

AB - 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≤u 0(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.

KW - Asymptotic convergence

KW - Kolmogorov, Petrovsky and Piscunov equation

KW - Travelling waves

UR - http://www.scopus.com/inward/record.url?scp=84856754825&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2012.01.018

DO - 10.1016/j.jmaa.2012.01.018

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AN - SCOPUS:84856754825

SN - 0022-247X

VL - 390

SP - 74

EP - 85

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -