Convergence to travelling waves for quasilinear Fisher–KPP type equations

Abstract

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.