Convergence to travelling waves for quasilinear Fisher–KPP type equations

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)=0φ(0)=0 and the equation may degenerate at u=0u=0 (in the case of φ′(0)=0φ′(0)=0). We consider the case of u0∈L∞(R)u0∈L∞(R), 0⩽u0(x)⩽10⩽u0(x)⩽1 a.e. x∈Rx∈R and the special case of ψ(u)=u−φ(u)ψ(u)=u−φ(u). We prove that the solution approaches the travelling wave solution (with speed c=1c=1), spreading either to the right or to the left, or to the two travelling waves moving in opposite directions.