Convergence to travelling waves for quasilinear Fisher–KPP type equations

We consider the Cauchy problem { u t = φ ( u ) x x + ψ ( u ) , ( t , x ) ∈ R + × R , u ( 0 , x ) = u 0 ( 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 u 0 ∈ 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.