Convergence to Sharp Traveling Waves of Solutions for Burgers-Fisher-KPP Equations with Degenerate Diffusion

Abstract

This paper is concerned with the convergence to sharp traveling waves of solutions with semi-compactly supported initial data for Burgers-Fisher-KPP equations with degenerate diffusion. We characterize the motion of the free boundary in the long-time asymptotic of the solution to Cauchy problem and the convergence to sharp traveling wave with almost exponential decay rates. Here a key difficulty lies in the intrinsic presence of nonlinear advection effect. After providing the analysis of the nonlinear advection effect on the asymptotic propagation speed of the free boundary, we construct sub- and super-solutions with semi-compact supports to estimate the motion of the free boundary. The new method overcomes the difficulties of the non-integrability of the generalized derivatives of sharp traveling waves at the free boundary.