Front formation and motion in quasilinear parabolic equations

This paper deals with the singular limit for Lɛu:=ut−F(u,ɛux)x−ɛ−1g(u)=0, where the function F is assumed to be smooth and uniformly elliptic, and g is a “bistable” nonlinearity. Denoting with um the unstable zero of g, for any initial datum u0 for which u0−um has a finite number of zeroes, and u0−um changes sign crossing each of them, we show the existence of solutions and describe the structure of the limiting function u0=limɛ→0+uɛ, where uɛ is the solution of a corresponding Cauchy problem. The analysis is based on the construction of travelling waves connecting the stable zeros of g and on the use of a comparison principle.