Front formation and motion in quasilinear parabolic equations

This paper deals with the singular limit for Lεu := ut −F(u,εux )x −ε−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 = limε→0+ uε, where uε 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.  2005 Published by Elsevier Inc.