Front formation and motion in quasilinear parabolic equations

This paper deals with the singular limit for L(epsilon)u : = u(1) - F(u, epsilon u(x))x -epsilon(-1) g(u) = 0, where the function F is assumed to be smooth and uniformly elliptic, and g is a "bistable" nonlinearity. Denoting with u(m) the unstable zero of g, for any initial datum no for which no - u(m) has a finite number of zeroes, and u(0) - u(m) changes sign crossing each of them, we show the existence of solutions and describe the structure of the limiting function u(0) = lim(epsilon -> 0) + u(epsilon), where u(epsilon) 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. (c) 2005 Published by Elsevier Inc.