Abstract :
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.
