Phase Transition Solutions With Spike For Fife-Greenlee Problem

We consider t.he Fife-Greenlee problem?2?u+(u-a(y))(1-u2)=0 in ?,(?)=0 on(?)?,where ? is a bounded domain in R2 with smooth boundary,?>0 is a small parameter,n denotes the unit outward normal of(?)?.Assume that ? = { y??:a(y)= 0} is a simple smooth curve intersecting orthogonally with(?)? at exactly two points,saying P1,P2,and dividing Q into two disjoint,nonempty components,saying ?+ and ?_.We assume that-1<a(y)<1 on ? and ?a?0 on ?,and also some admissibility conditions between the curves ?,(?)? and the inhomogeneity a(y)hold at P1,P2.We will show the existence of a solution u? with a transition layer near ? and a downward spike near the maximum points of a(y)whose profile looks like u??C<1 at a point P?,u??+1 in ?+\P?,u??-1 in ?_,as ??0.