Boundary and interior transition layer phenomena for pairs of second-order differential equations☆

Such systems arise, for example, in the study of steady-state configurations of a mixture of chemically reacting and diffusing substances. Principal attention will be given to layer-type qualitative features of the solution; that is, to the fact that families of solutions with E as parameter commonly exist which approach discontinuous functions of x as E --) 0. The solution when E is small but nonzero, being smooth, exhibits an abrupt but continuouslv differentiable transition at the location of the limit discontinuity. An interval where such an abrupt change takes place is loosel! called a “layer’‘-a “boundary layer” when it is adjacent to the boundary. It is possible for internal layers also to exist, and we give particular attention to such in Sections 4-6. If a family (u, , 21,) of solutions exists with a limit in some sense as l + 0, then one expects the limiting pair (L’, I/) to be a solution of the reduced problem .f( CT, I’) = 0; I;” == g( CT, F), with boundary conditions imposed