Nonmonotone Interior Layer Theory for Some Singularly Perturbed Quasilinear Boundary Value Problems with Turning Points

Abstract

The quasilinear singular perturbation problem $\varepsilon y'' = f(x,y)y' + g(x,y)$, $y( - 1,\varepsilon )A$, $y(1,\varepsilon ) = B$ is studied under the principal assumption that $f(0,y) = 0$ for all y, i.e., that $x = 0$ is a turning point for the function f. Under explicit conditions on f, g, A and B, solutions are shown to exhibit one of two types of nonmonotone interior layer behavior: (i) spike layer behavior or (ii) nonmonotone transition layer behavior. The results are obtained using a method based on the theory of differential inequalities. Applications and examples are discussed in detail.