Boundary layer behavior in perturbed second-order systems (Q762677)

We present here some results on the asymptotic behavior as \(\epsilon \to 0^+\) of solutions of the Dirichlet problem for the singularly perturbed second-order system in \({\mathbb{R}}^ n\) \({\mathcal E}y''={\mathcal F}(y)y'+g(x,y)\), \(a<x<b\), where \({\mathcal F}\) is an (n\(\times n)\)-matrix valued function. In particular, we are interested in providing sufficient conditions in order that a solution of this problem display boundary layer behavior at an endpoint \((x=a\) or \(x=b)\). Layer behavior simply means nonuniform dependence of the solution on x and \({\mathcal E}\) as \({\mathcal E}\to 0^+\), and very often it is this nonuniform or singular dependence which affords the most insight into natural phenomena modelled by the system.