Boundary and Corner Layer Behavior in Singularly Perturbed Semilinear Systems of Boundary Value Problems

Abstract

The existence and asymptotic behavior as $\varepsilon \to 0^ + $ of solutions of the boundary value problem $\varepsilon {\bf y}'' = {\bf h}(t,{\bf y})$, $a m_i > 0$ in the domain of interest ($m_i $ a constant, $i = 1, \cdots ,n$). A mild assumption on the reduced solution essentially decouples the system and allows the application of the scalar theory of singularly perturbed boundary value problems to each component of the system. The components of solutions are shown to exhibit essentially two types of asymptotic behavior: (i) boundary layer behavior when the reduced solution is smooth and/or (ii) corner layer behavior when the reduced solution has a discontinuous first derivative in $(a,b)$. Several illustrative examples of both types of behavior are discussed. The results are established by using the theory of differential inequalities for systems of second order boundary value pro...