On higher-dimensional contrast structure of singularly perturbed Dirichlet problem

In this paper, we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem. Based on the existence, an asymptotical analysis of a steplike contrast structure (i. e., an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection. In the framework of this paper, we propose a first integral condition, under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space. Then, the step-like contrast structure is constructed, and the internal transition time is determined. Meanwhile, the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained. Finally, an example is presented to illustrate the result.