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.