Asymptotics of the Solution to a Stationary Piecewise-Smooth Reaction-Diffusion-Advection Equation

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is considered. The existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is studied. An efficient algorithm for constructing the point itself and an asymptotic representation of arbitrary-order accuracy to the solution is proposed. For sufficiently small parameter values, the existence theorem is proved by the technique of matching asymptotic expansions. An example is given to show the effectiveness of their method.