Contrast structures in the reaction– advection–diffusion problem appearing in a drift–diffusion model of a semiconductor in the case of nonsmooth reaction

We consider the boundary-value problem for the singularly perturbed reaction-advection-diffusion equation in the case of a small nonlinear advection and a nonsmooth reaction; it appears in a drift–diffusion model of a semiconductor. The key feature of the problem is the discontinuity of the derivative of the reactive term with respect to a spatial coordinate at a preliminary known point lying inside the interval under consideration. Using the asymptotic method of differential inequalities, we show that the problem can have several solutions with an internal transition layer in a small neighborhood of the discontinuity point. Each of these solutions can be asymptotically Lyapunov stable and unstable; we formulate sufficient conditions for both cases. It follows from the results of the asymptotic study that in the presence of an external current in the semiconductor with the N-shaped dependence of the drift velocity on the electric field strength, two neighboring stationary electron-depletion layers can exists in a small neighborhood of an internal point if the equilibrium electron concentration is an insufficiently smooth function of the spatial coordinate at that point.