Solution with an inner transition layer of a two-dimensional boundary value reaction–diffusion–advection problem with discontinuous reaction and advection terms

We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the reaction–diffusion–advection equation assuming that the reaction and advection terms are comparable in size and have a jump along a smooth curve located inside the studied domain. The problem solution has a large gradient in a neighborhood of this curve. We prove theorems on the existence, asymptotic uniqueness, and Lyapunov asymptotic stability for such solutions using the method of upper and lower solutions. To obtain the upper and lower solutions, we use the asymptotic method of differential inequalities that consists in constructing them as modified asymptotic approximations in a small parameter of solutions of these problems. We construct the asymptotic approximation of a solution using a modified Vasil'eva method.