Existence and Stability of a Stationary Solution with a Boundary Layer in a Two-Dimensional System of Fast and Slow Reaction–Diffusion–Advection Equations with KPZ Nonlinearities

The paper considers a boundary-value problem for a singularly perturbed elliptic system of fast and slow equations, commonly referred to as a system of Tikhonov type. A distinctive feature of the problem is the presence of terms containing the squared gradient of the unknown function (KPZ nonlinearities). A boundary layer asymptotic expansion of the solution is constructed in the case of Dirichlet boundary conditions, the existence of a solution with the constructed asymptotics is proved, and its Lyapunov asymptotic stability is studied. The proof of the theorems is based on the asymptotic method of differential inequalities developed by N.N. Nefedov.