Existence and stability of a stationary solution of the system of diffusion equations in a medium with discontinuous characteristics under various quasimonotonicity conditions

Asymptotic analysis is used to study the existence, local uniqueness, and asymptotic stability in the sense of Lyapunov of a solution of a one-dimensional nonlinear system of reaction–diffusion equations with various types of quasimonotonicity of the functions describing reactions. A feature of the problem is the discontinuities (jumps) of these functions at a single point on the segment on which the problem is posed. The solution with a large gradient in the vicinity of the discontinuity point is studied. Sufficient conditions for the existence of a stable stationary solution of systems with various quasimonotonicity conditions are given. The asymptotic method of differential inequalities is used to prove the existence and stability theorems. The main distinctive features of this method for various types of quasimonotonicity are listed.