Existence and stability of the solution to a system of two nonlinear diffusion equations in a medium with discontinuous characteristics

Asymptotic analysis is used to study the existence, local uniqueness, and asymptotic Lyapunov stability of the solution to a one-dimensional nonlinear parabolic system of the activator–inhibitor type. A specific feature of the problem is the discontinuities of the first kind of the functions on the right-hand sides of the equations. The jump of the functions occurs at a single point of the interval on which the problem is considered. The solution with a large gradient in the vicinity of the discontinuity is studied. The existence and stability theorems are proved using the asymptotic method of differential inequalities.