Existence and Stability of Solutions with Internal Transition Layer for the Reaction-Diffusion-Advection Equation with a KPZ-Nonlinearity

We study a boundary value problem for a quasilinear reaction-diffusion-advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.