On the stability and domain of attraction of a stationary nonsmooth limit solution of a singularly perturbed parabolic equation

A stationary solution to the singularly perturbed parabolic equation $-u_t+\varepsilon^2u_{xx}-f(u,x)=0$ with Neumann boundary conditions is considered. The limit of the solution as $\varepsilon\to0$ is a nonsmooth solution to the reduced equation $f(u,x)=0$ that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.