Singularly perturbed two-dimensional parabolic problem in the case of intersecting roots of the reduced equation

The singularly perturbed parabolic equation $-u_t+\varepsilon^2\Delta u-f(u,x,\varepsilon)=0$, $x\in D\subset\mathbb R^2$, $t>0$ with Robin conditions on the boundary of $D$ is considered. The asymptotic stability as $t\to\infty$ and the global domain of attraction are analyzed for the stationary solution whose limit as $\varepsilon\to0$ is a nonsmooth solution to the reduced equation $f(u,x,0)=0$ that consists of two intersecting roots of this equation.