Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem

In this article we consider a singularly perturbed Allen-Cahn problem \begin{document}$ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $\end{document} , for \begin{document}$ (x,t)\in (0,1)\times\mathbb{R}^+ $\end{document} , supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities \begin{document}$ a(\cdot) $\end{document} and \begin{document}$ b(\cdot) $\end{document} are allowed to vanish at some points in \begin{document}$ (0,1) $\end{document} . Using the variational concept of \begin{document}$ \Gamma $\end{document} -convergence we prove that, for \begin{document}$ \epsilon $\end{document} small, such degeneracy of \begin{document}$ a(\cdot) $\end{document} and \begin{document}$ b(\cdot) $\end{document} induces the existence of stable stationary solutions which develop internal transition layer as \begin{document}$ \epsilon\to 0 $\end{document} .