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

<p style='text-indent:20px;'>In this article we consider a singularly perturbed Allen-Cahn problem <inline-formula><tex-math id="M1">\begin{document}$ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M2">\begin{document}$ (x,t)\in (0,1)\times\mathbb{R}^+ $\end{document}</tex-math></inline-formula>, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities <inline-formula><tex-math id="M3">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> are allowed to vanish at some points in <inline-formula><tex-math id="M5">\begin{document}$ (0,1) $\end{document}</tex-math></inline-formula>. Using the variational concept of <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence we prove that, for <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small, such degeneracy of <inline-formula><tex-math id="M8">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> induces the existence of stable stationary solutions which develop internal transition layer as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon\to 0 $\end{document}</tex-math></inline-formula>.</p>

Asymptotic solution of the inverse problem for restoring the modular type source in Burgers’ equation with modular advection

For a singularly perturbed Burgers’ type equation with modular advection that has a time-periodic solution with an internal transition layer, asymptotic analysis is applied to solve the inverse problem for restoring the function of the source using known information about the observed solution of a direct problem at a given time interval (period).