Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem
| Autor: | Maicon Sônego, Arnaldo Simal do Nascimento |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Discrete and Continuous Dynamical Systems - B. 27:3297 |
| ISSN: | 1553-524X 1531-3492 |
| DOI: | 10.3934/dcdsb.2021185 |
| Popis: | 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}. |
| Databáze: | OpenAIRE |
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