Phase transition layers for Fife-Greenlee problem on smooth bounded domain
Feifei Tang, Suting Wei, Jun Yang · Discrete and Continuous Dynamical Systems · 2018
We consider the Fife-Greenlee problem \begin{document}$ε^2\triangle u + \bigl(u-\mathbf{a}(y)\bigr)(1-u^2) =0 ~~~ \mbox{in}\ Ω,~~~~~~~\frac{\partial u}{\partialν} = 0 ~~~ \mbox{on}\ \partialΩ,$ \end{document} where $Ω$ is a bounded domain in ${\mathbb R}^2$ with smooth boundary, $\epsilon>0$ is a small parameter, $ν$ denotes the unit outward normal of $\partialΩ$. Let $Γ = \{y∈ Ω: \mathbf{a}(y) = 0 \}$ be a simple smooth curve intersecting orthogonally with $\partialΩ$ at exactly two points and dividing $Ω$ into two disjoint nonempty components. We assume that $-1\,<\,\mathbf{a}(y)\,<1$ on $Ω$ and $\triangledown\mathbf{a}≠ 0$ on $Γ$, and also some admissibility conditions between the curves $Γ$, $\partialΩ$ and the inhomogeneity ${\mathbf a}$ hold at the connecting points. We can prove that there exists a solution $u_{\epsilon}$ such that: as $\epsilon → 0$, $u_{\epsilon}$ approaches $+1$ in one part, while tends to $-1$ in the other part, except a small neighborhood of $Γ$.