Transition layers for an inhomogeneous Allen-Cahnequation in Riemannian manifolds
Zhuoran Du, Baishun Lai · Discrete and Continuous Dynamical Systems · 2012
Let $(\mathcal{M}, \tilde{g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the problem$$ \varepsilon^2 Δ_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0 in \mathcal{M},$$where $\varepsilon>0$ is a small parameter and $V$ is a positive, smooth function in $\mathcal{M}$. Let$ \mathcal{K}\subset \mathcal{M}$ be an $(N-1)$-dimensional smoothsubmanifold that divides $\mathcal{M}$ into two disjoint components$\mathcal{M}_{\pm}$. We assume $\mathcal{K}$ is stationary andnon-degenerate relative to the weighted area functional$\int_{\mathcal{K}}V^{\frac{1}{2}}$. We prove that there exist two transition layer solutions $u_\varepsilon^{(1)},u_\varepsilon^{(2)}$ when $\varepsilon$ is sufficiently small. Thefirst layer solution $u_\varepsilon^{(1)}$ approaches $-1$ in$\mathcal{M}_{-}$ and $+1$ in $\mathcal{M}_{+}$ as $\varepsilon$tends to 0, while the other solution $u_\varepsilon^{(2)}$ exhibitsa transition layer in the opposite direction.