American Institute of Mathematical Sciences

April  2013, 33(4): 1407-1429. doi: 10.3934/dcds.2013.33.1407

Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds

 1 College of Mathematics and Econometrics, Hunan University, Changsha 410082 2 Institute of Contemporary Mathematics, Henan University, School of Mathematics and Information Science, Henan University, Kaifeng 475004

Received  July 2011 Revised  August 2012 Published  October 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 smooth submanifold that divides $\mathcal{M}$ into two disjoint components $\mathcal{M}_{\pm}$. We assume $\mathcal{K}$ is stationary and non-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. The first layer solution $u_\varepsilon^{(1)}$ approaches $-1$ in $\mathcal{M}_{-}$ and $+1$ in $\mathcal{M}_{+}$ as $\varepsilon$ tends to 0, while the other solution $u_\varepsilon^{(2)}$ exhibits a transition layer in the opposite direction.
Citation: Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
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