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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show all references

References:
[1]

Cal. Var. PDE, 11 (2000), 233-306.  Google Scholar

[2]

Acta. Metall., 27 (1979), 1084-1095. Google Scholar

[3]

Math. Res. Lett., 3 (1996), 117-131.  Google Scholar

[4]

J. Diff. Eqns., 194 (2003), 382-405.  Google Scholar

[5]

J. Funct. Anal., 258 (2010), 458-503.  Google Scholar

[6]

Comm. Pure Appl. Math., 70 (2007), 113-146.  Google Scholar

[7]

Archive Rational Mechanical Analysis, 190 (2008), 141-187.  Google Scholar

[8]

Didcrete Contin. Dunam. Systems, A, 28 (2010), 975-1006. doi: 10.3934/dcds.2010.28.975.  Google Scholar

[9]

Geom. Funct. Anal., 20 (2010), 918-957. doi: 10.1007/s00039-010-0083-6.  Google Scholar

[10]

J. Diff. Eqns., 238 (2007), 87-117.  Google Scholar

[11]

J. Diff. Eqns., 249 (2010), 215-239.  Google Scholar

[12]

J. Diff. Eqns., 169 (2001), 190-207.  Google Scholar

[13]

Proc. Royal Soc. Edinburgh, 11A (1989), 69-84.  Google Scholar

[14]

Annali di Matematica Pura et Aplicata, (4), 184 (2005), 17-52. doi: 10.1007/s10231-003-0088-y.  Google Scholar

[15]

Geom. Funct. Anal., 16 (2006), 924-958.  Google Scholar

[16]

Duke Math. J., 124 (2004), 105-143.  Google Scholar

[17]

Pacific J. Math., 229 (2007), 447-468.  Google Scholar

[18]

J. Fixed Point Theory Appl., 1 (2007), 305-336.  Google Scholar

[19]

Arch. Rat. Mech. Anal., 98 (1987), 357-383.  Google Scholar

[20]

J. Diff. Eqns., 191 (2003), 234-276.  Google Scholar

[21]

Ann. Inst. H. Poincaré Anal. NonLinéaire, 20 (2003), 107-143.  Google Scholar

[22]

J. Diff. Geom., 64 (2003), 359-423.  Google Scholar

[23]

Comm. Pure Appl. Math., 51 (1998), 551-579.  Google Scholar

[24]

Commun. Pure Appl. Math., 56 (2003), 1078-1134.  Google Scholar

[25]

Calc. Var. Partial Differential Equations, 21 (2004), 157-207.  Google Scholar

[26]

Arch. Rational Mech. Anal., 141 (1998), 375-400.  Google Scholar

[27]

Asymptot. Anal., 69 (2010), 175-218.  Google Scholar

[28]

Commun. Pure Appl. Anal., 1 (2013), 303-340. Google Scholar

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