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May  2020, 19(5): 2575-2616. doi: 10.3934/cpaa.2020113

## Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation

 1 Department of Mathematics, South China Agricultural University, Guangzhou 510642, China, 2 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 3 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

* Corresponding author

Received  December 2018 Revised  November 2019 Published  March 2020

Fund Project: Jun Yang is supported by National Natural Science Foundation of China (Grant No. 11771167 and No. 11831009)

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation
 $\epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega,$
where
 $\Omega$
is a bounded domain in
 $\mathbb R^2$
with smooth boundary,
 $\epsilon$
is a small positive parameter,
 $\nu$
denotes the unit outward normal of
 $\partial \Omega$
,
 $V$
is a positive smooth function on
 $\bar\Omega$
. Let
 $\Gamma\subset\Omega$
be a smooth curve dividing
 $\Omega$
into two disjoint regions and intersecting orthogonally with
 $\partial\Omega$
at exactly two points
 $P_1$
and
 $P_2$
. Moreover, by considering
 ${\mathbb R}^2$
as a Riemannian manifold with the metric
 $g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2)$
, we assume that: the curve
 $\Gamma$
is a non-degenerate geodesic in the Riemannian manifold
 $({\mathbb R}^2, g)$
, the Ricci curvature of the Riemannian manifold
 $({\mathbb R}^2, g)$
along the normal
 $\mathbf{n}$
of
 $\Gamma$
is positive at
 $\Gamma$
, the generalized mean curvature of the submanifold
 $\partial\Omega$
in
 $({\mathbb R}^2, g)$
vanishes at
 $P_1$
and
 $P_2$
. Then for any given integer
 $N\geq 2$
, we construct a solution exhibiting
 $N$
-phase transition layers near
 $\Gamma$
(the zero set of the solution has
 $N$
components, which are curves connecting
 $\partial\Omega$
and directed along the direction of
 $\Gamma$
) with mutual distance
 $O(\epsilon|\log \epsilon|)$
, provided that
 $\epsilon$
stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.
Citation: Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
##### References:

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##### References:
$\begin{array}{*{20}{l}} {{ Curves}{\rm{: }}{C_1} = \left( {{y_1},{\varphi _1}\left( {{y_1}} \right)} \right),\;\;\;{\kern 1pt} {C_2} = \left( {{y_1},{\varphi _2}\left( {{y_1}} \right)} \right), - {\delta _0} < {y_1} < {\delta _0},}\\ {{ Points}{\rm{: }}{P_1} = (0,\varphi (0)),{P_2} = \left( {0,{\varphi _2}(0)} \right).} \end{array}$
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