*Published Paper*

**Inserted:** 31 jan 2019

**Last Updated:** 31 jan 2019

**Journal:** J. Differential Equations

**Volume:** 256

**Number:** 1

**Pages:** 157--205

**Year:** 2014

**Doi:** 10.1016/j.jde.2013.08.018

**Abstract:**

In this work we construct a family of entire bounded solutions for the singularly perturbed inhomogeneous Allen-Cahn equation $\varepsilon^2\textrm{div}(a(x)\nabla u) -a(x)F'(u)=0$ in $\mathbb{R}^2$, where $\varepsilon\to 0$. The nodal set of these solutions is close to a ``nondegenerate" curve which is asymptotically two non-parallel straight lines. Here $F'$ is a double-well potential and $a$ is a smooth positive function. We also provide examples of curves and functions $a$ where our result applies. This work is in connection with the results found in [Z.Du and B.Lai, *Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian manifolds*, Preprint], [Z.Du and C.Gui, *Interior layers for an inhomogeneous Allen-Cahn equation*, J. Differential Equations, 249 (2010), pp 215-239] and [F. Pacard and M. RitorĂ©, *From the constant mean curvature hypersurfaces to the gradient theory of phase transitions*, J. Differential Geom. 64 (2003), pp. 359-423], handling the compact case.

**Keywords:**
phase transitions, entire solution, inhomogeneity, noncompact nodal set, Allen-Cahn equation

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