# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021185
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## Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem

 1 Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil 2 Departamento de Matemática - DM, Universidade Federal de São Carlos, SP, Brazil

Received  January 2021 Revised  May 2021 Early access July 2021

In this article we consider a singularly perturbed Allen-Cahn problem $u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3)$, for $(x,t)\in (0,1)\times\mathbb{R}^+$, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities $a(\cdot)$ and $b(\cdot)$ are allowed to vanish at some points in $(0,1)$. Using the variational concept of $\Gamma$-convergence we prove that, for $\epsilon$ small, such degeneracy of $a(\cdot)$ and $b(\cdot)$ induces the existence of stable stationary solutions which develop internal transition layer as $\epsilon\to 0$.

Citation: Maicon Sônego, Arnaldo Simal do Nascimento. Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021185
##### References:
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##### References:
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