American Institute of Mathematical Sciences

Stable transition layers in an unbalanced bistable equation

 Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil

Received  July 2020 Revised  October 2020 Published  December 2020

In this paper we are concerned with the existence of stable stationary solutions for the problem $u_t = \epsilon^2(k_1^2(x) u_x)_x+k_2^2(x)g(u,x)$, $(t,x)\in\mathbb{R}^+\times (0,1)$ subject to Neumann boundary condition. We suppose that $k_1,k_2\in C^1(0,1)$ are positive functions and $g$ is an unbalanced bistable function. We prove the existence of a family of stable stationary solutions developing internal transition layers in a specific sub-interval of $(0,1)$. For this, we provide a general variational method inspired by the $\Gamma$-convergence theory.

Citation: Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020370
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$u_{\epsilon}$ developing two internal transition layer with interfaces at $\overline{x}_1$ and $\overline{x}_2$ (isolated local minima of $\gamma$ in $[x_1,x_2]$)