Stable transition layers in an unbalanced bistable equation

Maicon Sônego

doi:10.3934/dcdsb.2020370

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2021, Volume 26, Issue 10: 5627-5640. Doi: 10.3934/dcdsb.2020370

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Stable transition layers in an unbalanced bistable equation

Maicon Sônego

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

Received: June 30, 2020

Revised: September 30, 2020

Early access: December 2020

Published: October 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B25, 35B35; Secondary: 35B36, 35B40.

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Figure 1. $ 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] $)

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