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October 2021, 26(10): 5627-5640. doi: 10.3934/dcdsb.2020370
Stable transition layers in an unbalanced bistable equation
Maicon Sônego
Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil
Received July 2020 Revised October 2020 Published October 2021 Early access December 2020
Figure(1)
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.
Keywords: Internal transition layers, unbalanced case, stability, semi-linear problem.
Mathematics Subject Classification: Primary: 35B25, 35B35; Secondary: 35B36, 35B40.
Citation: Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5627-5640. doi: 10.3934/dcdsb.2020370
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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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