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.

中文翻译：

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不平衡双稳态方程中的稳定过渡层

在本文中，我们关注问题 $ 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) $ 服从诺依曼边界条件。我们假设 $ k_1,k_2\in C^1(0,1) $ 是正函数，$ g $ 是不平衡双稳态函数。我们证明了在 $ (0,1) $ 的特定子区间内开发内部过渡层的稳定平稳解系列的存在。为此，我们提供了一种受 $\Gamma $-收敛理论启发的通用变分方法。