Abstract

We consider the Cahn–Hilliard equation in the case where its solution depends on two spatial variables, with homogeneous Dirichlet and Neumann boundary conditions, and also periodic boundary conditions. For these three boundary value problems, we study the problem of local bifurcations arising when changing stability by spatially homogeneous equilibrium states. We show that the nature of bifurcations that lead to spatially inhomogeneous solutions is strongly related to the choice of boundary conditions. In the case of homogeneous Dirichlet boundary conditions, spatially inhomogeneous equilibrium states occur in a neighborhood of a homogeneous equilibrium state, depending on both spatial variables. An alternative scenario is realized in analyzing the Neumann problem and the periodic boundary value problem. In these, as a result of bifurcations, invariant manifolds formed by spatially inhomogeneous solutions occur. The dimension of these manifolds ranges from 1 to 3. In analyzing three boundary value problems, we use methods of infinite-dimensional dynamical system theory and asymptotic methods. Using the integral manifold method together with the techniques of normal form theory allows us to analyze the stability of bifurcating invariant manifolds and also to derive asymptotic formulas for spatially inhomogeneous solutions forming these manifolds.

中文翻译：

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具有两个空间变量的 Cahn-Hilliard 方程。图案形成

摘要

我们考虑 Cahn-Hilliard 方程的解取决于两个空间变量，具有齐次 Dirichlet 和 Neumann 边界条件，以及周期性边界条件。对于这三个边值问题，我们研究了空间均匀平衡状态改变稳定性时出现的局部分叉问题。我们表明，导致空间非均匀解的分岔的性质与边界条件的选择密切相关。在均匀狄利克雷边界条件的情况下，空间非均匀平衡状态出现在均匀平衡状态的邻域中，这取决于两个空间变量。在分析诺依曼问题和周期边值问题时实现了另一种情况。在这些中，由于分叉，由空间非均匀解形成的不变流形出现。这些流形的维数从1到3不等。在分析三个边值问题时，我们使用了无限维动力系统理论和渐近方法。将积分流形方法与范式理论的技术结合使用使我们能够分析分岔不变流形的稳定性，并推导出形成这些流形的空间非齐次解的渐近公式。