标题： 局部二维图案从空间异质性中出现的全局分岔 显示英文标题

标题： Global bifurcation of localised 2D patterns emerging from spatial heterogeneity

摘要： 我们提出了一种通用方法，以证明在包含紧致空间非均匀性的偏微分方程中，多维完全局域模式的存在性，无论是在局部还是全局振幅上。 虽然由空间非均匀性引起的单维局域模式已经被很好地研究过，但在更高维度中从Turing不稳定性出现的完全局域模式的存在性仍然是模式形成中的一个关键开放问题。 为了展示该方法，我们考虑了二维Swift-Hohenberg方程，其线性分岔参数被一个径向对称的势函数扰动。 在这种情况下，平凡状态在原点的紧邻域内不稳定，并在外部线性稳定。 我们证明了完全局域模式的局部分岔分支的存在性，描述了它们的稳定性和分岔结构，并通过解析全局分岔理论严格地将解延续到大振幅。 值得注意的是，在Swift-Hohenberg方程中，主要分岔分支根据空间非均匀性的宽度在轴对称斑点和非轴对称的“二极”图案之间交替。 显示英文摘要

摘要： We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional localised patterns induced by spatial heterogeneities have been well-studied, proving the existence of fully localised patterns emerging from a Turing instability in higher dimensions remains a key open problem in pattern formation. In order to demonstrate the approach, we consider the two-dimensional Swift-Hohenberg equation, whose linear bifurcation parameter is perturbed by a radially-symmetric potential function. In this case, the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue solutions to large amplitude via analytic global bifurcation theory. Notably, the primary bifurcating branch in the Swift-Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity.