非局部sinh-Gordon方程边界层的区域尺寸效应
Domain-size effects on boundary layers of a nonlocal sinh-Gordon equation
摘要：
本文研究了球中带奇摄动参数的非局部sinh-Gordon方程。在Robin边界条件下，解在薄的环形区域内渐近地形成一个相当陡的边界层，而在该区域外则迅速地变成一条平坦的曲线。{更具体地集中在该区域中薄环形层的结构上，更精确地评估涉及域大小的逐点渐近展开，其中域大小正好出现在渐近展开的第二项中。}应该强调的是，匹配渐近展开的标准参数是有限的，因为该模型具有依赖于未知解的非局部系数。一种新的方法依赖于集成思想的基础上的狄利克雷到诺依曼映射的渐近框架。薄层结构的严格渐近展开也与数值结果吻合良好。此外，薄环形层的各种边界浓度现象被精确地显示。
英文摘要：
This work investigates a nonlocal sinh-Gordon equation with a singularly perturbed parameter in a ball. Under the Robin boundary condition, the solution asymptotically forms a quite steep boundary layer in a thin annular region, and rapidly becomes a flat curve outside this region. {Focusing more particularly on the structure of the thin annular layer in this region, the pointwise asymptotic expansion involving the domain-size is evaluated more sharply, where the domain-size exactly appears in the second term of the asymptotic expansion.} It should be stressed that the standard argument of matching asymptotic expansions is limited because the model has a nonlocal coefficient depending on the unknown~solution. A new approach relies on integrating ideas based on a Dirichlet-to-Neumann map in an asymptotic framework. The rigorous asymptotic expansions for the thin layer structure also matches well with the numerical results. Furthermore, various boundary concentration phenomena of the thin annular layer are precisely demonstrated.
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