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.

这项工作研究了一个非局部sinh-Gordon方程，其中球中的参数具有奇异摄动。在Robin边界条件下，解渐近地在一个薄的环形区域中形成一个非常陡峭的边界层，并迅速在该区域之外变为平坦曲线。更特别地集中在该区域中的薄环形层的结构上，涉及域尺寸的逐点渐进扩展被更清晰地评估，其中域尺寸恰好出现在渐进扩展的第二项中。应该强调的是，由于模型具有取决于未知解的非局部系数，因此匹配渐近展开的标准参数受到限制。一种新方法依赖于在渐近框架中整合基于Dirichlet到Neumann图的思想。薄层结构的严格渐近展开也与数值结果非常吻合。此外，精确地证明了薄环形层的各种边界集中现象。