# Domain-size effects on boundary layers of a nonlocal sinh-Gordon equation

## TL;DR

This paper analyzes a nonlocal sinh-Gordon equation with a small parameter, revealing how the boundary layer structure depends on domain size, using novel asymptotic methods and confirming results with numerical simulations.

## Contribution

It introduces a new asymptotic approach incorporating domain size effects for nonlocal boundary layer analysis, overcoming limitations of standard matching techniques.

## Key findings

- Boundary layer forms a steep annular region near the boundary
- Domain size explicitly appears in the second term of the asymptotic expansion
- Numerical results confirm the accuracy of the asymptotic analysis

## Abstract

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.