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

This work investigates a nonlocal sinh-Gordon equation with singularly perturbed parameter in ball. Under the Robin boundary condition, solution asymptotically forms quite steep layer thin annular region, and rapidly becomes flat curve outside this region. {Focusing more particularly on structure of pointwise asymptotic expansion involving domain-size is evaluated sharply, where exactly appears second term expansion.} It should be stressed that standard argument matching expansions limited because model has coefficient depending unknown~solution. A new approach relies integrating ideas based Dirichlet-to-Neumann map an framework. The rigorous for also matches well numerical results. Furthermore, various concentration phenomena are precisely demonstrated.