Department: Institute of Computational and Modeling Science            PI's Name: Chiun-Chang Lee

Journal: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Title: Uniqueness and asymptotics of singularly perturbed equations involving implicit boundary conditions

Abstract: This works investigates a class of one-dimensional convection–diffusion equations with a singularly perturbed parameter :

where the boundary condition

is a nonlocal type depending on the properties of the unknown solution   in the subdomain  of  . These conditions describe how the solution of the equation is influenced at the boundary and interior behaviors through an implicit interaction. Such nonlocal equations usually lack a variational structure. In recent years, the related investigations on these equations mostly consists of numerical computation results. Although these numerical methods have been continuously improved over the years, they are still only useful for some special forms of nonlocal equations and cannot be generalized. The standard elliptic theory, such as variational methods and the maximum principle, cannot be directly applied to the study of these nonlocal equations. The highlight of this work is mainly based on rigorous singular perturbation analysis and combined with the fixed point theory to prove the existence and uniqueness of the global boundedness solution for such nonlocal equations as   is sufficiently small. We further analyze the effect of nonlocal boundary conditions on the solution and obtain the global asymptotic behavior of the solution. Such arguments are based upon establishing a mapping corresponding to the boundary conditions, proving the existence of fixed points through refined asymptotic estimates (depending on ε), while also obtaining the asymptotic behavior of the solution.