Uniqueness and asymptotics of singularly perturbed equations involving implicit boundary conditions

A class of one-dimensional convection–diffusion equations with a singularly perturbed parameter in a bounded domain is presented, where the boundary condition is nonlocal type with an implicit form related to the unknown solutions. In general, the validity of the maximum principle of this type equation is unassurable. Employing a singular perturbations method as a natural tool, we establish the uniqueness and maximum principle as the singularly perturbed parameter is sufficiently small. Such an argument is different from the standard fixed point approaches. Moreover, as this parameter tends to zero, the boundary and interior asymptotics of solutions is obtained.