On the uniqueness of linear convection--diffusion equations with integral boundary conditions

TL;DR

This paper investigates how the size of the domain influences the existence and uniqueness of solutions to linear convection--diffusion equations with non-local integral boundary conditions, providing conditions for uniqueness and examples of non-uniqueness.

Contribution

It offers a new approach transforming integral boundary conditions into Dirichlet conditions to analyze solution uniqueness based on domain size.

Findings

Uniqueness holds for sufficiently large or small domains.

Existence of multiple or no solutions depends on boundary data and domain size.

A method to convert integral boundary conditions into Dirichlet conditions is proposed.

Abstract

This work contributes to an understanding of the domain size's effect on the existence and uniqueness of the linear convection--diffusion equation with integral-type boundary conditions, where boundary conditions depend non-locally on unknown solutions. Generally, the uniqueness result of this type of equation is unclear. In this preliminary study, a uniqueness result is verified when the domain is sufficiently large or small. The main approach has an advantage of transforming the integral boundary conditions into new Dirichlet boundary conditions so that we can obtain refined estimates, and the comparison theorem can be applied to the equations. Furthermore, we show a domain such that under different boundary data, the equation in this domain can have infinitely numerous solutions or no solution.