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This work contributes to an understanding of the domain size's effect on theexistence and uniqueness of the linear convection--diffusion equation withintegral-type boundary conditions, where boundary conditions depend non-locallyon unknown solutions. Generally, the uniqueness result of this type of equationis unclear. In this preliminary study, a uniqueness result is verified when thedomain is sufficiently large or small. The main approach has an advantage oftransforming the integral boundary conditions into new Dirichlet boundaryconditions so that we can obtain refined estimates, and the comparison theoremcan be applied to the equations. Furthermore, we show a domain such that underdifferent boundary data, the equation in this domain can have infinitelynumerous solutions or no solution.

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