Geodesic
Equations aux dérivées partielles
On the uniqueness of linear convection–diffusion equations with integral boundary conditions
Lee, Chiun-Chang1  ; Mizuno, Masashi2 ; Moon, Sang-Hyuck3
1 Institute for Computational and Modeling Science, National Tsing Hua University, Hsinchu 30013, Taiwan
2 Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-Ku, Tokyo, 101-8308, Japan
3 Department of Mathematical Sciences, College of Natural Sciences, Ulsan National Institute of Science and Technology, Republic of Korea
Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 191-206

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We investigate a class of convection–diffusion equations in an expanding domain involving a parameter, where we consider integral boundary conditions that depend non-locally on unknown solutions. Generally, the uniqueness result of this type of equation is unclear. In this work, we obtain a uniqueness result when the domain is sufficiently large or small. This approach has the 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. This work may contribute to the first understanding of the domain size’s effect on the existence and uniqueness of the linear convection–diffusion equation with integral-type boundary conditions.

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DOI : 10.5802/crmath.396
Classification : 34B10, 34D15, 34E05, 34K26, 35J25

Lee, Chiun-Chang 1 ; Mizuno, Masashi 2 ; Moon, Sang-Hyuck 3

1 Institute for Computational and Modeling Science, National Tsing Hua University, Hsinchu 30013, Taiwan
2 Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-Ku, Tokyo, 101-8308, Japan
3 Department of Mathematical Sciences, College of Natural Sciences, Ulsan National Institute of Science and Technology, Republic of Korea
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On the uniqueness of linear convection{\textendash}diffusion equations with integral boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {191--206},
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Lee, Chiun-Chang; Mizuno, Masashi; Moon, Sang-Hyuck. On the uniqueness of linear convection–diffusion equations with integral boundary conditions. Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 191-206. doi: 10.5802/crmath.396

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