Comptes Rendus
Volume articles
Partial differential equations
On the uniqueness of linear convection–diffusion equations with integral boundary conditions
Chiun-Chang Lee1 ; Masashi Mizuno2; Sang-Hyuck Moon3
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, Volume 361 (2023), pp. 191-206.

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.

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

Chiun-Chang Lee 1; Masashi Mizuno 2; Sang-Hyuck Moon 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{CRMATH_2023__361_G1_191_0,
     author = {Chiun-Chang Lee and Masashi Mizuno and Sang-Hyuck Moon},
     title = {On the uniqueness of linear convection{\textendash}diffusion equations with integral boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {191--206},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.396},
     language = {en},
}
TY  - JOUR
AU  - Chiun-Chang Lee
AU  - Masashi Mizuno
AU  - Sang-Hyuck Moon
TI  - On the uniqueness of linear convection–diffusion equations with integral boundary conditions
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 191
EP  - 206
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.396
LA  - en
ID  - CRMATH_2023__361_G1_191_0
ER  - 
%0 Journal Article
%A Chiun-Chang Lee
%A Masashi Mizuno
%A Sang-Hyuck Moon
%T On the uniqueness of linear convection–diffusion equations with integral boundary conditions
%J Comptes Rendus. Mathématique
%D 2023
%P 191-206
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.396
%G en
%F CRMATH_2023__361_G1_191_0
Chiun-Chang Lee; Masashi Mizuno; Sang-Hyuck Moon. On the uniqueness of linear convection–diffusion equations with integral boundary conditions. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 191-206. doi : 10.5802/crmath.396. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.396/

[1] Martin Berdau; Georgii G. Yelenin; Andrzej Karpowicz; Mohammed Ehsasi; Klaus Christmann; Jochen H. Block Macroscopic and mesoscopic characterization of a bistable reaction system: CO oxidation on Pt(111) surface, J. Chem. Phys., Volume 110 (1999) no. 23, pp. 11551-11573 | DOI

[2] Leonid V. Berlyand; Petru Mironescu Ginzburg–Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices, J. Funct. Anal., Volume 239 (2006) no. 1, pp. 76-99 | DOI | MR | Zbl

[3] A. Boucherif Second-order boundary value problems with integral boundary conditions, Nonlinear Analysis, Volume 70 (2009), pp. 364-371 | DOI | MR | Zbl

[4] Musa Cakir; G. M. Amiraliyev A finite difference method for the singularly perturbed problem with non-local boundary condition, Appl. Math. Comput., Volume 160 (2005), pp. 539-549 | DOI | Zbl

[5] Musa Cakir; Derya Arslan A new numerical approach for a singularly perturbed problem with two integral boundary conditions, Comput. Appl. Math., Volume 40 (2021) no. 6, 189, 17 pages | DOI | MR | Zbl

[6] Edward B. Davies The equivalence of certain heat kernel and Green function bounds, J. Funct. Anal., Volume 71 (1987), pp. 88-103 | DOI | MR | Zbl

[7] Lawrence C. Evans; Ronald F. Gariepy Measure theory and fine properties of functions, Textbooks in Mathematics, CRC Press, 2015 | DOI | Zbl

[8] Meiqiang Feng Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions, Appl. Math. Lett., Volume 24 (2011) no. 8, pp. 1419-1427 | DOI | MR | Zbl

[9] Avner Friedman Monotonic decay of solutions of parabolic equations with non-local boundary conditions, Q. Appl. Math., Volume 44 (1986), pp. 401-407 | DOI | Zbl

[10] David Gilbarg; Neil S. Trudinger Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, 2001 | DOI | Zbl

[11] Chiun-Chang Lee Asymptotic analysis of charge conserving Poisson–Boltzmann equations with variable dielectric coefficients, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 6, pp. 3251-3276 | DOI | MR | Zbl

[12] Chiun-Chang Lee Nontrivial boundary structure in a Neumann problem on balls with radii tending to infinity, Ann. Mat. Pura Appl., Volume 199 (2020) no. 3, pp. 1123-1146 | MR | Zbl

[13] Lishan Liu; Xinan Hao; Yonghong Wu Positive solutions for singular second order differential equations with integral boundary conditions, Math. Comput. Modelling, Volume 57 (2013) no. 3-4, pp. 836-847 | DOI | MR | Zbl

[14] Vladimir G. Maz’ya Sobolev Spaces, Springer, 1985 | DOI | Zbl

[15] A. Saadatmandi; M. Dehghan Numerical solution of the one-dimensional wave equation with an integral condition, Numer. Methods Partial Differ. Equations, Volume 23 (2007) no. 2, pp. 282-292 | DOI | MR | Zbl

[16] Tim P. Schulze; M. Grae Worster Weak convection, liquid inclusions and the formation of chimneys in mushy layers, J. Fluid Mech., Volume 388 (1999), pp. 197-215 | DOI | Zbl

[17] René P. Sperb Optimal bounds in semilinear elliptic problems with nonlinear boundary conditions, Z. Angew. Math. Phys., Volume 44 (1993) no. 4, pp. 639-653 | DOI | MR | Zbl

[18] Paweł J. Zuk; Marek Kochańczyk; Joanna Jaruszewicz; Witold Bednorz; Tomasz Lipniacki Dynamics of a stochastic spatially extended system predicted by comparing deterministic and stochastic attractors of the corresponding birth–death process, Phys. Biol., Volume 9 (2012) no. 5, 055002, 12 pages | DOI

Cited by Sources:

Comments - Policy