TY - JOUR
T1 - On the evolutionary dynamics of the cahn-hilliard equation with cut-off mass source
AU - Lee, Chaeyoung
AU - Kim, Hyundong
AU - Yoon, Sungha
AU - Park, Jintae
AU - Kim, Sangkwon
AU - Yang, Junxiang
AU - Kim, Junseok
N1 - Funding Information:
The first author (C. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A13094308). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053). The authors thank the editor and the reviewers for their constructive and helpful comments on the revision of this article.
Publisher Copyright:
© 2021 Global-Science Press.
PY - 2020/10
Y1 - 2020/10
N2 - We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cut-off logistic source term and have good results.
AB - We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cut-off logistic source term and have good results.
KW - Cahn-Hilliard equation
KW - Finite difference method
KW - Logistic source
KW - Tumor growth application
UR - http://www.scopus.com/inward/record.url?scp=85094646937&partnerID=8YFLogxK
U2 - 10.4208/NMTMA.OA-2020-0051
DO - 10.4208/NMTMA.OA-2020-0051
M3 - Article
AN - SCOPUS:85094646937
SN - 1004-8979
VL - 14
SP - 242
EP - 260
JO - Numerical Mathematics
JF - Numerical Mathematics
IS - 1
ER -