Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis
In this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. The special case where n=3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into con...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2020-01-01
|
| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2020/9683982 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849690736674471936 |
|---|---|
| author | Shihe Xu Fangwei Zhang |
| author_facet | Shihe Xu Fangwei Zhang |
| author_sort | Shihe Xu |
| collection | DOAJ |
| description | In this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. The special case where n=3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into consideration. Tumor cell division takes a certain length of time, thus we assume that the proliferation process leg behind as compared to the process of apoptosis. The angiogenesis is reflected as the time dependent Robin boundary condition in the model. Global existence and uniqueness of the nonnegative solution of the problem is proved. When c>0 is sufficiently small, the stability of the steady state solution is studied, where c is the ratio of the time scale of diffusion to the tumor doubling time scale. Under some conditions, the results show that the magnitude of the delay does not affect the final dynamic behavior of the solutions. An application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given. |
| format | Article |
| id | doaj-art-eea4237136454e85bb92acc8d0b3b00a |
| institution | DOAJ |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-eea4237136454e85bb92acc8d0b3b00a2025-08-20T03:21:13ZengWileyComplexity1076-27871099-05262020-01-01202010.1155/2020/96839829683982Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of AngiogenesisShihe Xu0Fangwei Zhang1School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, ChinaSchool of Civil and Environmental Engineering, Ningbo University, Ningbo 315211, ChinaIn this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. The special case where n=3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into consideration. Tumor cell division takes a certain length of time, thus we assume that the proliferation process leg behind as compared to the process of apoptosis. The angiogenesis is reflected as the time dependent Robin boundary condition in the model. Global existence and uniqueness of the nonnegative solution of the problem is proved. When c>0 is sufficiently small, the stability of the steady state solution is studied, where c is the ratio of the time scale of diffusion to the tumor doubling time scale. Under some conditions, the results show that the magnitude of the delay does not affect the final dynamic behavior of the solutions. An application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given.http://dx.doi.org/10.1155/2020/9683982 |
| spellingShingle | Shihe Xu Fangwei Zhang Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis Complexity |
| title | Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis |
| title_full | Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis |
| title_fullStr | Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis |
| title_full_unstemmed | Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis |
| title_short | Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis |
| title_sort | analysis of a delayed free boundary problem with application to a model for tumor growth of angiogenesis |
| url | http://dx.doi.org/10.1155/2020/9683982 |
| work_keys_str_mv | AT shihexu analysisofadelayedfreeboundaryproblemwithapplicationtoamodelfortumorgrowthofangiogenesis AT fangweizhang analysisofadelayedfreeboundaryproblemwithapplicationtoamodelfortumorgrowthofangiogenesis |