International Journal of Partial Differential Equations and Applications
ISSN (Print): 2376-9548 ISSN (Online): 2376-9556 Website: Editor-in-chief: Mahammad Nurmammadov
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International Journal of Partial Differential Equations and Applications. 2020, 7(1), 1-7
DOI: 10.12691/ijpdea-7-1-1
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Existence and Uniqueness of a Chemotaxis System Influenced by Cancer Cells

Gang Li1, Hui Min Hu1, Xi Chen1 and Fei Da Jiang1,

1College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Pub. Date: March 04, 2020

Cite this paper:
Gang Li, Hui Min Hu, Xi Chen and Fei Da Jiang. Existence and Uniqueness of a Chemotaxis System Influenced by Cancer Cells. International Journal of Partial Differential Equations and Applications. 2020; 7(1):1-7. doi: 10.12691/ijpdea-7-1-1


We present a mathematical analysis of a reaction-diffusion model in a bounded open domain which describes vascular endothelial growth factor(VEGF), endothelial cells and oxygen. We use the parabolic theory to prove the existence of the solution in the function space under the homogeneous Neumann conditions. Then we get the existence of nonnegative solution in by using the global Schauder estimation.

existence and uniqueness parabolic system chemotaxis system influenced by cancer

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[1]  A.R.A. Anderson, M.A.J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60(1998), 857-899.
[2]  N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl., 25(2015), 1663-1763.
[3]  M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Models Methods Appl., 18(2005), 1685-1734.
[4]  M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Net. Hetero. Med., 1(2006), 399-439.
[5]  Y. Li. K. Lin, C. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxis haptotaxis model in high dimensions, Appl. Math. Lett., 50(2015), 91-97.
[6]  C. Stinner, C. Surulescu, G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80(2015), 1300-1321.
[7]  C. Stinner, C. Surulescu, A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Models Methods Appl., 26(2016), 2163-2201.
[8]  A.R.A. Anderson, M.A.J. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett., 11(1998), 109-116.
[9]  M.A.J. Chaplain, A.M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10(1993), 149-168.
[10]  H.A. Levine, B.D. Sleeman, M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42(2001), 195-238.
[11]  N. Paweletz, M. Knierim, Tumor related angiogenesis, Crit. Rev. Oncal. Hematol., 9(1989), 197-242.
[12]  B.D. Sleeman, Mathematical modelling of tumor growth and angiogenesis, Adv. in Exp. Med. Bio., 428(1997), 671-677.
[13]  A. Friedman, Mathematical biology. Modeling and analysis. CBMS Regional Conference Series in Mathematics, 127. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2018.
[14]  E. Keller, L. Segel, Model for chemotaxis, J. Theor. Biol., 30(1970), 225-234.
[15]  Y.Z. Chen, Second order parabolic partial differential equation(In Chinese), Peking University Press, 2003.
[16]  R. Adams, Sobolev spaces. New York: Academic press, 1975.
[17]  O. Ladyzhenskaky, V. Solonnikov, N. Uraltseva, Linear and quasilinear equations of parabolic type, Amer. Math., 1968.
[18]  A. C. Fassoni, Mathematical modeling in cancer addressing the early stage and treatment of avascular tumors, PhD thesis, University of Campinas, 2016.
[19]  A. Friedman, Partial differential equations of parabolic type, Prentice-Hall. Inc., Englewood Cliffs, N.J., 1964.