---

Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 473--485 | DOI:10.5890/DNC.2022.09.009

V.N. Deiva Mani, S. Marshal Anthoni

Department of Mathematics, Anna University Regional Campus Coimbatore, India, 641 046

Download Full Text PDF

 

Abstract

This work deals with the existence of the system consists of coupling dynamics of the two types of tumor cells among the density of epithelial cells (ECs) and the mesenchymal cells (MCs) with the proteins matrix metalloproteinases (MMPs) and extra cellular matrix (ECM) which involved in the invasion and the intravasation processes. Along with square integrable mesenchymal epithelial transition function, the existence and uniqueness of mathematical model illustrated under Faedo-Galerkin approximation method which governed by the invasion model along with EMT and MET process which contains nonlinear terms due to acidification and interactions.

References

1. [1]	Shangerganesh, L., Deiva Mani, V.N., and Karthikeyan, S. (2020), Existence of solutions of cancer invasion parabolic system with integrable data, Afrika Matematika, 31, 1359-1378.

2. [2]	Shangerganesh, L., Nyamoradi, N., Deiva Mani, V.N., and Karthikeyan, S. (2018), On the existence of weak solutions of nonlinear degenerate parabolic system with variable exponents, Computers and Mathematics with Applications, 75, 322-334.

3. [3]	Shangerganesh, L., Deiva Mani, V.N., and Karthikeyan, S. (2017), Existence of global weak solutions for cancer invasion system with nonlinear diffusion, Communications in Applied Analysis, 21, 607-629.

4. [4]	Shangerganesh, L., Deiva Mani, V.N., and Karthikeyan, S. (2017), Renormalized and entropy solutions of tumor growth model with nonlinear acid production, Mathematical Modelling and Analysis, 22, 695-716.

5. [5]	Tao, Y. and Winkler, M. (2020), A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy, Nonlinear Analysis, 198, 111870.

6. [6]	Tao, Y. and Winkler, M. (2020), Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy, Journal of Differential Equations, 268, 4973-4997.

7. [7]	Tao, Y. (2009), A free boundary problem modeling the cell cycle and cell movement in multi cellular tumor spheroids, Journal of Differential Equations, 247, 49-68.

8. [8]	Baleanu, D., Jajarmi, A., Sajjadi, S.S., and Mozyrska, D. (2019), A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29, 083127.

9. [9]	Qureshi, S. and Aziz, S. (2020), Fractional modeling for a chemical kinetic reaction in a batch reactor via nonlocal operator with power law kernel, Physica A, 542, 123494.

10. [10]	Qureshi, S., Yusuf, A., Shaikh, A.A., Inc, M., and Baleanu, D. (2019), Fractional modeling of blood ethanol concentration system with real data application, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29, 013143.

11. [11]	Qureshi, S. and Atangana, A. (2020), Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data, Chaos, Solitons and Fractals, 136, 109812.

12. [12]	Sfakianakis, N., Madzvamuse, A., and Chaplain, M.A.J. (2020), A hybrid multiscale model for cancer invasion of the extracellular matrix, Multiscale Modeling and Simulation, 18, 824-850.

13. [13]	Blanchard, D. and Murat, F. (1997), Renormalized solutions of nonlinear parabolic problems with $L_1$ data, existence and uniqueness, Proceedings of the Royal Society of Edinburgh Section A, 127, 1137-1152.

14. [14]	Blanchard, D., Murat, F., and Redwane, H. (2001), Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, Journal of Differential Equations, 177, 331-374.

15. [15]	Boccardo, L., Porzio, M.M., and Primo, A. (2009), Summability and existence results for nonlinear parabolic equations, Nonlinear Analysis, 71, 978-990.

16. [16]	Meral, G., Stinner, C., and Surulescu, C. (2015), A multiscale model for acid-mediated tumor invasion: therapy approaches, Journal of Coupled Systems and Multiscale Dynamics, 3, 135-142.

17. [17]	Li, C., Kaushik, A., and Yin, G. (2014), Global existence of classical solutions to an acid-mediated invasion model for tumor-stromal interactions, Applied Mathematics and Computation, 234, 599-605.

18. [18]	Tao, Y. and Winkler, M. (2011), A chemotaxis-hatotaxis model: the roles of nonlinear diffusion and logistic source, SIAM Journal on Mathematical Analysis, 43, 685-704.

19. [19]	Xu, X. (2015), Existence theorems for the quantum drift-diffusion equations with mixed boundary conditions, Communications in Contemporary Mathematics, 1550048(21 pages).

20. [20]	Bendahmane, M. and Karlsen, K.H. (2006), Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Networks and Heterogeneous Media, 1, 185-218.

21. [21]	Bendahmane, M. (2010), Weak and classical solutions to predator-prey system with cross-diffusion, Nonlinear Analysis, 73, 2489-2503.