Article Contents
Article Contents

# The existence of solutions for tumor invasion models with time and space dependent diffusion

• We shall show the existence of a solution for a nonlinear parabolic system. This system is a tumor invasion model which has the time and space dependent diffusion coefficient. In this paper, we apply an existence result for Quasi-Variational Inequalities. Quasi-Variational Inequality is a problem to find a function which satisfies a variational inequality in which the constraint depends upon the unknown function. In this paper, I shall show how to approach to our tumor invasion model by Quasi-Variational inequality, and obtain a solution for it.
Mathematics Subject Classification: Primary: 35K45; Secondary: 35K50.

 Citation:

•  [1] A. Bensoussan and J.-L. Lions, Nouvelle formulation de problémes de contrôle impulsionnel et applications, C. R. Acad. Sci. Paris Sér. A, 276 (1973), 1189-1192. [2] A. Bensoussan, M. Goursat and J.-L.Lions, Contrôle impulsionnel et inéquations quasivariationnelles stationnaires, C. R. Acad. Sci. Paris Sér. A, 276 (1973), 1279-1284. [3] M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion, in "Cancer Modelling and Simulation," Chapman & Hall/CRC Math. Biol. Med. Ser., Chapman & Hall/CRC, Boca Raton, FL, (2003), 269-297. [4] J. L. Joly and U. Mosco, Sur les inéquations quasi-variationnelles, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 449-502. [5] J. L. Joly and U. Mosco, A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Funct. Anal., 34 (1979), 107-137.doi: 10.1016/0022-1236(79)90028-4. [6] R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces, in "Recent Advances in Nonlinear Analysis," World Sci. Publ., Hackensack, NJ, (2008), 149-169.doi: 10.1142/9789812709257_0010. [7] R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 175-194.doi: 10.4064/bc86-0-11. [8] R. Kano, A. Ito, K. Yamamoto and H. Nakayama, Quasi-variational inequality approach to tumor invasion models with constraints, GAKUTO International Series Mathematical Sciences and Applications, 32 (2010), 365-388. [9] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-86. [10] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in "Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. IV," Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2007), 203-298.doi: 10.1016/S1874-5733(07)80007-6. [11] M. Kunze and J. F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications, Math. Methods Appl. Sci., 23 (2000), 897-908.doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. [12] U. Stefanelli, Nonlocal quasivariational evolution problems, J. Differential Equations, 229 (2006), 204-228.doi: 10.1016/j.jde.2006.05.004. [13] Z. Szymańska, J. Urbański and A. Marciniak-Czochra, Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue, J. Math. Biol., 58 (2009), 819-844.doi: 10.1007/s00285-008-0220-0.