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

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February 2014, 7(1): 63-74. doi: 10.3934/dcdss.2014.7.63

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

1.	Department of Mathematics, Faculty of Education, Kochi University, Akebonotyo 2-5-1, Kochi, Kochi, 780-8019, Japan

Received March 2012 Revised May 2012 Published July 2013

Abstract
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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.

Keywords: evolution equation, heat shock protein., Quasi-variational inequality, subdifferential, tumer invasion model.

Mathematics Subject Classification: Primary: 35K45; Secondary: 35K5.

Citation: Risei Kano. The existence of solutions for tumor invasion models with time and space dependent diffusion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 63-74. doi: 10.3934/dcdss.2014.7.63

References:

[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. Google Scholar
[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. Google Scholar
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[4]	J. L. Joly and U. Mosco, Sur les inéquations quasi-variationnelles,, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 449. Google Scholar
[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. doi: 10.1016/0022-1236(79)90028-4. Google Scholar
[6]	R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces,, in, (2008), 149. doi: 10.1142/9789812709257_0010. Google Scholar
[7]	R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints,, Banach Center Publ., 86 (2009), 175. doi: 10.4064/bc86-0-11. Google Scholar
[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. Google Scholar
[9]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. Google Scholar
[10]	N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, in, (2007), 203. doi: 10.1016/S1874-5733(07)80007-6. Google Scholar
[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. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. Google Scholar
[12]	U. Stefanelli, Nonlocal quasivariational evolution problems,, J. Differential Equations, 229 (2006), 204. doi: 10.1016/j.jde.2006.05.004. Google Scholar
[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. doi: 10.1007/s00285-008-0220-0. Google Scholar

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References:

[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. Google Scholar
[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. Google Scholar
[3]	M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion,, in, (2003), 269. Google Scholar
[4]	J. L. Joly and U. Mosco, Sur les inéquations quasi-variationnelles,, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 449. Google Scholar
[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. doi: 10.1016/0022-1236(79)90028-4. Google Scholar
[6]	R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces,, in, (2008), 149. doi: 10.1142/9789812709257_0010. Google Scholar
[7]	R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints,, Banach Center Publ., 86 (2009), 175. doi: 10.4064/bc86-0-11. Google Scholar
[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. Google Scholar
[9]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. Google Scholar
[10]	N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, in, (2007), 203. doi: 10.1016/S1874-5733(07)80007-6. Google Scholar
[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. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. Google Scholar
[12]	U. Stefanelli, Nonlocal quasivariational evolution problems,, J. Differential Equations, 229 (2006), 204. doi: 10.1016/j.jde.2006.05.004. Google Scholar
[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. doi: 10.1007/s00285-008-0220-0. Google Scholar

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