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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

Risei Kano 1,

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

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), 1189-1192. 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), 1279-1284. Google Scholar

[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.  Google Scholar

[4]

J. L. Joly and U. Mosco, Sur les inéquations quasi-variationnelles, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 449-502. 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), 107-137. 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 "Recent Advances in Nonlinear Analysis," World Sci. Publ., Hackensack, NJ, (2008), 149-169. 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, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 175-194. 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), 365-388. Google Scholar

[9]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-86. Google Scholar

[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.  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), 897-908. 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), 204-228. 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), 819-844. doi: 10.1007/s00285-008-0220-0.  Google Scholar

show all references

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), 1189-1192. 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), 1279-1284. Google Scholar

[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.  Google Scholar

[4]

J. L. Joly and U. Mosco, Sur les inéquations quasi-variationnelles, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 449-502. 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), 107-137. 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 "Recent Advances in Nonlinear Analysis," World Sci. Publ., Hackensack, NJ, (2008), 149-169. 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, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 175-194. 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), 365-388. Google Scholar

[9]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-86. Google Scholar

[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.  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), 897-908. 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), 204-228. 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), 819-844. doi: 10.1007/s00285-008-0220-0.  Google Scholar

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