American Institute of Mathematical Sciences

Advanced
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

[1]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523
[2]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[3]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1791-1801. doi: 10.3934/dcdss.2020105
[4]

Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447
[5]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[6]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[7]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[8]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423
[9]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363
[10]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[11]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control & Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365
[12]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[13]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545
[14]

Akisato Kubo, Hiroki Hoshino, Katsutaka Kimura. Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model. Conference Publications, 2015, 2015 (special) : 733-744. doi: 10.3934/proc.2015.0733
[15]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895
[16]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015
[17]

Sophie Guillaume. Evolution equations governed by the subdifferential of a convex composite function in finite dimensional spaces. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 23-52. doi: 10.3934/dcds.1996.2.23
[18]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360
[19]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[20]

Kentarou Fujie, Akio Ito, Michael Winkler, Tomomi Yokota. Stabilization in a chemotaxis model for tumor invasion. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 151-169. doi: 10.3934/dcds.2016.36.151

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences