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December  2020, 13(6): 1175-1191. doi: 10.3934/krm.2020042

Global existence theorem for a model governing the motion of two cell populations

Brock C. Price and  Xiangsheng Xu

Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, USA

Received  April 2020 Revised  July 2020 Published  September 2020

This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is $ 0 $. The existence assertion is established by exploring the fact that the total population density satisfies a porous media equation.

Keywords: Cross-diffusion systems, reaction-diffusion, tissue growth models.
Mathematics Subject Classification: Primary: 35B45, 35K57, 35K55, 35K65, 35Q92; Secondary: 76N10.
Citation: Brock C. Price, Xiangsheng Xu. Global existence theorem for a model governing the motion of two cell populations. Kinetic & Related Models, 2020, 13 (6) : 1175-1191. doi: 10.3934/krm.2020042
References:
[1]

M. Bertsch, M. E. Gurtin and D. Hilhorst, On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities, Nonlinear Anal., 11 (1987), 493-499. doi: 10.1016/0362-546X(87)90067-8.  Google Scholar

[2]

M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biology, 23 (1985), 1-13.  doi: 10.1007/BF00276555.  Google Scholar

[3]

F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen, Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues, Arch. Rational Mech. Anal., 236 (2020), 735-766.  doi: 10.1007/s00205-019-01479-1.  Google Scholar

[4]

H. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24 (1996), 1-17.  doi: 10.1016/S0895-7177(96)00174-4.  Google Scholar

[5]

H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison, Journal of mathematical biology, 58 (2009), 657-687.  doi: 10.1007/s00285-008-0212-0.  Google Scholar

[6]

J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes & segregation in reaction-(cross-)diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695-5718.  doi: 10.1137/17M1158379.  Google Scholar

[7]

X. Chen, E. S. Daus and A. Jüngel, Global existence analysis of cross-diffusion population systems for multiple species, Arch. Ration. Mech. Anal., 227 (2018), 715-747.  doi: 10.1007/s00205-017-1172-6.  Google Scholar

[8]

X. Chen and A. Jüngel, When do cross-diffusion systems have an entropy structure? arXiv: 1908.06873, [math.AP], 2019. Google Scholar

[9]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[10]

L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS #74, American Mathematical Society, 1990. Third printing, 2002. doi: 10.1090/cbms/074.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

M. E. Gurtin and A. C. Pipkin, A note on interacting populations that disperse to avoid crowding, Quarterly Appl. Math., 42 (1984), 87-94.  doi: 10.1090/qam/736508.  Google Scholar

[13]

P. Gwiazda, B. Perthame and A. Świerczewska-Gwiazdak, A two species hyperbolic-parabolic model of tissue growth, Comm. Partial Differential Equations, 44 (2019), 1605-1618.  doi: 10.1080/03605302.2019.1650064.  Google Scholar

[14]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.  Google Scholar

[15]

Q. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Tran. Math. Monographs, Vol. 23, AMS, Providence, RI, 1968.  Google Scholar

[16]

T. Lorenzi, A. Lorz and B. Perthame, On interfaces between cell populations with different mobilities, Kinetic and Related Models, 10 (2017), 299-311.  doi: 10.3934/krm.2017012.  Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

show all references

References:
[1]

M. Bertsch, M. E. Gurtin and D. Hilhorst, On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities, Nonlinear Anal., 11 (1987), 493-499. doi: 10.1016/0362-546X(87)90067-8.  Google Scholar

[2]

M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biology, 23 (1985), 1-13.  doi: 10.1007/BF00276555.  Google Scholar

[3]

F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen, Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues, Arch. Rational Mech. Anal., 236 (2020), 735-766.  doi: 10.1007/s00205-019-01479-1.  Google Scholar

[4]

H. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24 (1996), 1-17.  doi: 10.1016/S0895-7177(96)00174-4.  Google Scholar

[5]

H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison, Journal of mathematical biology, 58 (2009), 657-687.  doi: 10.1007/s00285-008-0212-0.  Google Scholar

[6]

J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes & segregation in reaction-(cross-)diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695-5718.  doi: 10.1137/17M1158379.  Google Scholar

[7]

X. Chen, E. S. Daus and A. Jüngel, Global existence analysis of cross-diffusion population systems for multiple species, Arch. Ration. Mech. Anal., 227 (2018), 715-747.  doi: 10.1007/s00205-017-1172-6.  Google Scholar

[8]

X. Chen and A. Jüngel, When do cross-diffusion systems have an entropy structure? arXiv: 1908.06873, [math.AP], 2019. Google Scholar

[9]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[10]

L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS #74, American Mathematical Society, 1990. Third printing, 2002. doi: 10.1090/cbms/074.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

M. E. Gurtin and A. C. Pipkin, A note on interacting populations that disperse to avoid crowding, Quarterly Appl. Math., 42 (1984), 87-94.  doi: 10.1090/qam/736508.  Google Scholar

[13]

P. Gwiazda, B. Perthame and A. Świerczewska-Gwiazdak, A two species hyperbolic-parabolic model of tissue growth, Comm. Partial Differential Equations, 44 (2019), 1605-1618.  doi: 10.1080/03605302.2019.1650064.  Google Scholar

[14]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.  Google Scholar

[15]

Q. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Tran. Math. Monographs, Vol. 23, AMS, Providence, RI, 1968.  Google Scholar

[16]

T. Lorenzi, A. Lorz and B. Perthame, On interfaces between cell populations with different mobilities, Kinetic and Related Models, 10 (2017), 299-311.  doi: 10.3934/krm.2017012.  Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

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