# American Institute of Mathematical Sciences

June  2020, 40(6): 3117-3142. doi: 10.3934/dcds.2019226

## A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation

 1 Dipartimento di Matematica, University of Rome Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy 2 Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, Rome, Italy 3 CNRS, Laboratoire de Mathématique, Analyse Numérique et EDP, Université de Paris-Sud, F-91405 Orsay Cedex, France 4 Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki, 889-2192, Japan 5 Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo, 135-8181, Japan 6 Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan 7 Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyushu, 804-8550, Japan

* Corresponding author: Hirofumi Izuhara

Received  April 2018 Revised  October 2018 Published  June 2019

We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

Citation: Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226
##### References:
 [1] M. Abercrombie, Contact inhibition in tissue culture, In Vitro, 6 (1970), 128-142.  doi: 10.1007/BF02616114. [2] L. Ambrosio, F. Bouchut and C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. Partial Diff. Equ., 29 (2004), 1635-1651.  doi: 10.1081/PDE-200040210. [3] M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplifies tumour growth model of contact inhibition, Interfaces Free Boundaries, 12 (2010), 235-250.  doi: 10.4171/IFB/233. [4] M. Bertsch, D. Hilhorst, H. Izuhara and and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Diff. Eq. Appl., 12 (2010), 235-250.  doi: 10.7153/dea-04-09. [5] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth, European J. Appl. Math., 26 (2015), 297-323.  doi: 10.1017/S0956792515000042. [6] M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves, Netw. Heterog. Media, 8 (2012), 131-147.  doi: 10.3934/nhm.2013.8.131. [7] M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, (in preparation). [8] M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, Standing and traveling waves in a parabolic-hyperbolic system, to appear in Discret. Contin. Dyn. Syst. Ser. A. doi: 10.3934/nhm.2013.8.131. [9] Z. Biró, Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type, Advanced Nonlinear Studies, 2 (2002), 357-371.  doi: 10.1515/ans-2002-0402. [10] F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0. [11] H. Brezis, Analyse Fonctionnelle, Masson, 1983. [12] J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction cross-diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695-5718.  doi: 10.1137/17M1158379. [13] M. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math. Med. Biol., 23 (2006), 197-229.  doi: 10.1093/imammb/dql009. [14] C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, in Handbook of Differential Equations: Evolutionary Differential Equations, (2006), 277–382. doi: 10.1016/S1874-5717(07)80007-7. [15] E. Dibenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118.  doi: 10.1512/iumj.1983.32.32008. [16] J. Goncerzewicz and D. Hilhorst, Large time behavior of a class of solutions of second order conservation laws, Banach Center Publ., 52 (2000), 119-132. [17] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23, Amer. Math. Soc., Providence, R.I. 1968. [18] L. A. Peletier, The porous media equation, in Application of Nonlinear Analysis in the Physical Sciences, Pitman, Boston, 1981,229–241. [19] J. Sherratt, Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 2365-2386.  doi: 10.1098/rspa.2000.0616. [20] J. Sherratt and M. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.  doi: 10.1007/s002850100088.

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##### References:
 [1] M. Abercrombie, Contact inhibition in tissue culture, In Vitro, 6 (1970), 128-142.  doi: 10.1007/BF02616114. [2] L. Ambrosio, F. Bouchut and C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. Partial Diff. Equ., 29 (2004), 1635-1651.  doi: 10.1081/PDE-200040210. [3] M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplifies tumour growth model of contact inhibition, Interfaces Free Boundaries, 12 (2010), 235-250.  doi: 10.4171/IFB/233. [4] M. Bertsch, D. Hilhorst, H. Izuhara and and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Diff. Eq. Appl., 12 (2010), 235-250.  doi: 10.7153/dea-04-09. [5] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth, European J. Appl. Math., 26 (2015), 297-323.  doi: 10.1017/S0956792515000042. [6] M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves, Netw. Heterog. Media, 8 (2012), 131-147.  doi: 10.3934/nhm.2013.8.131. [7] M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, (in preparation). [8] M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, Standing and traveling waves in a parabolic-hyperbolic system, to appear in Discret. Contin. Dyn. Syst. Ser. A. doi: 10.3934/nhm.2013.8.131. [9] Z. Biró, Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type, Advanced Nonlinear Studies, 2 (2002), 357-371.  doi: 10.1515/ans-2002-0402. [10] F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0. [11] H. Brezis, Analyse Fonctionnelle, Masson, 1983. [12] J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction cross-diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695-5718.  doi: 10.1137/17M1158379. [13] M. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math. Med. Biol., 23 (2006), 197-229.  doi: 10.1093/imammb/dql009. [14] C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, in Handbook of Differential Equations: Evolutionary Differential Equations, (2006), 277–382. doi: 10.1016/S1874-5717(07)80007-7. [15] E. Dibenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118.  doi: 10.1512/iumj.1983.32.32008. [16] J. Goncerzewicz and D. Hilhorst, Large time behavior of a class of solutions of second order conservation laws, Banach Center Publ., 52 (2000), 119-132. [17] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23, Amer. Math. Soc., Providence, R.I. 1968. [18] L. A. Peletier, The porous media equation, in Application of Nonlinear Analysis in the Physical Sciences, Pitman, Boston, 1981,229–241. [19] J. Sherratt, Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 2365-2386.  doi: 10.1098/rspa.2000.0616. [20] J. Sherratt and M. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.  doi: 10.1007/s002850100088.
Snapshots of dynamics in (1) with compactly supported initial data. The parameter values are $\alpha = 4$, $\beta = 3$, $\gamma = 1$ and $k = 0.5$. The solid and dashed curves indicate $u$ and $v$, respectively
The relation between the parameter $k$ and the wave velocity $c_k^*$, where the horizontal and vertical axes indicate $k$ and $c_k^*$, respectively. The other parameter values are $\alpha = 4$, $\beta = 3$ and $\gamma = 1$
$(\varphi, \psi)$-phase planes for (40) with $k = 2$ and $\gamma = 1$ : (0) $c = 0$, (i) $c = 0.8$, (ii) $c = 1$, (iii) $c = 1.2$
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