# American Institute of Mathematical Sciences

October  2019, 39(10): 5603-5635. doi: 10.3934/dcds.2019246

## Standing and travelling waves in a parabolic-hyperbolic system

 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 Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki, 889-2192, Japan 4 Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo, 135-8181, Japan 5 Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan 6 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  March 2018 Revised  March 2019 Published  July 2019

We consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product $uv$ vanishes.

Citation: Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246
##### References:
 [1] M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces Free Bound., 12 (2010), 235-250.  doi: 10.4171/ifb/233.  Google Scholar [2] M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl., 4 (2012), 137-157.  doi: 10.7153/dea-04-09.  Google Scholar [3] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Travelling 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.  Google Scholar [4] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic Fisher KPP equation, to appear in Discret. Contin. Dyn. Syst. Ser. A. Google Scholar [5] M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves, Netw. Heterog. Media, 8 (2013), 131-147.  doi: 10.3934/nhm.2013.8.131.  Google Scholar [6] Z. Biró, Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type, Adv. Nonlinear Stud., 2 (2002), 357-371.  doi: 10.1515/ans-2002-0402.  Google Scholar [7] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), ⅳ+190 pp. doi: 10.1090/memo/0285.  Google Scholar [8] 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.  Google Scholar [9] M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Mathematical Medicine and Biology, 23 (2006), 197-229.  doi: 10.1093/imammb/dql009.  Google Scholar [10] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.   Google Scholar [11] S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reactiondiffusion equation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei(9) Mat. Appl., 15 (2004), 271–280.  Google Scholar [12] A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Série Internationale A, 1 (1937), 1-26.   Google Scholar [13] J. D. Murray, Mathematical Biology. I, Springer-Verlag, New York, 2002.  Google Scholar [14] J. D. Murray, Mathematical Biology. II, Springer-Verlag, New York, 2003.  Google Scholar [15] J. A. 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.  Google Scholar [16] J. A. Sherratt and M. A. J. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.  doi: 10.1007/s002850100088.  Google Scholar

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##### References:
 [1] M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces Free Bound., 12 (2010), 235-250.  doi: 10.4171/ifb/233.  Google Scholar [2] M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl., 4 (2012), 137-157.  doi: 10.7153/dea-04-09.  Google Scholar [3] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Travelling 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.  Google Scholar [4] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic Fisher KPP equation, to appear in Discret. Contin. Dyn. Syst. Ser. A. Google Scholar [5] M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves, Netw. Heterog. Media, 8 (2013), 131-147.  doi: 10.3934/nhm.2013.8.131.  Google Scholar [6] Z. Biró, Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type, Adv. Nonlinear Stud., 2 (2002), 357-371.  doi: 10.1515/ans-2002-0402.  Google Scholar [7] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), ⅳ+190 pp. doi: 10.1090/memo/0285.  Google Scholar [8] 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.  Google Scholar [9] M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Mathematical Medicine and Biology, 23 (2006), 197-229.  doi: 10.1093/imammb/dql009.  Google Scholar [10] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.   Google Scholar [11] S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reactiondiffusion equation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei(9) Mat. Appl., 15 (2004), 271–280.  Google Scholar [12] A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Série Internationale A, 1 (1937), 1-26.   Google Scholar [13] J. D. Murray, Mathematical Biology. I, Springer-Verlag, New York, 2002.  Google Scholar [14] J. D. Murray, Mathematical Biology. II, Springer-Verlag, New York, 2003.  Google Scholar [15] J. A. 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.  Google Scholar [16] J. A. Sherratt and M. A. J. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.  doi: 10.1007/s002850100088.  Google Scholar
Dependency of the wave velocity of segregated travelling wave solutions on $\alpha$. The horizontal and the vertical axes are respectively the wave velocity $\overline c$ and the value $\alpha$. The parameter values are $\gamma = 1$ and $k = 2$
Typical profiles of segregated TWs. The parameter values are $\gamma = 1$ and $k = 2$, the same as the ones in Figure 1. The solid and the dashed curves respectively denote $U$ and $V$. (a) $\overline{c} = 0.2238529804$ (b) $\overline{c} = 0$ (c) $\overline{c} = -0.2895584564$
Profiles of overlapping TWs. The solid and the dashed curves respectively denote $U$ and $V$ and the gray curve denotes $U+V$. The parameter values are $\alpha = 1$, $k = 2$ and $\gamma = 1$. For this parameter setting, the velocity of segregated TW is $\overline{c} = 0.4094908611$
Profiles of overlapping TWs. The solid and the dashed curves respectively denote $U$ and $V$ and the gray curve denotes $U+V$. The parameter values are $\alpha = 4$, $k = 2$ and $\gamma = 1$. For this parameter setting, the velocity of segregated TW is $\overline{c} = -0.2895584564$. See also Figure 2(c)
Profiles of overlapping TWs. The solid and the dashed curves respectively denote $U$ and $V$ and the gray curve denotes $U+V$. The parameter values are $\alpha = 4$, $k = 2$, $\gamma = 0.4$ and $c = 1$
Profiles of $t_\gamma$ in cases (ⅰ), (ⅱ) and (ⅲ) of Lemma 2.1
Profiles of $\Omega_1$ and $\Omega_2$ in cases (ii) and (iii) of Lemma 2.2
The case $\lambda_1<\lambda_2$ and $\gamma>k(\alpha-1)/(k-1)$: $(\gamma, c) = (10, 1)$
The case $\lambda_1>\lambda_2$ and $\gamma>k(\alpha-1)/(k-1)$: $(\gamma, c) = (9, 2)$
The case $\lambda_1<\lambda_2$ and $0<\gamma<k(\alpha-1)/(k-1)$: $(\gamma, c) = (2, 1)$ ($k = 4$ and $\alpha = 10$)
The case $\lambda_1>\lambda_2$ and $0<\gamma<k(\alpha-1)/(k-1)$: $(\gamma, c) = (3, 1)$ ($k = 4$ and $\alpha = 10$)
The case $\Lambda_1<\Lambda_2$ and $\gamma>\alpha(k-1)/(\alpha-1)$: $(\gamma, c) = (10, 1)$
The case $\Lambda_1>\Lambda_2$ and $\gamma>\alpha(k-1)/(\alpha-1)$: $(\gamma, c) = (25, 0.5)$ ($k = 8$ and $\alpha = 20$)
The case $\Lambda_1<\Lambda_2$ and $0<\gamma<\alpha(k-1)/(\alpha-1)$: $(\gamma, c) = (0.00066, 0.02475)$ ($k = 1.0004$ and $\alpha = 2.5020$)
The case $\Lambda_1>\Lambda_2$ and $0<\gamma<\alpha(k-1)/(\alpha-1)$: $(\gamma, c) = (3, 1)$ ($k = 4$ and $\alpha = 10$)
Global portrait of the system (Ⅰ) $(\gamma, c) = (10, 1)$
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