# 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 and Continuous Dynamical Systems, 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. [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. [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. [13] J. D. Murray, Mathematical Biology. I, Springer-Verlag, New York, 2002. [14] J. D. Murray, Mathematical Biology. II, Springer-Verlag, New York, 2003. [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. [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.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. [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. [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. [13] J. D. Murray, Mathematical Biology. I, Springer-Verlag, New York, 2002. [14] J. D. Murray, Mathematical Biology. II, Springer-Verlag, New York, 2003. [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. [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.
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) $(\gamma, c) = (10, 1)$
 [1] 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 [2] Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815 [3] Aijun Zhang. Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022061 [4] M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827 [5] Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure and Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1 [6] Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227 [7] Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure and Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97 [8] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [9] A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251 [10] Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925 [11] Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069 [12] Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 [13] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [14] Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801 [15] Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087 [16] Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687 [17] Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 [18] Elisabetta Rocca, Giulio Schimperna. Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1193-1214. doi: 10.3934/dcds.2006.15.1193 [19] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure and Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 [20] Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719

2021 Impact Factor: 1.588