April  2015, 35(4): 1479-1501. doi: 10.3934/dcds.2015.35.1479

Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem

1. 

University of Oviedo, Spain, Spain, Spain

Received  July 2013 Revised  August 2014 Published  November 2014

We study the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=div(a_iu_i\nabla (u_1+u_2))+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder $Q=\Omega\times (0,T]$. It is assumed that the functions $f_1(r,0)$, $f_2(0,s)$ are locally Lipschitz-continuous and $f_1(0,s)=0$, $f_2(r,0)=0$. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique and derive the equation of motion of the surface $\Gamma$ which separates the parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $\Gamma$ is a modification of the Darcy law in filtration theory.
Citation: Gonzalo Galiano, Sergey Shmarev, Julian Velasco. Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1479-1501. doi: 10.3934/dcds.2015.35.1479
References:
[1]

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

[2]

M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves,, Netw. Heterog. Media, 8 (2013), 131. doi: 10.3934/nhm.2013.8.131. Google Scholar

[3]

M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition,, Interfaces and Free Boundaries, 12 (2010), 235. doi: 10.4171/IFB/233. Google Scholar

[4]

M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth,, Diff. Equ. Appl., 4 (2012), 137. doi: 10.7153/dea-04-09. Google Scholar

[5]

S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration,, J. Math. Biol., 16 (1983), 181. doi: 10.1007/BF00276056. Google Scholar

[6]

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,, Math. Med. Biol., 23 (2006), 197. doi: 10.1093/imammb/dql009. Google Scholar

[7]

J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology,, Arch. Ration. Mech. Anal., 194 (2009), 75. doi: 10.1007/s00205-008-0164-y. Google Scholar

[8]

J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets. II. A quasilinear equation in climatology,, J. Math. Anal. Appl., 352 (2009), 475. doi: 10.1016/j.jmaa.2008.09.046. Google Scholar

[9]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89. doi: 10.1007/s002050050146. Google Scholar

[10]

G. Galiano and V. Selgas, On a cross-diffusion segregation problem arising from a model of interacting particles,, Nonlinear Anal. Real World Appl., 18 (2014), 34. doi: 10.1016/j.nonrwa.2014.02.001. Google Scholar

[11]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). Google Scholar

[12]

M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding,, Q. Appl. Math., 42 (1984), 87. Google Scholar

[13]

A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis. Vol. 2: Measure. The Lebesgue Integral. Hilbert Space,, Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm Graylock Press, (1960). Google Scholar

[14]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Quasillinear Equations of Parabolic Type,, Translations of Mathematical Monographs 23, (1968). Google Scholar

[15]

O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (1968). Google Scholar

[16]

S. Shmarev, Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension,, Trends in partial differential equations of mathematical physics, 61 (2005), 257. doi: 10.1007/3-7643-7317-2_19. Google Scholar

[17]

S. Shmarev, Interfaces in multidimensional diffusion equations with absorption terms,, Nonlinear Anal., 53 (2003), 791. doi: 10.1016/S0362-546X(03)00034-8. Google Scholar

[18]

S. Shmarev and J. L. Vazquez, The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates,, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 465. doi: 10.1007/BF01193831. Google Scholar

show all references

References:
[1]

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

[2]

M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves,, Netw. Heterog. Media, 8 (2013), 131. doi: 10.3934/nhm.2013.8.131. Google Scholar

[3]

M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition,, Interfaces and Free Boundaries, 12 (2010), 235. doi: 10.4171/IFB/233. Google Scholar

[4]

M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth,, Diff. Equ. Appl., 4 (2012), 137. doi: 10.7153/dea-04-09. Google Scholar

[5]

S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration,, J. Math. Biol., 16 (1983), 181. doi: 10.1007/BF00276056. Google Scholar

[6]

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,, Math. Med. Biol., 23 (2006), 197. doi: 10.1093/imammb/dql009. Google Scholar

[7]

J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology,, Arch. Ration. Mech. Anal., 194 (2009), 75. doi: 10.1007/s00205-008-0164-y. Google Scholar

[8]

J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets. II. A quasilinear equation in climatology,, J. Math. Anal. Appl., 352 (2009), 475. doi: 10.1016/j.jmaa.2008.09.046. Google Scholar

[9]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89. doi: 10.1007/s002050050146. Google Scholar

[10]

G. Galiano and V. Selgas, On a cross-diffusion segregation problem arising from a model of interacting particles,, Nonlinear Anal. Real World Appl., 18 (2014), 34. doi: 10.1016/j.nonrwa.2014.02.001. Google Scholar

[11]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). Google Scholar

[12]

M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding,, Q. Appl. Math., 42 (1984), 87. Google Scholar

[13]

A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis. Vol. 2: Measure. The Lebesgue Integral. Hilbert Space,, Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm Graylock Press, (1960). Google Scholar

[14]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Quasillinear Equations of Parabolic Type,, Translations of Mathematical Monographs 23, (1968). Google Scholar

[15]

O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (1968). Google Scholar

[16]

S. Shmarev, Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension,, Trends in partial differential equations of mathematical physics, 61 (2005), 257. doi: 10.1007/3-7643-7317-2_19. Google Scholar

[17]

S. Shmarev, Interfaces in multidimensional diffusion equations with absorption terms,, Nonlinear Anal., 53 (2003), 791. doi: 10.1016/S0362-546X(03)00034-8. Google Scholar

[18]

S. Shmarev and J. L. Vazquez, The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates,, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 465. doi: 10.1007/BF01193831. Google Scholar

[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 & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226

[2]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185

[3]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193

[4]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[5]

Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725

[6]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435

[7]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239

[8]

Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145

[9]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152

[10]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719

[11]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1193-1200. doi: 10.3934/dcds.2003.9.1193

[12]

Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145

[13]

Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147

[14]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597

[15]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875

[16]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

[17]

Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745

[18]

Daniel Ryan, Robert Stephen Cantrell. Avoidance behavior in intraguild predation communities: A cross-diffusion model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1641-1663. doi: 10.3934/dcds.2015.35.1641

[19]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019198

[20]

Jing Yang. Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1785-1805. doi: 10.3934/cpaa.2017087

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]