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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

Gonzalo Galiano 1, , Sergey Shmarev 1, and  Julian Velasco 1,

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.
Keywords: Nonlinear parabolic system, contact inhibition., cross-diffusion, Lagrangian coordinates, segregated solutions.
Mathematics Subject Classification: Primary: 35K55, 35K65, 35R35, 92C5.
Citation: Gonzalo Galiano, Sergey Shmarev, Julian Velasco. Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem. Discrete & Continuous Dynamical Systems, 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), 1-13. 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), 131-147. 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), 235-250. doi: 10.4171/IFB/233.  Google Scholar

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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), 137-157. doi: 10.7153/dea-04-09.  Google Scholar

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S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration, J. Math. Biol., 16 (1983), 181-198. doi: 10.1007/BF00276056.  Google Scholar

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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), 197-229. doi: 10.1093/imammb/dql009.  Google Scholar

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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), 75-103. doi: 10.1007/s00205-008-0164-y.  Google Scholar

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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), 475-495. doi: 10.1016/j.jmaa.2008.09.046.  Google Scholar

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G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. 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), 34-49. doi: 10.1016/j.nonrwa.2014.02.001.  Google Scholar

[11]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp.  Google Scholar

[12]

M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94.  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, Albany, N.Y. 1961. ix+128 pp.  Google Scholar

[14]

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

[15]

O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968. xviii+495 pp.  Google Scholar

[16]

S. Shmarev, Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension, Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 61 (2005), 257-273. 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), 791-828. 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), 465-497. 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), 1-13. 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), 131-147. 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), 235-250. 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), 137-157. 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), 181-198. 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), 197-229. 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), 75-103. 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), 475-495. 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), 89-118. 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), 34-49. doi: 10.1016/j.nonrwa.2014.02.001.  Google Scholar

[11]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp.  Google Scholar

[12]

M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94.  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, Albany, N.Y. 1961. ix+128 pp.  Google Scholar

[14]

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

[15]

O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968. xviii+495 pp.  Google Scholar

[16]

S. Shmarev, Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension, Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 61 (2005), 257-273. 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), 791-828. 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), 465-497. doi: 10.1007/BF01193831.  Google Scholar

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