# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp. [12] M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp. [12] M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94. [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. [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. [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. [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. [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. [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.
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