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

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  • 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.
    Mathematics Subject Classification: Primary: 35K55, 35K65, 35R35, 92C50.


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  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


    M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94.


    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.


    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.


    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.


    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.


    S. Shmarev, Interfaces in multidimensional diffusion equations with absorption terms, Nonlinear Anal., 53 (2003), 791-828.doi: 10.1016/S0362-546X(03)00034-8.


    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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