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A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation

  • * Corresponding author: Hirofumi Izuhara

    * Corresponding author: Hirofumi Izuhara 
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  • We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

    Mathematics Subject Classification: Primary: 35M31, 35A01, 35R35, 35C07, 35K65; Secondary: 35Q92, 92C17.

    Citation:

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  • Figure 1.  Snapshots of dynamics in (1) with compactly supported initial data. The parameter values are $ \alpha = 4 $, $ \beta = 3 $, $ \gamma = 1 $ and $ k = 0.5 $. The solid and dashed curves indicate $ u $ and $ v $, respectively

    Figure 2.  The relation between the parameter $ k $ and the wave velocity $ c_k^* $, where the horizontal and vertical axes indicate $ k $ and $ c_k^* $, respectively. The other parameter values are $ \alpha = 4 $, $ \beta = 3 $ and $ \gamma = 1 $

    Figure 3.  $ (\varphi, \psi) $-phase planes for (40) with $ k = 2 $ and $ \gamma = 1 $ : (0) $ c = 0 $, (i) $ c = 0.8 $, (ii) $ c = 1 $, (iii) $ c = 1.2 $

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