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Standing and travelling waves in a parabolic-hyperbolic system

  • * Corresponding author: Hirofumi Izuhara

    * Corresponding author: Hirofumi Izuhara 
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  • We consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product $ uv $ vanishes.

    Mathematics Subject Classification: Primary: 35C07, 35M30, 70K05; Secondary: 35Q92, 92C17.

    Citation:

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  • Figure 1.  Dependency of the wave velocity of segregated travelling wave solutions on $ \alpha $. The horizontal and the vertical axes are respectively the wave velocity $ \overline c $ and the value $ \alpha $. The parameter values are $ \gamma = 1 $ and $ k = 2 $

    Figure 2.  Typical profiles of segregated TWs. The parameter values are $ \gamma = 1 $ and $ k = 2 $, the same as the ones in Figure 1. The solid and the dashed curves respectively denote $ U $ and $ V $. (a) $ \overline{c} = 0.2238529804 $ (b) $ \overline{c} = 0 $ (c) $ \overline{c} = -0.2895584564 $

    Figure 3.  Profiles of overlapping TWs. The solid and the dashed curves respectively denote $ U $ and $ V $ and the gray curve denotes $ U+V $. The parameter values are $ \alpha = 1 $, $ k = 2 $ and $ \gamma = 1 $. For this parameter setting, the velocity of segregated TW is $ \overline{c} = 0.4094908611 $

    Figure 4.  Profiles of overlapping TWs. The solid and the dashed curves respectively denote $ U $ and $ V $ and the gray curve denotes $ U+V $. The parameter values are $ \alpha = 4 $, $ k = 2 $ and $ \gamma = 1 $. For this parameter setting, the velocity of segregated TW is $ \overline{c} = -0.2895584564 $. See also Figure 2(c)

    Figure 5.  Profiles of overlapping TWs. The solid and the dashed curves respectively denote $ U $ and $ V $ and the gray curve denotes $ U+V $. The parameter values are $ \alpha = 4 $, $ k = 2 $, $ \gamma = 0.4 $ and $ c = 1 $

    Figure 6.  Profiles of $ t_\gamma $ in cases (ⅰ), (ⅱ) and (ⅲ) of Lemma 2.1

    Figure 7.  Profiles of $ \Omega_1 $ and $ \Omega_2 $ in cases (ii) and (iii) of Lemma 2.2

    Figure 8.  The case $ \lambda_1<\lambda_2 $ and $ \gamma>k(\alpha-1)/(k-1) $: $ (\gamma, c) = (10, 1) $

    Figure 9.  The case $ \lambda_1>\lambda_2 $ and $ \gamma>k(\alpha-1)/(k-1) $: $ (\gamma, c) = (9, 2) $

    Figure 10.  The case $ \lambda_1<\lambda_2 $ and $ 0<\gamma<k(\alpha-1)/(k-1) $: $ (\gamma, c) = (2, 1) $ ($ k = 4 $ and $ \alpha = 10 $)

    Figure 11.  The case $ \lambda_1>\lambda_2 $ and $ 0<\gamma<k(\alpha-1)/(k-1) $: $ (\gamma, c) = (3, 1) $ ($ k = 4 $ and $ \alpha = 10 $)

    Figure 12.  The case $ \Lambda_1<\Lambda_2 $ and $ \gamma>\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (10, 1) $

    Figure 13.  The case $ \Lambda_1>\Lambda_2 $ and $ \gamma>\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (25, 0.5) $ ($ k = 8 $ and $ \alpha = 20 $)

    Figure 14.  The case $ \Lambda_1<\Lambda_2 $ and $ 0<\gamma<\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (0.00066, 0.02475) $ ($ k = 1.0004 $ and $ \alpha = 2.5020 $)

    Figure 15.  The case $ \Lambda_1>\Lambda_2 $ and $ 0<\gamma<\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (3, 1) $ ($ k = 4 $ and $ \alpha = 10 $)

    Figure 16.  Global portrait of the system (Ⅰ) $ (\gamma, c) = (10, 1) $

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