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Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects

  • * Corresponding author: S. Mancini

    * Corresponding author: S. Mancini
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  • In this paper we study a degenerate parabolic system of reaction-diffusion equations arising in cellular biology models. Its specificity lies in the fact that one of the concentrations does not diffuse. Under realistic conditions on the reaction term, we prove existence and uniqueness of a nonnegative solution to the considered system, and we study its regularity. Moreover, we discuss the existence and linear stability of the steady solutions (equilibria), and give a sufficient condition on the reaction term for Turing-like instabilities to be triggered. These results are finally illustrated by some numerical simulations.

    Mathematics Subject Classification: Primary: 35K65, 82B21; Secondary: 35B35.


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  • Figure 1.  Time evolutions of $\min_{x\in\Omega} u(t,x)$ and $\max_{x\in\Omega} u(t,x)$. Both curves converge to the value 0.5943519.

    Figure 2.  Time evolutions of $\min_{x\in\Omega} v(t,x)$ and $\max_{x\in\Omega} v(t,x)$. Both curves converge to the value 0.4015822.

    Figure 3.  Time evolution of $\max_{x\in \Omega}u(t,x) - \min_{x\in\Omega} u(t,x)$.

    Figure 4.  Time evolutions of $\min_{x\in\Omega} u(t,x)$ and $\max_{x\in\Omega} u(t,x)$. Both curves converge to the value $U^*\cong 0.4536921$.

    Figure 5.  Time evolutions of $\min_{x\in\Omega} v(t,x)$ and $\max_{x\in\Omega} v(t,x)$. The $\min_{x\in\Omega} v(t,x)$ curve converges to the value $V_1^*\cong 0.0073564$, while the $\max_{x\in\Omega} v(t,x)$ one converges to $V_2^*\cong 0.7526096$.

    Figure 6.  The bi-dimensional distribution $v(T,x)$ for T very large. Patterns induced by the initial data defined by 35 clearly appear.

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