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Front propagation in diffusion-aggregation models with bi-stable reaction

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  • In this paper, necessary and sufficient conditions are given for the existence of travelling wave solutions of the reaction-diffusion-aggregation equation

    $v_\tau=(D(v)v_x)_{x}+f(v), $

    where the diffusivity $D$ changes sign twice in the interval $(0,1)$ (from positive to negative and again to positive) and the reaction $f$ is bi-stable. We show that classical travelling waves with decreasing profile do exist for a single admissible value of their speed of propagation which can be either positive or negative, according to the behavior of $f$ and $D$. An example is given, illustrating the employed techniques. The results are then generalized to a diffusivity $D$ with $2n$ sign changes.

    Mathematics Subject Classification: Primary: 35K25, 92D25; Secondary: 34B40, 34B16.


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