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arXiv · 1110.3242

Hyperbolic traveling waves driven by growth

Emeric Bouin·Vincent Calvez·Grégoire Nadin
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Abstract

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $ε^{-1}$ ($ε\textgreater{}0$), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter $ε$: for small $ε$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $ε$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $ε^{-1}$. The traveling fronts with minimal speed are linearly stable in weighted $L^2$ spaces. We also prove local nonlinear stability of the traveling front with minimal speed when $ε$ is smaller than the transition parameter.

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Emeric Bouin, Vincent Calvez, Grégoire Nadin. 2016-11-21. Hyperbolic traveling waves driven by growth. https://doi.org/10.1142/s0218202513500802

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