Hyperbolic traveling waves driven by growth

Emeric Bouin 1, 2 Vincent Calvez 1, 2, * Grégoire Nadin 3
* Corresponding author
2 NUMED - Numerical Medicine
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées, Inria Grenoble - Rhône-Alpes
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 $\epsilon^{-1}$ ($\epsilon>0$), 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 $\epsilon$: for small $\epsilon$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $\epsilon$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $\epsilon^{-1}$. 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 $\epsilon$ is smaller than the transition parameter.
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Emeric Bouin, Vincent Calvez, Grégoire Nadin. Hyperbolic traveling waves driven by growth. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014, 24 (6), http://www.worldscientific.com/doi/abs/10.1142/S0218202513500802. ⟨10.1142/S0218202513500802⟩. ⟨hal-00632597v2⟩

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