Hyperbolic traveling waves driven by growth

Details Hyperbolic traveling waves driven by growth Hyperbolic traveling waves driven by growth

2011-10-14

arxiv.org/abs/1110.3242v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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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