Hyperbolic traveling waves driven by growth

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 (ε > 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. The traveling fronts with minimal speed are linearly stable in weighted L spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ε is smaller than the transition parameter. Key-words: traveling waves; Fisher-KPP equation; telegraph equation; nonlinear stability.