Analysis and instabilities of travelling waves solutions for a free boundary problem with non-homogeneous KPP reaction, with degenerate diffusion and with non-linear advection

Travelling waves (TWs) instabilities to a degenerate diffusion problem with heterogeneous Fisher-KPP problem have not been previously analysed. The intention along this paper is to study existence, uniqueness and TW instability for a high order diffusion heterogeneous reaction Fisher-KPP problem with advection. The TW profiles are obtained analytically in the proximity of the stationary points, making use of the geometric perturbation theory. In addition, we examine a characterization of a local in time positive inner region where the TW behaves monotonically in contrast with an outer region of instabilities. Furthermore, a numerical exercise determines an accurate estimation of a local time to ensure the existence of the positive inner region, given a certain TW propagation speed.