Invasive-invaded system of non-Lipschitz porous medium equations with advection.

Abstract

This work provides analytical results towards applications in the field of invasive-invaded systems modelled with non-linear diffusion and with advection. The results focus on showing regularity, existence and uniqueness of weak solutions using the condition of a non-linear slightly positive parabolic operator and the reaction-absorption monotone properties. The coupling in the reaction-absorption terms, that characterizes the species interaction, impedes the formulation of a global comparison principle that is shown to exist locally. Additionally, the present work provides analytical solutions obtained as selfsimilar minimal and maximal profiles. A propagating diffusive front is shown to exist until the invaded specie notes the existence of the invasive. When the desertion of the invaded starts, the diffusive front vanishes globally and the non-linear diffusion concentrates only on the propagating tail which exhibits finite speed. Finally, the invaded specie is shown to exhibit an exponential decay along a characteristic curve. Such exponential decay is not trivial in the non-linear diffusion case and confirms that the invasive continues to feed on the invaded during the desertion.