Invasive-invaded system of non-Lipschitz porous medium equations with advection.
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2021
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International Journal of Biomathematics
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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.
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Non-linearity, Reaction, Absorption, Coupled system, Diffusion, Porous media