Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion.
Autor: Díaz Palencia, José Luis
Resumen: It is the objective to provide a mathematical treatment of a model to predict the behaviour
of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of
the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation
with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and
uniqueness of solutions together with a comparison principle. Once the regularity principles are shown,
the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in
the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW
profile follows a potential law in the stable connection that converges to the stationary solution. Such
potential law suggests that the pressure induced by the invasive over the hostile area increases over
time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent
invasion.
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