Analysis of selfsimilar solutions and a comparison principle for an heterogeneous diffusion cooperative system with advection and non-linear reaction

The present analysis introduces a system of cooperative species formulated with a high order parabolic operator, a Fisher-KPP reaction and a linear advection. Firstly, the oscillatory behaviour of solutions is shown to exist with a shooting method approach. It is to be highlighted that the existence of oscillatory patterns (also called instabilities) is an inherent property of high order operators. Afterwards, existence and uniqueness results are provided. The most remarkable result, obtained during the existence exercise, is related with the finding of a particular time-degenerate bound for the advection term that ensures positivity of solutions. This is one of the main results as such positivity property does not hold for high order operators in general. Indeed, high-order operators provide oscillatory solutions that may induce such solutions to be negative in the proximity of the null state introduced by the Fisher-KPP reaction term. As a consequence, a comparison principle does not hold as formulated in order two operators. Further, a positive maximal kernel with similar asymptotic behaviour compared to the high order kernel has been shown to exist and a precise assessment has been done with a computational exercise. Eventually, such a positive maximal kernel permits to show the existence of a comparison principle.