Existence, uniqueness and positivity on a free-boundary high order diffusion cooperative system

Abstract

We study existence, uniqueness and positivity conditions for a cooperative system formulated with a high order diffusion. In addition, we show and characterize the instabilities due to the high order diffusion for which a positivity and a comparison principle hold after rescaling. Instabilities shall be understood as the oscillatory behaviour of solutions acting as an inherent feature driven by complex exponential bundles of solutions. Firstly, a shooting method approach is used to show the existence of such instabilities. Afterwards, the existence of solutions is assessed in the self-similar approach and characterized by analytical and numerical evidences for a single equation in and for the complete system. Finally, a positive kernel is shown to exist and a sharp estimation is obtained.

Keywords

• Bi-laplacian
• Instabilities
• Positivity
• Kernel
• Uniqueness