A parabolic system with strong absorption modeling dry-land vegetation

Abstract

We consider a variant of a nonlinear parabolic system, proposed by Gilad, von Hardenberg, Provenzale, Shachak and Meron, in desertification studies, in which there is a strong absorption. The system models the mutual interaction between the biomass, the soil-water content w and the surface-water height which is diffused by means of the degenerate operator $\Delta h^m$ with m≥ 2. The main novelty in this article is that the absorption is given in terms of an exponent $\alpha \in (0,1)$, in contrast to the case $\alpha =1$ considered in the previous literature. Thanks to this, some new qualitative behavior of the dynamics of the solutions can be justified.

Keywords

• nonlinear parabolic system
• dry-land vegetation
• positive solution
• free boundary problem