On an elliptic system related to desertification studies

Abstract

In this communication, we consider the stationary problem of a non-linear parabolic system which arises in the context of dry-land vegetation. In the first part, we examine the existence and multiplicity of biomass stationary solutions, in terms of the precipitation rate parameter p, for a localized simplification of the system, with non-homogeneous rate of biomass loss. In fact, we show that under appropriate conditions on fixed parameters of the problem, multiple positive solutions exist for a range of the parameter p. In the second part, we consider the case of an idealized “oasis”, ω ⊂⊂ Ω, where we study the transition of the surface-water height in a neighborhood of the set ω