A parabolic system with strong absorption modeling dry-land vegetation
Diaz, Jesus Ildefonso
Texas State University, Department of Mathematics
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 Δhm with m ≥ 2. The main novelty in this article is that the absorption is given in terms of an exponent α ∈ (0, 1), in contrast to the case α = 1 considered in the previous literature. Thanks to this, some new qualitative behavior of the dynamics of the solutions can be justified. After proving the existence of non-negative solutions for the system with Dirichlet and Neumann boundary conditions, we demonstrate the possible extinction in finite time and the finite speed of propagation for the surface-water height component h(t, x). Also, we prove, for the associate stationary problem, that if the precipitation datum p(x) grows near the boundary of the domain ∂Ω as d(x, ∂Ω) 2α/m-α then hm (x) grows, at most, as d(x, ∂Ω) 2/m-α. This property also implies the infinite waiting time property when the initial datum h0(x) grows at fast as d(x, ∂S(h0)) 2m/m-α near the boundary of its support S(h0).
Nonlinear parabolic system, Dry-land vegetation, Positive solution, Free boundary problem
Díaz, J. I., Hilhorst, D., & Kyriazopoulos, P. (2021). A parabolic system with strong absorption modeling dry-land vegetation. <i>Electronic Journal of Differential Equations, 2021</i>(08), pp. 1-19.