Electron. J. Differential Equations, Vol. 2021 (2021), No. 08, pp. 1-19.

A parabolic system with strong absorption modeling dry-land vegetation

Jesus Ildefonso Diaz, Danielle Hilhorst, Paris Kyriazopoulos

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. 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 $\partial \Omega $ as $d(x,\partial \Omega )^{\frac{2\alpha }{m-\alpha }}$ then $h^m(x)$ grows, at most, as $d(x,\partial \Omega )^{\frac{2}{m-\alpha }}$. This property also implies the infinite waiting time property when the initial datum $h_0(x)$ grows at fast as $d(x,\partial S(h_0))^{\frac{2m}{m-\alpha }}$ near the boundary of its support $S(h_0)$.

Submitted October 12, 2020. Published February 20, 2021.
Math Subject Classifications: 35K55, 35K65, 35B05.
Key Words: Nonlinear parabolic system; dry-land vegetation; positive solution; free boundary problem.

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Jes\ús Ildefonso Díiaz
Instituto de Matemática Interdisciplinar
Univ. Complutense de Madrid
Plaza de las Ciencias 3 28040
Madrid, Spain
email: jidiaz@ucm.es
Danielle Hilhorst
CNRS and Laboratoire de Mathématiques
University Paris-Saclay
Orsay Cedx 91405, France
email: danielle.hilhorst@universite-paris-saclay.fr
Paris Kyriazopoulos
Department of Mathematics
University of the Aegean
832 00 Karlovassi, Samos, Greece
email: pkyriazopoulos@aegean.gr

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