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 with m≥ 2. The main novelty in this article is that the absorption is given in terms of an exponent , in contrast to the case 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 grows near the boundary of the domain as then grows, at most, as . This property also implies the infinite waiting time property when the initial datum grows at fast as near the boundary of its support .
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
| Danielle Hilhorst |
CNRS and Laboratoire de Mathématiques
Orsay Cedx 91405, France
| Paris Kyriazopoulos |
Department of Mathematics
University of the Aegean
832 00 Karlovassi, Samos, Greece
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