A parabolic system with strong absorption modeling dry-land vegetation
| dc.contributor.author | Diaz, Jesus Ildefonso | |
| dc.contributor.author | Hilhorst, Danielle | |
| dc.contributor.author | Kyriazopoulos, Paris | |
| dc.date.accessioned | 2021-08-19T20:27:55Z | |
| dc.date.available | 2021-08-19T20:27:55Z | |
| dc.date.issued | 2021-02-20 | |
| dc.description.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 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). | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 19 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | 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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/14405 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2021, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Nonlinear parabolic system | |
| dc.subject | Dry-land vegetation | |
| dc.subject | Positive solution | |
| dc.subject | Free boundary problem | |
| dc.title | A parabolic system with strong absorption modeling dry-land vegetation | en_US |
| dc.type | Article |