Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term

Dao, Nguyen Anh; Díaz, Jesús Ildefonso Díaz; Nguyen, Quan Ba Hong

doi:10.1515/ans-2020-2076

Cited by 4 publications

(1 citation statement)

References 49 publications

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“…It seems possible to extend most of the results of this paper to the case in which α ∈ (−1, 0]. See, e.g., the treatment made in [11] for a scalar equation.…”

Section: Qualitative Properties Of the Surface Water Componentmentioning

confidence: 76%

parabolic system with strong absorption modeling dry-land vegetation

Díaz

Hilhorst

Kyriazopoulos

2021

ejde

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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 $m\geq 2$. 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 $S(h_0)$. For more information see https://ejde.math.txstate.edu/Volumes/2021/08/abstr.html

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“…It seems possible to extend most of the results of this paper to the case in which α ∈ (−1, 0]. See, e.g., the treatment made in [11] for a scalar equation.…”

Section: Qualitative Properties Of the Surface Water Componentmentioning

confidence: 76%

parabolic system with strong absorption modeling dry-land vegetation

Díaz

Hilhorst

Kyriazopoulos

2021

ejde

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Monotone continuous dependence of solutions of singular quenching parabolic problems

Díaz

Giacomoni

2022

Rend. Circ. Mat. Palermo, II. Ser

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We prove the continuous dependence, with respect to the initial datum of solutions of the “quenching parabolic problem” $$\partial _{t}u-\Delta u+\chi _{\{u>0\}}u^{-\beta }=\lambda u^{p}$$ ∂ t u - Δ u + χ { u > 0 } u - β = λ u p , with zero Dirichlet boundary conditions, when $$\beta \in (0,1),p\in (0,1],\lambda \ge 0$$ β ∈ ( 0 , 1 ) , p ∈ ( 0 , 1 ] , λ ≥ 0 and $$\chi _{\{u>0\}}$$ χ { u > 0 } denotes the characteristic function of the set of points (x, t) where $$u(x,t)>0$$ u ( x , t ) > 0 . Notice that the absorption term $$\chi _{\{u>0\}}u^{-\beta }$$ χ { u > 0 } u - β is singular and monotone decreasing which does not allow the application of standard monotonicity arguments.

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Instantaneous shrinking of the support of solutions to parabolic equations with a singular absorption

Dao

2020

RACSAM

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Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term

Cited by 4 publications

References 49 publications

parabolic system with strong absorption modeling dry-land vegetation

parabolic system with strong absorption modeling dry-land vegetation

Monotone continuous dependence of solutions of singular quenching parabolic problems

Instantaneous shrinking of the support of solutions to parabolic equations with a singular absorption

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