Free boundaries touching the boundary of the domain for some reaction–diffusion problems

Díaz, Jesús Ildefonso Díaz; Mingazzini, Tommaso

doi:10.1016/j.na.2014.10.016

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…The Signorini boundary conditions can be also formulated in terms of multivalued nonlinear maximal monotone graphs (see [2]). Some results analyzing the set in which the solution vanishes on the boundary for other different nonlinear boundary conditions was the main goal of the paper [12]. See also [5] for the case of singular nonlinear boundary conditions.…”

Section: Final Remarks and Related Workmentioning

confidence: 99%

Some remarks on the coincidence set for the Signorini problem

Delgado¹,

Díaz²

2019

Opuscula Math.

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Section: Final Remarks and Related Workmentioning

confidence: 99%

Some remarks on the coincidence set for the Signorini problem

Delgado¹,

Díaz²

2019

Opuscula Math.

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“…Advances in connection with existence theory of dead-core solutions, formation of dead-core regions and decay estimates at infinity were obtained in [7], [21] and [27]. Other properties of solutions such as growth of interfaces, shrink and estimates of/on the support and finite extinction in reaction-diffusion problems together with other qualitative properties may be found in Díaz et al's fundamental articles [1], [3], [16], [17], the Antontsev et al's classical book [4], the survey [15] and references therein. However, the lack of quantitative properties for p parabolic dead-core problems constitutes our starting point in this research.…”

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confidence: 99%

Regularity for degenerate evolution equations with strong absorption

Silva

Ochoa

Silva

2018

Journal of Differential Equations

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In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2  p < •) under a strong absorption condition:This model is interesting because it yields the formation of dead-core sets, i.e, regions where nonnegative solutions vanish identically. We shall prove sharp and improved parabolic C a regularity estimates along the set F 0 (u, W T ) = ∂ {u > 0} \ W T (the free boundary), where a = p p 1 q 1 + 1 p 1 . Some weak geometric and measure theoretical properties as non-degeneracy, positive density, porosity and finite speed of propagation are proved. As an application, we prove a Liouville-type result for entire solutions. A specific analysis for Blow-up type solutions will be done as well. The results are new even for dead-core problems driven by the heat operator.2000 Mathematics Subject Classification. 35B53, 35B65, 35J60, 35K55, 35K65. Key words and phrases. p-Laplacian type operators, dead-core problems, sharp and improved intrinsic regularity, Liouville type results.

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

Dao

Díaz

Nguyen

2020

Advanced Nonlinear Studies

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We consider the high-dimensional equation {\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0}, extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution {u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))}, with {u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)}, {\delta(x)=d(x,\partial\Omega)}, we prove some pointwise gradient estimates for a certain range of the dimension N, {m\geq 1} and {\beta\in(0,m)}, mainly when the absorption dominates over the diffusion ({1\leq m<2+\beta}). In particular, a new kind of universal gradient estimate is proved when {m+\beta\leq 2}. Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.

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Free boundaries touching the boundary of the domain for some reaction–diffusion problems

Cited by 3 publications

References 18 publications

Some remarks on the coincidence set for the Signorini problem

Some remarks on the coincidence set for the Signorini problem

Regularity for degenerate evolution equations with strong absorption

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

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