Regularity of solutions to the Dirichlet problem for fast diffusion equations

Jin, Tianling; Xiong, Jingang

doi:10.48550/arxiv.2201.10091

Cited by 8 publications

(16 citation statements)

References 41 publications

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“…Their results especially answer the regularity problem proposed by Berryman and Holland [4]. It is worth pointing out that the degeneracy of weight in [28] is located at the boundary. By contrast, the degeneracy or singularity of the weights considered in this paper lies in the interior.…”

Section: The Nonlinear Parabolic Equations With Anisotropic Weightssupporting

confidence: 66%

“…For the fast diffusion problem (1.6), the regularity of solution and its asymptotic behaviour near extinction time have been extensively studied, for example, see [7, 12, 15, 17-19, 27-29, 35] for the regularity and [1,[4][5][6]23] for the extinction behaviour, respectively. In particular, Jin and Xiong recently established a priori Hölder estimates for the solution to a weighted nonlinear parabolic equation in Theorem 3.1 of [28], which is critical to the establishment of optimal global regularity for fast diffusion equation with any 1 < p < ∞. Their results especially answer the regularity problem proposed by Berryman and Holland [4].…”

Section: The Nonlinear Parabolic Equations With Anisotropic Weightsmentioning

confidence: 99%

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Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

Miao

Zhao²

2023

Proceedings of the Edinburgh Mathematical Society

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The main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

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Section: The Nonlinear Parabolic Equations With Anisotropic Weightssupporting

confidence: 66%

Section: The Nonlinear Parabolic Equations With Anisotropic Weightsmentioning

confidence: 99%

Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

Miao

Zhao²

2023

Proceedings of the Edinburgh Mathematical Society

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“…Later the equivalence between GNS and smoothing effects was established in [27,75], see also [50]. In the nonlinear case, the Nash method does not work, and the classical alternative is provided by the celebrated Moser iteration, which was first introduced for linear parabolic equations [92,93], then extended by various authors to the nonlinear setting, see [80,54,32,77,95,29,83,21,84]. Another classical possibility is the DeGiorgi method, which can be adapted to the nonlinear setting.…”

Section: Smoothing Effects Vs Gagliardo-nirenberg-sobolev Inequalitiesmentioning

confidence: 99%

Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

Bonforte¹,

Endal²

2022

Preprint

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We establish boundedness estimates for solutions of generalized porous medium equations of the form ∂tu + (−L)[u m ] = 0 in R N × (0, T ), where m ≥ 1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L 1 -L ∞ -smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I − L.In the linear case m = 1, it is well-known that the L 1 -L ∞ -smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques.We establish a similar scenario in the nonlinear setting m > 1. First, we can show that operators for which ultracontractivity holds, also provide L 1 -L ∞ -smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L = I − J * . They do not regularize when m = 1, but we show that surprisingly enough they do so when m > 1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator.Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.

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“…From Aronson-Peletier [1], we also know that the solution v of (4) with −1 < p < 0 satisfies (5) as well after certain waiting time. Therefore, the linearized equation of ( 4), which plays an important role in proving optimal regularity of solutions to (4) in [19,20,21], falls into a form of the equation (2). In our earlier work [19], we have obtained many properties for equations like (2) with p > 0, such as well-posedness, local boundedness and Schauder estimates.…”

Section: Introductionmentioning

confidence: 99%