Optimal existence and uniqueness theory for the fractional heat equation

被引：75

作者：

Bonforte, Matteo ^{[1
]}

Sire, Yannick ^{[2
,3
]}

Luis Vazquez, Juan ^{[1
]}

机构：

[1] Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain

[2] Johns Hopkins Univ, Baltimore, MD USA

[3] Univ Aix Marseille, Marseille, France

来源：

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS | 2017年 / 153卷

关键词：

Fractional heat equation; Existence and uniqueness theory; Optimal data; Pointwise estimates; Harnack inequalities; POROUS-MEDIUM EQUATION; NONLOCAL PARABOLIC EQUATIONS; NONLINEAR DIFFUSION; REGULARITY; OPERATORS; SPACES; KERNEL; GUIDE;

D O I：

10.1016/j.na.2016.08.027

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation partial derivative(t)u + (-Delta)(s)u = 0, 0 < s < 1, posed in the whole space RN with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions, which is given in one direction by the representation formula, in the other one by the initial trace. We review many of the typical properties of the solutions, in particular we prove optimal pointwise estimates and new Harnack inequalities. (C) 2016 Elsevier Ltd. All rights reserved.

引用

页码：142 / 168

页数：27

共 48 条

[31] Symmetric stable laws and stable-like jump-diffusions
Kolokoltsov, V
[J]. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 2000, 80 : 725 - 768
[32] Landkof N., 1972, Foundations of Modern Potential Theory. Grundlehren der mathematischen Wissenschaften
[33] Regularity for solutions of non local parabolic equations
Lara, Hector Chang
Davila, Gonzalo
[J]. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2014, 49 (1-2) : 139 - 172
[34] Optimal estimates for fractional fast diffusion equations
Luis Vazquez, Juan
Volzone, Bruno
[J]. JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, 2015, 103 (02): : 535 - 556
[35] Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
Luis Vazquez, Juan
[J]. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 2014, 16 (04) : 769 - 803
[36] The random walk's guide to anomalous diffusion: a fractional dynamics approach
Metzler, R
Klafter, J
[J]. PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS, 2000, 339 (01): : 1 - 77
[37] HARNACK INEQUALITY FOR PARABOLIC DIFFERENTIAL EQUATIONS
MOSER, J
[J]. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1964, 17 (01) : 101 - &
[38] Rodino L., 1993, LINEAR PARTIAL DIFFE, P251
[39] Talenti G., 1979, ANN MAT PUR APPL, V4, P160
[40] Tychonoff A., 1935, MAT SB, V42, P199

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