Optimal existence and uniqueness theory for the fractional heat equation. (April 2017)

• Record Type: Journal Article
• Authors: Bonforte, Matteo; Sire, Yannick; Vázquez, Juan Luis
• Is Part Of: Nonlinear analysis. Volume 153(2017)
• Page Start: 142
• Page End: 168
• Publication Date: 2017-04
• Subjects: 35A01 -- 35A02 -- 35K99; Fractional heat equation -- Existence and uniqueness theory -- Optimal data -- Pointwise estimates -- Harnack inequalities; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.08.027
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation ∂ t u + ( − Δ ) s u = 0, 0 < s < 1, posed in the whole space R N 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.