On a fractional thin film equation

Advances in Nonlinear Analysis

article

Antonio Segatti¹  Juan Luis Vázquez² 

[1] Dipartimento di Matematica “F. Casorati, Università di Pavia;Universidad Autónoma de Madrid, Campus de Cantoblanco

关键词: Fractional operators;    thin film equations;    self-similar solutions;    obstacle problem;   

DOI  :  10.1515/anona-2020-0065

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

【 授权许可】

CC BY