On a fractional thin film equation

Antonio Segatti,Juan Luis Vázquez

doi:10.1515/anona-2020-0065

Antonio Segatti, Juan Luis Vázquez

Open Access

https://doi.org/10.1515/anona-2020-0065

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Abstract

AbstractThis paper deals with a nonlinear degenerate parabolic equation of orderαbetween 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit valueα= 4 while the Porous Medium Equation is the limitα= 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.

Highlights

In this paper we are mainly interested in the analysis of the following system of partial di erential equations∂t u − div (m(u)∇p) =, in Rd × (, T)p = Ls u, in Rd × (, T) (1.1)u(x, ) = u (x), in Rd, where Ls := (−∆)s, s ∈ (, ), is the fractional Laplacian, the dimension d ≥, and the mobility function m is linear, namely m(u) = u
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
Our aim in this paper is to develop a basic theory for System (1.1)

Summary

Introduction

It is interesting to note that an analogous identity holds the weak solutions of the fractional porous medium equation (1.2) constructed in [21] (see Lemma 5.5). This is the case, as we prove in this paper, under a connectedness condition on the positivity set of the cluster points for large times of the weak solutions of the nonlocal Fokker-Planck equation.

Results

Conclusion