On a fractional thin film equation

Abstract

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.2010 Mathematics Subject Classification. 35R11 (35G25, 35K46, 35K65 35C06).

A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate

“…From the analytical point of view, the literature offers a wide range of reports that focus on the extension of integer-order methods and results for the fractional case. For example, there are various articles that tackle the existence, uniqueness, regularity, and asymptotic behavior of the solution for the fractional porous medium equation [42], nonlinear fractional diffusion equations [43], nonlinear fractional heat equations [44], the Fisher-Kolmogorov-Petrovskii-Piscounov equation with nonlinear fractional diffusion [45], fractional thinfilm equations [46], and the fractional Schrödinger equation with general nonnegative potentials [47].…”

Universal self-similar attractor in the bending-driven levelling of thin viscous films

“…The dynamics of such systems is often described within the framework of lubrication theory [35][36][37][38][39], which for small plate deformations leads to the bending-driven thin-film equation. Since thin-film equations are high-order nonlinear diffusive-like equations, they have attracted substantial attention from the applied mathematical community [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. In particular, the general solution in terms of the heat kernel to the nth-order linear equation has been presented in [55], providing a detailed discussion on its increasingly oscillatory behaviour with increasing n and its asymptotic nature, which has been applied in, for example, quantum electrodynamics [56].…”

Fractional high order thin film equation: gradient flow approach

“…The equation in (1.1) is a fractional version of a Thin Film type Equation. The case s ∈ (0, 1) was recently studied in [21] for existence of weak solutions and their asymptotic behavior. The case s = 1 in dimension d = 1 has been treated in [20] and [12], where the gradient flow formulation has been highlighted.…”

“…The aim of this paper is to prove an existence result for problem (1.1), for all order s ∈ (0, +∞), using the gradient flow interpretation in the space of Probability measures endowed with the Wasserstein distance. The problem of the proof of existence of weak solutions of (1.1) using a gradient flow technique has been raised in Section 7 of [21].…”