Study of a family of higher order nonlocal degenerate parabolic equations: from the porous medium equation to the thin film equation

TL;DR

This paper investigates a family of nonlocal degenerate parabolic equations of order between 2 and 4, extending models like the porous medium and thin film equations, proving existence of solutions under various conditions.

Contribution

It generalizes existing models by analyzing a broader class of nonlocal equations, establishing solution existence for different parameter ranges and initial data.

Findings

Existence of solutions for 0<α<2 with nonnegative initial data.

Construction of solutions for α > 1 under weaker assumptions.

Adaptation of compactness arguments to different Sobolev embeddings.

Abstract

In this paper, we study a nonlocal degenerate parabolic equation of order {\alpha} + 2 for 0<{\alpha}<2. The equation is a generalization of the one arising in the modeling of hydraulic fractures studied by Imbert and Mellet in 2011. Using the same approach, we prove the existence of solutions for this equation for 0<{\alpha}<2 and for nonnegative initial data satisfying appropriate assumptions. The main difference is the compactness results due to different Sobolev embeddings. Furthermore, for {\alpha} > 1, we construct a nonnegative solution for nonnegative initial data under weaker assumptions.