Bernis estimates for higher-dimensional doubly-degenerate non-Newtonian thin-film equations

Abstract

For the doubly-degenerate parabolic non-Newtonian thin-film equation

we derive (local versions) of Bernis estimates of the form

for functions $u\in W_p^2(\Omega )$ with Neumann boundary condition, where $2\le p<\frac{19}{3}$ and $n$ lies in a certain range. Here, $\Omega \subset {\mathbb{R}}^d$ is a smooth convex domain with $d<3p$. A particularly important consequence is the estimate

The methods used in this article follow the approach of Grün [Z. Anal. Anwendungen 20 (2001), no. 4, 987–998] for the Newtonian case, while addressing the specific challenges posed by the nonlinear higher-order term $|\nabla \Delta u{|}^{p-2}\nabla \Delta u$ and the additional degeneracy. The derived estimates are key to establishing further qualitative results, such as the existence of weak solutions, finite propagation of support, and the appearance of a waiting-time phenomenon.