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

Abstract

For the doubly-degenerate parabolic non-Newtonian thin-film equation $$u_t+\text{div}(u^n|\nabla \Delta u{|}^{p-2}\nabla \Delta u)=0,$$ we derive (local versions) of Bernis estimates of the form $${\int }_{\Omega }{u}^{n-2p}|\nabla u{|}^{3p}dx+{\int }_{\Omega }{u}^{n-\frac{p}{2}}|\Delta u{|}^{\frac{3p}{2}}dx\le c(n,p,d){\int }_{\Omega }{u}^n|\nabla \Delta u{|}^{p}dx,$$ 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 $${\int }_{\Omega }|\nabla \Delta (u^{\frac{n+p}{p}}){|}^{p}dx\le c(n,p,d){\int }_{\Omega }{u}^n|\nabla \Delta u{|}^{p}dx.$$ The methods used in this article follow the approach of [Grü01] 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.

24 pages, 1 figure