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

For the doubly-degenerate parabolic non-Newtonian thin-film equation \begin{equation*} u_t + \mathrm{div}(u^n |\nabla \Delta u|^{p-2} \nabla \Delta u)=0, \end{equation*} we derive (local versions) of Bernis estimates of the form \begin{equation*} \int_{\Omega} u^{n-2p} |\nabla u|^{3p} \mathrm{d} x +\int_{\Omega} u^{n-\frac{p}{2}} |\Delta u|^{\frac{3p}{2}}\mathrm{d} x \leq c(n,p,d) \int_{\Omega} u^n|\nabla \Delta u|^p \mathrm{d} x \end{equation*} for functions u \in W^{2}_{p}(\Omega) with Neumann boundary condition, where 2 \leq 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 \begin{equation*} \int_{\Omega} |\nabla \Delta (u^{\frac{n+p}{p}})|^p\mathrm{d} x \leq c(n,p,d) \int_{\Omega} u^n|\nabla \Delta u|^p \mathrm{d} x. \end{equation*} 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.