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

For the doubly-degenerate parabolic non-Newtonian thin-film equation $$u_t+\text{div}\left(u^n\mathrm{\mid }\mathrm{\nabla }\mathrm{\Delta }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }\mathrm{\Delta }u\right)=0,$$ we derive (local versions) of Bernis estimates of the form $${\int }_{\mathrm{\Omega }}{u}^{n-2p}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{3p}dx+{\int }_{\mathrm{\Omega }}{u}^{n-\frac{p}{2}}\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^{\frac{3p}{2}}dx\le c(n,p,d){\int }_{\mathrm{\Omega }}{u}^n\mathrm{\mid }\mathrm{\nabla }\mathrm{\Delta }u{\mathrm{\mid }}^{p}dx,$$ for functions $u\in W_p^2(\mathrm{\Omega })$ with Neumann boundary condition, where $2\le p<\frac{19}{3}$ and $n$ lies in a certain range. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a smooth convex domain with $d<3p$. A particularly important consequence is the estimate $${\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }\mathrm{\Delta }(u^{\frac{n+p}{p}}){\mathrm{\mid }}^{p}dx\le c(n,p,d){\int }_{\mathrm{\Omega }}{u}^n\mathrm{\mid }\mathrm{\nabla }\mathrm{\Delta }u{\mathrm{\mid }}^{p}dx\mathrm{.}$$ 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 $\mathrm{\mid }\mathrm{\nabla }\mathrm{\Delta }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }\mathrm{\Delta }u$ 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.