Steady states and dynamics of a higher dimensional thin film equation

Abstract

We study a higher-dimensional thin film equation that incorporates competitive effects between aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion, with a degenerate diffusion exponent $m>0$. We first conduct a systematic analysis of the existence and geometric properties of steady-state solutions for all $m>0$, revealing a critical threshold $m^*=(d+2)/(d-2)$ for variational compactness and solution structure. For $0 < m < m^*$, we then prove that, under natural regularity constraints, radially decreasing steady states coincide with both the extremals of the Gagliardo-Nirenberg-Sobolev inequality and the global minimizers of the free energy. Moreover, we establish the uniqueness of such steady states for $m \neq 1 + 2/d$. Furthermore, in the supercritical regime $1 + 2/d < m < m^*$, we identify a sharp threshold given by the $L^{m+1}$ norm of the unique radial steady-state solution, which distinguishes between global existence for initial data below the threshold and finite-time blow-up for initial data above the threshold. The main contribution of this work is to use steady-state solutions as a theoretical pivot to construct a unified analytical framework that connects parameter classification, variational structure and dynamic behavior. This framework elucidates how the regularity barrier prevents infinite energy descent and selects stable equilibrium states, and thus predicts the global evolution of the system, thereby providing a unified variational principle for understanding steady-state selection and dynamic bifurcations in such higher-order degenerate diffusion equations.