Sharp threshold for a one-dimensional thin film equation in the supercritical case

We study a one-dimensional thin film equation combining competitive effects of aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion with exponent $m>0$. Under natural regularity constraints, we prove that for every $m>0$, there exists a unique (up to the mass-critical case $m=3$) nonnegative, radially decreasing steady state $U_*$ which coincides with the extremal function of the sharp Sz.-Nagy inequality and is simultaneously the global minimizer of the free energy. Using this variational characterization in the supercritical regime $3<m<\infty$, we show that finite-time blow-up occurs for all initial data whose initial free energy lies below the positive threshold $F(U_*)$, provided the $L^{m+1}$-norm of the initial datum exceeds that of $U_*$. Conversely, if the $L^{m+1}$-norm is below that of $U_*$, the solution exists globally and its second moment diverges as $t\to\infty$. This sharp criterion significantly extends the previously known blow-up condition requiring negative free energy to a much wider class of initial data (see \cite{BP00}). Our results identify the steady state $U_*$ as the critical pivot linking variational structure to dynamical behavior, and provide a constructive method to determine blow-up versus global existence via an explicit $L^{m+1}$-norm comparison.