Asymptotic Stability of Solutions to the Forced Higher-order Degenerate Parabolic Equations

We study a class of fourth-order quasilinear degenerate parabolic equations under both time-dependent and time-independent inhomogeneous forces, modeling non-Newtonian thin-film flow over a solid surface in the "complete wetting" regime. Using regularity theory for higher-order parabolic equations and energy methods, we establish the global existence of positive weak solutions and characterize their long-time behavior. Specifically, for power-law thin-film problem with the time-dependent force $f(t,x)$, we prove that the weak solution converges to $ \bar{u}_0 + \frac{1}{|\Omega|}\int_{0}^t \int_{\Omega} f(s,x) \, {\rm d}x \, {\rm d}s$, and provide the convergence rate, where $\bar{u}_0$ is the spatial average of the initial data. Compared with the homogeneous case in \cite{JJCLKN} (Jansen et al., 2023), this result clearly demonstrates the influence of the inhomogeneous force on the convergence rate of the solution. For the time-independent force $f(x)$, we prove that the difference between the weak solution and the linear function $\bar{u}_0 + \frac{t}{|\Omega|}\int_\Omega f(x)\, {\rm d}x$ is uniformly bounded. For the constant force $f_0$, we show that in the case of shear thickening, the weak solution coincides exactly with $\bar{u}_0 + tf_0$ in a finite time. In both shear-thinning and Newtonian cases, the weak solution approaches $\bar{u}_0 + tf_0$ at polynomial and exponential rates, respectively. Later, for the Ellis law thin-film problem, we find that its solutions behave like those of Newtonian fluids. Finally, we conduct numerical simulations to confirm our main analytical results.