Asymptotic Stability of Solutions to the Forced Higher-order Degenerate Parabolic Equations
Jinhong Zhao, Bin Guo
TL;DR
This work develops a rigorous analysis of forced fourth-order, degenerate parabolic equations modeling non-Newtonian thin-film flows under inhomogeneous forcing. By combining regularity theory for higher-order parabolic equations with energy methods, the authors establish global existence of positive weak solutions for power-law and Ellis fluids and derive precise long-time behavior and convergence rates that depend on the rheology parameter $\alpha$ and the nature of the forcing. The results reveal how time-dependent and time-independent forces alter convergence toward reference profiles, including explicit polynomial, exponential, or finite-time behaviors, as well as the special Newtonian and Ellis-law cases. Numerical simulations corroborate the theoretical predictions and illustrate the influence of forcing and rheology on the dynamics, offering insights into non-Newtonian thin-film evolution in complete wetting regimes.
Abstract
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}{|Ω|}\int_{0}^t \int_Ω 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 $\bar{u}_0 + \frac{t}{|Ω|}\int_Ωf(x)\, {\rm d}x$ 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.