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.

翻译：我们研究一类受时间依赖与时间无关非齐次力作用下的四阶拟线性退化抛物方程，该方程模拟了‘完全润湿’状态下非牛顿薄层流体在固体表面的流动。利用高阶抛物方程的正则性理论与能量方法，我们建立了正弱解的全局存在性并刻画了其长时间行为。具体而言，对于含时间依赖外力$f(t,x)$的幂律薄层问题，我们证明了弱解收敛于$ \bar{u}_0 + \frac{1}{|\Omega|}\int_{0}^t \int_{\Omega} f(s,x) \, {\rm d}x \, {\rm d}s$，并给出了收敛速率，其中$\bar{u}_0$为初始数据的空间平均值。与文献\\cite{JJCLKN}（Jansen等人，2023）中的齐次情形相比，该结果清晰地揭示了非齐次外力对解收敛速率的影响。对于时间无关外力$f(x)$，我们证明了弱解与线性函数$\bar{u}_0 + \frac{t}{|\Omega|}\int_\Omega f(x)\, {\rm d}x$的差是一致有界的。对于常值外力$f_0$，我们证明在剪切增稠情形下，弱解在有限时间内精确等于$\bar{u}_0 + tf_0$；在剪切稀化与牛顿流体情形中，弱解分别以多项式速率与指数速率趋近于$\bar{u}_0 + tf_0$。进一步，对于埃利斯定律薄层问题，我们发现其解的行为与牛顿流体相似。最后，我们通过数值模拟验证了主要理论结果。