Qualitative Behavior of Solutions to a Forced Nonlocal Thin-Film Equation

We study a one-dimensional nonlocal degenerate fourth-order parabolic equation with inhomogeneous forces relevant to hydraulic fracture modeling. Employing a regularization scheme, modified energy/entropy methods, and novel differential inequality techniques, we establish global existence and long-time behavior results for weak solutions under both time-and space-dependent and time-and space-independent inhomogeneous forces. Specifically, for the time-and space-dependent force $S(t, x)$, we prove that the solution converges to $\bar{u}_0+\frac{1}{|\Omega|}\int_0^\infty \int_\Omega S(r, x)\, dxdr$, where $\bar{u}_0=\frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)\,dx$ is the spatial average of the initial data, and we provide bilateral estimates for the convergence rate. For the time-and space-independent force $S_0$, we show that the solution approaches the linear function $\bar{u}_0 + tS_0$ at an exponential rate.