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

Abstract: 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-dependent and time-independent inhomogeneous forces. Specifically, for the time-dependent force $S(t, x)$, we prove that the solution converges in $H^s (\Omega )$ to $\bar{u}0+\frac{1}{|\Omega|}\int_0^t \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. For the time-independent force $S(x)$, we prove that the difference between the weak solution and the linear function $\bar{u}0 + \frac{t}{|\Omega|}\int\Omega S(x)\, dx$ remains uniformly bounded in $H^s (\Omega )$.