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-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 ${\bar{u}}_0+t{S}_0$ at an exponential rate.

28pages