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

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal, degenerate fourth-order parabolic equation modeling hydraulic fractures, establishing global existence and convergence results using advanced mathematical techniques.

Contribution

It introduces novel analytical methods to prove global existence and detailed convergence rates for solutions under various inhomogeneous forcing conditions.

Findings

Solutions converge to specific steady states or linear functions depending on the force type.

Exponential convergence rate is established for space-independent forces.

Bilateral estimates for convergence rates are provided.

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 $\bar{u}_0 + tS_0$ at an exponential rate.