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

Jinhong Zhao, Bin Guo

TL;DR

The paper analyzes a one-dimensional nonlocal, degenerate thin-film-type equation with inhomogeneous forcing, formulated with the nonlocal operator $I=-(-\Delta)^s$ for $s\in(0,1)$. It develops a regularization framework together with energy-entropy methods and novel differential-inequality tools to prove global existence of weak solutions and to characterize their long-time behavior under both time-dependent and time-independent forces. For time-dependent forcing, solutions converge in $H^s(\Omega)$ to the moving mean $\bar{u}_0+|\Omega|^{-1}\int_0^t\int_\Omega S(r,x)\,dx\,dr$, while for time-independent forcing the deviation from the linear-in-time profile is uniformly bounded in $H^s(\Omega)$. In the special case of a constant force $S_0$, the dynamics exhibit either exponential relaxation toward $\bar{u}_0+tS_0$ or finite-time extinction when $S_0<0$, highlighting the model’s qualitative dependence on forcing. These results advance the understanding of nonlocal thin-film dynamics in hydraulic-fracturing contexts by linking asymptotics to external forcings through quantitative inequalities and regularized variational techniques.

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 $H^s (Ω)$ to $\bar{u}_0+\frac{1}{|Ω|}\int_0^t \int_ΩS(r, x)\, dxdr $, where $\bar{u}_0=\frac{1}{|Ω|}\int_Ωu_{0}(x)\,dx$ 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 $\bar{u}_0 + \frac{t}{|Ω|}\int_ΩS(x)\, dx$ remains uniformly bounded in $H^s (Ω)$.

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

TL;DR

The paper analyzes a one-dimensional nonlocal, degenerate thin-film-type equation with inhomogeneous forcing, formulated with the nonlocal operator for . It develops a regularization framework together with energy-entropy methods and novel differential-inequality tools to prove global existence of weak solutions and to characterize their long-time behavior under both time-dependent and time-independent forces. For time-dependent forcing, solutions converge in to the moving mean , while for time-independent forcing the deviation from the linear-in-time profile is uniformly bounded in . In the special case of a constant force , the dynamics exhibit either exponential relaxation toward or finite-time extinction when , highlighting the model’s qualitative dependence on forcing. These results advance the understanding of nonlocal thin-film dynamics in hydraulic-fracturing contexts by linking asymptotics to external forcings through quantitative inequalities and regularized variational techniques.

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 , we prove that the solution converges in to , where is the spatial average of the initial data. For the time-independent force , we prove that the difference between the weak solution and the linear function remains uniformly bounded in .
Paper Structure (13 sections, 19 theorems, 213 equations)

This paper contains 13 sections, 19 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Fractional Laplacian operator
  5. Energy/entropy methods -- Integral inequalities
  6. Notations
  7. Time-dependent force $S(t, x)$
  8. Regularized problem
  9. Steady-state problem
  10. Proof of Proposition \ref{['exist.regular']}
  11. Proof of Theorem \ref{['ex1']}
  12. Time-independent force $S(x)$
  13. Constant force $S_0$

Key Result

Theorem 1.1

(Time-dependent force $S(t, x)$) Let $n > 1$, $s \in (\frac{1}{2},1)$, $0 \le u_0 \in H^s(\Omega)$ and $S(t, x) \in L^{2}(0, \infty;H^s(\Omega) )$. If there exists a positive constant $C=C(\Omega)$ satisfying Then for every fixed $T>0$, problem 1.1 has a global positive weak solution $u$ in the sense of Definition def.eq2. Meanwhile, $u$ satisfies for almost every $t\in (0, T)$, and where the

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • ...and 16 more