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

2025/10/23 by Zhao, Jinhong, Guo, Bin
#35B40 #35K35 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2510.20289

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 u₀+(1)/(|Ω|)∫₀^t ∫_ΩS(r, x) dxdr , where u₀=(1)/(|Ω|)∫_Ωu₀(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 u₀ + (t)/(|Ω|)∫_ΩS(x) dx remains uniformly bounded in H^s (Ω).

Citations

• Study of a family of higher order nonlocal degenerate parabolic equations: from the porous medium equation to the thin film equation
• Hitchhiker's guide to the fractional Sobolev spaces
• Long-scale evolution of thin liquid films

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