Preservation of strict positivity for the cubic thin-film equation
Let $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ and consider the thin-film equation with cubic mobility,
$$\partial_t h + \partial_x\bigl(h^3 \partial_x^3 h\bigr) = 0, \qquad h(0,x) = h_0(x),$$ with no disjoining-pressure, gravitational or stochastic terms, for $h_0 \in C^\infty(\mathbb{T})$ with $\min h_0 > 0$. For the mobility $h^n$ in one space dimension, Bernis and Friedman constructed nonnegative weak solutions and proved that positivity is preserved when $n \ge 4$ (1990), and Beretta, Bertsch and Dal Passo proved that strict positivity is preserved when $n \ge 7/2$ and that it can fail when $n < 1/2$ (1995); Bertozzi and Pugh obtained the threshold $7/2$ independently with $\alpha$-entropy estimates (1996).
Problem. Does the positive classical solution extend to every finite time and satisfy $$\inf_{(t,x) \in [0,T] \times \mathbb{T}} h(t,x) > 0 \quad \text{for every } T < \infty?$$ The question concerns strict positivity at every time, rather than nonnegative weak solutions or positivity at almost every time.
Source: Goes back to the positivity results of Bernis and Friedman (1990) and Beretta, Bertsch and Dal Passo (1995) for thin-film equations with mobility h^n; see Kunick (2022), §1.3 and §3.10, on whether the deterministic thin-film equation with cubic mobility preserves positivity. Stated in this form, on the circle, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, degenerate parabolic equations, thin-film equations
References
@phdthesis{kunick2022,
author = {Kunick, Florian Johannes},
title = {Analysis and Numerics of Stochastic Gradient Flows},
school = {Universit{\"a}t Leipzig},
year = {2022},
url = {https://d-nb.info/1296492575/34},
}
@article{bernis1990,
author = {Bernis, Francisco and Friedman, Avner},
title = {Higher order nonlinear degenerate parabolic equations},
journal = {J. Differential Equations},
volume = {83},
number = {1},
year = {1990},
pages = {179--206},
doi = {10.1016/0022-0396(90)90074-Y},
}
@article{beretta1995,
author = {Beretta, Elena and Bertsch, Michiel and Dal Passo, Roberta},
title = {Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation},
journal = {Arch. Rational Mech. Anal.},
volume = {129},
number = {2},
year = {1995},
pages = {175--200},
doi = {10.1007/BF00379920},
}
@misc{aim-thin-film-positivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Preservation of strict positivity for the cubic thin-film equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/081-thin-film-positivity.md},
note = {Entry 081 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00648, "Preservation of strict positivity for the cubic thin-film equation", clopen.solutions.