Point-source solutions of fractional porous-medium diffusion on manifolds
Let $(M,g)$ be a complete, connected, noncompact Riemannian manifold of dimension $N \ge 2$ with $\operatorname{Ric} \ge -(N-1)k$ for some $k \ge 0$, satisfying the uniform Faber–Krahn inequality $\lambda_1(D) \ge c \operatorname{Vol}(D)^{-2/N}$, with $c > 0$, for every relatively compact smooth domain $D \subset M$, where $\lambda_1(D)$ is the first Dirichlet eigenvalue of $-\Delta_M$ on $D$. Fix $m > 1$ and $0 < s < 1$ and consider $$\partial_t u + (-\Delta_M)^s (u^m) = 0,$$ with the spectral fractional Laplace–Beltrami operator; integrals are with respect to Riemannian volume. For $o \in M$ and $A > 0$, a point-source solution with initial trace $A\delta_o$ is a nonnegative function $u$ on $(0,\infty) \times M$ with $$u \in C((0,\infty); L^1(M)) \cap L^\infty_{\mathrm{loc}}((0,\infty) \times M), \qquad 0 < \sup_{0 < t < T} \|u(t)\|_{L^1(M)} \le A \quad (T > 0),$$ and $u^m \in L^1_{\mathrm{loc}}((0,\infty) \times M)$, such that $$\int_0^\infty\!\!\int_M u\, \partial_t \phi = \int_0^\infty\!\!\int_M u^m (-\Delta_M)^s \phi$$ for every $\phi \in C_c^\infty((0,\infty) \times M)$, with an integrable right-hand side, and $\lim_{t \downarrow 0} \int_M u(t) \psi = A \psi(o)$ for every bounded continuous $\psi$ on $M$.
Berchio, Bonforte, Grillo and Muratori constructed weak dual solutions on these manifolds for nonnegative initial data in $L^1(M)$, and in a larger space weighted by the fractional Green function, and proved $L^1$–$L^\infty$ smoothing estimates for them. On $\mathbb{R}^N$, Vázquez proved the existence and uniqueness of self-similar fundamental solutions of the fractional porous medium equation.
Problem. For every $o \in M$ and $A > 0$, does a point-source solution with initial trace $A\delta_o$ exist? No radial symmetry or homogeneity of $M$ is assumed.
Source: Berchio, Bonforte, Grillo and Muratori (2024), §7, under large time behaviour, who ask for the existence of fundamental solutions on the manifolds of their Assumption 2.1; also Grillo (2025). Stated in this form by AIM — Open Applied Problems, for distributional solutions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, fractional porous medium equations, fundamental solutions, Riemannian manifolds
References
@article{berchio2024,
author = {Berchio, Elvise and Bonforte, Matteo and Grillo, Gabriele and Muratori, Matteo},
title = {The fractional porous medium equation on noncompact {R}iemannian manifolds},
journal = {Math. Ann.},
volume = {389},
number = {4},
year = {2024},
pages = {3603--3651},
doi = {10.1007/s00208-023-02731-6},
eprint = {2109.10732},
archivePrefix = {arXiv},
}
@article{grillo2025,
author = {Grillo, Gabriele},
title = {Fractional nonlinear diffusions on manifolds: well-posedness and smoothing effects},
journal = {Oberwolfach Rep.},
volume = {22},
number = {1},
year = {2025},
pages = {597--598},
doi = {10.4171/OWR/2025/14},
note = {Report No.~14/2025, Mini-Workshop: Hardy Inequalities in Discrete and Continuum Settings},
}
@article{vazquez2014,
author = {V{\'a}zquez, Juan Luis},
title = {Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {16},
number = {4},
year = {2014},
pages = {769--803},
doi = {10.4171/JEMS/446},
eprint = {1205.6332},
archivePrefix = {arXiv},
}
@misc{aim-manifold-fractional-pme-point-source,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Point-source solutions of fractional porous-medium diffusion on manifolds}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/474-manifold-fractional-pme-point-source.md},
note = {Entry 474 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct solutions of the spectral fractional porous medium equation starting from a point mass on a complete noncompact Riemannian manifold. The assumptions are a lower Ricci curvature bound and a uniform Euclidean Faber--Krahn inequality. Every exponent $m>1$, fractional order $0<s<1$, prescribed source point and positive mass is allowed. The solution is strongly continuous in $L^1$ at positive times, locally …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00810, "Point-source solutions of fractional porous-medium diffusion on manifolds", clopen.solutions.