Uniqueness for a point source in time-fractional porous-medium flow
Fix $d \ge 2$, $m > 1$, $0 < \alpha < 1$ and $M > 0$, and consider the time-fractional porous medium equation $D_t^\alpha u = \Delta(u^m)$ on $\mathbb{R}^d$, with $D_t^\alpha$ the Caputo derivative of order $\alpha$. A nonnegative mass-$M$ weak solution with initial measure $M\delta_0$ is a nonnegative $u \in C((0,\infty); L^1(\mathbb{R}^d))$ with $u^m \in L^1_{\mathrm{loc}}([0,\infty) \times \mathbb{R}^d)$ and $\int u(t) = M$, such that $u(t,x)\,dx \rightharpoonup M\delta_0$ narrowly as $t \downarrow 0$ and, for each $\phi \in C_c^\infty(\mathbb{R}^d)$ and almost every $t > 0$, $$\int u(t,x)\phi(x)\,dx = M\phi(0) + \frac{1}{\Gamma(\alpha)} \int_0^t (t-r)^{\alpha-1} \int u(r,x)^m \Delta\phi(x)\,dx\,dr.$$ This Volterra identity fixes the Caputo convention for measure data.
Gómez-Castro, Płociniczak and Vázquez constructed, for every $d \ge 1$ and $m > (d-2)_+/d$, a self-similar solution $u(t,x) = t^{-a} U(t^{-b}|x|)$ of mass $M$ with initial trace $M\delta_0$, where $b = \alpha/(d(m-1)+2)$ and $a = db$. For $m > 1$ its profile is compactly supported and is the unique nonnegative solution of mass $M$ of their integral equation for profiles; for $d \ge 2$, $u^m$ has the singularity of the Newtonian potential at the origin, so $u$ is unbounded there. For $d = 1$ and $m > 1$ such solutions had been studied by Caballero, Okrasińska-Płociniczak, Płociniczak and Sadarangani. For $m = 1$, Gómez-Castro, Płociniczak and Vázquez proved that the very weak solution with initial datum $M\delta_0$ is unique.
Problem. Is there at most one nonnegative mass-$M$ weak solution with initial measure $M\delta_0$? Self-similarity and radial symmetry are not imposed.
Source: Gómez-Castro, Płociniczak and Vázquez (2026), §8, who expect uniqueness of weak solutions with Dirac initial data. Stated with this weak formulation, for d ≥ 2, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, Caputo derivatives, fundamental solutions, porous medium equations
References
@misc{gomezcastro2026,
author = {G{\'o}mez-Castro, David and P{\l}ociniczak, {\L}ukasz and V{\'a}zquez, Juan Luis},
title = {Self-similar solutions to the time-fractional porous-medium equation},
year = {2026},
eprint = {2604.09281},
archivePrefix = {arXiv},
}
@article{caballero2026,
author = {Caballero, Josefa and Okrasi{\'n}ska-P{\l}ociniczak, Hanna and P{\l}ociniczak, {\L}ukasz and Sadarangani, Kishin},
title = {Barenblatt solutions for the time-fractional porous medium equation: approach via integral equations},
journal = {Fract. Calc. Appl. Anal.},
volume = {29},
number = {3},
year = {2026},
pages = {1516--1546},
doi = {10.1007/s13540-026-00514-9},
eprint = {2507.19217},
archivePrefix = {arXiv},
}
@misc{aim-time-fractional-pme-dirac-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness for a point source in time-fractional porous-medium flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/475-time-fractional-pme-dirac-uniqueness.md},
note = {Entry 475 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-00650, "Uniqueness for a point source in time-fractional porous-medium flow", clopen.solutions.