Barenblatt attraction for porous-medium diffusion with time memory
Fix $d \ge 1$, $m > 1$ and $0 < \alpha < 1$, and let $u_0 \in L^1 \cap L^\infty(\mathbb{R}^d)$ be nonnegative and compactly supported, of mass $M > 0$. Consider nonnegative weak solutions of the time-fractional porous medium equation $D_t^\alpha u = \Delta(u^m)$ on $\mathbb{R}^d$: $u \in C([0,\infty); L^1(\mathbb{R}^d))$ with $u(0) = u_0$, $u^m \in L^1_{\mathrm{loc}}([0,\infty) \times \mathbb{R}^d)$ and $\int u(t) = M$ for every $t$, satisfying in spatial distributions the Volterra identity $$u(t) = u_0 + \frac{1}{\Gamma(\alpha)} \int_0^t (t-r)^{\alpha-1} \Delta\bigl(u(r)^m\bigr)\,dr,$$ which specifies the Caputo derivative based at time zero. Set $b = \alpha/(d(m-1)+2)$ and $a = db$, and let $U_{\alpha,m,M}$ be the nonnegative radial self-similar profile of mass $M$ constructed by Gómez-Castro, Płociniczak and Vázquez, so that $t^{-a} U_{\alpha,m,M}(t^{-b}x)$ is a solution with initial datum $M\delta_0$.
For $m > 1$ this profile is compactly supported and is the unique nonnegative solution of mass $M$ of their integral equation for profiles, and the canonical (unit) profile converges to the Barenblatt profile as $\alpha \to 1$. For $\alpha = 1$ the convergence below is the classical Barenblatt asymptotics of the porous medium equation. For $m = 1$, Cortázar, Quirós and Wolanski proved that solutions with integrable data behave for large times, in the characteristic scale $|x| \sim t^{\alpha/2}$, like $M$ times the fundamental solution.
Problem. Does every such $u$ satisfy $$\lim_{t \to \infty} \bigl\| t^a u(t, t^b\,\cdot) - U_{\alpha,m,M} \bigr\|_{L^1(\mathbb{R}^d)} = 0?$$ The rescaling changes only the observation variables; it does not replace the original Caputo memory by a derivative with a shifted lower limit.
Source: Gómez-Castro, Płociniczak and Vázquez (2026), §8, who ask whether their self-similar profiles are long-time global attractors for the Cauchy problem. Stated by AIM — Open Applied Problems for m > 1, compactly supported bounded data and convergence in L¹.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, Caputo derivatives, large-time behaviour, 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},
}
@book{vazquez2007,
author = {V{\'a}zquez, Juan Luis},
title = {The Porous Medium Equation: Mathematical Theory},
series = {Oxford Mathematical Monographs},
publisher = {Oxford University Press},
address = {Oxford},
year = {2007},
doi = {10.1093/acprof:oso/9780198569039.001.0001},
}
@article{cortazar2021,
author = {Cort{\'a}zar, Carmen and Quir{\'o}s, Fernando and Wolanski, Noemi},
title = {A heat equation with memory: large-time behavior},
journal = {J. Funct. Anal.},
volume = {281},
number = {9},
year = {2021},
pages = {109174},
doi = {10.1016/j.jfa.2021.109174},
eprint = {2005.02860},
archivePrefix = {arXiv},
}
@misc{aim-time-fractional-pme-barenblatt-attraction,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Barenblatt attraction for porous-medium diffusion with time memory}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/476-time-fractional-pme-barenblatt-attraction.md},
note = {Entry 476 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-00649, "Barenblatt attraction for porous-medium diffusion with time memory", clopen.solutions.