Logarithmic convexity for two coupled subdiffusion orders
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and fix $0 < \alpha_1 < \alpha_2 < 1$ and $T > 0$. Consider, with homogeneous Dirichlet boundary conditions, $$\partial_t^{\alpha_1} u_1 = \Delta u_1 + u_1 + u_2, \qquad \partial_t^{\alpha_2} u_2 = \Delta u_2 + u_1 + u_2,$$ with initial pair $U_0 \in L^2(\Omega)^2$, where each derivative is the Caputo derivative $\partial_t^\alpha h(t) = \Gamma(1-\alpha)^{-1} \int_0^t (t-s)^{-\alpha} h'(s)\, ds$. Write $U = (u_1, u_2)$ and $\|\cdot\|$ for the product norm of $L^2(\Omega)^2$.
For a single equation $\partial_t^\alpha u = A u$, $0 < \alpha \le 1$, with $A$ self-adjoint, bounded above and with compact resolvent, Chorfi, Maniar and Yamamoto proved that $\|u(t)\| \le M \|u(0)\|^{1 - t/T} \|u(T)\|^{t/T}$ for $0 \le t \le T$, with a constant $M \ge 1$.
Problem. Is there a constant $C = C(\Omega, \alpha_1, \alpha_2, T)$, independent of $U_0$, such that $$\|U(t)\| \le C \|U_0\|^{1 - t/T} \|U(T)\|^{t/T} \qquad (0 < t < T)$$ for every mild solution?
Source: Chorfi (2025), §5, open problem 1, who asks whether logarithmic convexity extends to coupled systems such as this one; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, backward problems, logarithmic convexity, time-fractional diffusion equations
References
@incollection{chorfi2025,
author = {Chorfi, Salah-Eddine},
title = {Logarithmic convexity of evolution equations and application to inverse problems},
booktitle = {Methusalem Seminars at Ghent Analysis and PDE Center},
series = {Trends in Mathematics},
publisher = {Springer, Cham},
year = {2026},
pages = {21--29},
doi = {10.1007/978-3-032-22493-4_3},
eprint = {2506.19954},
archivePrefix = {arXiv},
}
@article{chorfi2024,
author = {Chorfi, Salah-Eddine and Maniar, Lahcen and Yamamoto, Masahiro},
title = {The backward problem for time-fractional evolution equations},
journal = {Appl. Anal.},
volume = {103},
number = {12},
year = {2024},
pages = {2194--2212},
doi = {10.1080/00036811.2023.2290273},
eprint = {2211.16493},
archivePrefix = {arXiv},
}
@misc{aim-mixed-order-fractional-logarithmic-convexity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Logarithmic convexity for two coupled subdiffusion orders}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/402-mixed-order-fractional-logarithmic-convexity.md},
note = {Entry 402 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove a logarithmic convexity estimate for a pair of coupled Caputo subdiffusion equations with different time orders. For every bounded smooth domain, every $0<\alpha_1<\alpha_2<1$, and every final time $T>0$, the product $L^2$ norm at time $t$ is bounded by a constant times the geometric interpolation of the initial and final norms, with exponent $t/T$. The estimate holds for arbitrary …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Logarithmic convexity for a coupled system with two subdiffusion orders v1 recorded from AIM @ 24edaad, 402-mixed-order-fractional-logarithmic-convexity/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00537, "Logarithmic convexity for two coupled subdiffusion orders", clopen.solutions.