The local limit of superlinear multispecies nonlocal diffusion
Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded connected domain, let $n \ge 2$ and $s > 1$, and for $u = (u_1, \dots, u_n) \ge 0$ set $$P_i(u) = u_i \Bigl(a_{i0} + \sum_{j=1}^n a_{ij} u_j^s\Bigr), \qquad f_i(u) = b_{i0} - \sum_{j=1}^n b_{ij} u_j,$$ where $a_{i0} > 0$, $b_{i0}, b_{ij} \ge 0$, and the symmetric matrix $(a_{ij})$ has nonnegative entries and satisfies $$a_{ii} > C(s) \sum_{j \ne i} a_{ij}, \qquad C(s) = \begin{cases} \dfrac{s^2}{2(s+1)(s-1)}, & 1 < s < 2, \\[6pt] \dfrac{4(s-1)^2 + 1}{4(s+1)(s-1)}, & s \ge 2. \end{cases}$$ Let $J$ be a smooth, nonnegative, radial, compactly supported kernel on $\mathbb{R}^d$, positive near $0$, normalized by $\frac12 \int_{\mathbb{R}^d} J(z) z_1^2\, dz = 1$, and put $J_\varepsilon(z) = \varepsilon^{-d} J(z/\varepsilon)$. Fix nonnegative $u_i^0 \in L^\infty(\Omega) \cap BV(\Omega)$. Hirvonen, Jüngel and Pollino proved that for every $\varepsilon > 0$ the regional nonlocal system $$\partial_t u_i^\varepsilon(t,x) = \varepsilon^{-2} \int_\Omega J_\varepsilon(x - y) \bigl[P_i(u^\varepsilon(t,y)) - P_i(u^\varepsilon(t,x))\bigr]\, dy + u_i^\varepsilon f_i(u^\varepsilon), \qquad u_i^\varepsilon(0) = u_i^0,$$ has a unique nonnegative strong solution $u^\varepsilon$ on every finite time interval, with $u_i^\varepsilon \in W^{1,\infty}(0,T; L^\infty(\Omega)) \cap C([0,T]; L^\infty(\Omega) \cap BV(\Omega))$. Since $\varepsilon^{-2} \int_\Omega J_\varepsilon(x - y) [g(y) - g(x)]\, dy$ approximates $\Delta g(x)$, the system is a nonlocal approximation of the Shigesada–Kawasaki–Teramoto cross-diffusion system below. For two species and $s = 1$, Galiano and Velasco proved the corresponding limit.
Problem. As $\varepsilon \downarrow 0$, does a subsequence of $u^\varepsilon$ converge strongly in $L^1((0,T) \times \Omega)^n$, for every finite $T$, to a global weak solution of $$\partial_t u_i = \Delta P_i(u) + u_i f_i(u), \qquad \partial_\nu P_i(u) = 0 \ \text{ on } \partial\Omega, \qquad u_i(0) = u_i^0?$$ The nonlinear fluxes and reaction terms, not only the densities, must pass to the limit in the weak formulation.
Source: Hirvonen, Jüngel and Pollino (2026), §1.1, on extending to s ≠ 1 the localization limit that Galiano and Velasco proved for s = 1 and two species. Stated in this form by AIM — Open Applied Problems, for s > 1 under the coefficient condition of Hirvonen, Jüngel and Pollino's existence theorem.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, cross-diffusion systems, nonlocal diffusion, Shigesada–Kawasaki–Teramoto model
References
@misc{hirvonen2026,
author = {Hirvonen, Peter and J{\"u}ngel, Ansgar and Pollino, Annamaria},
title = {Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules},
year = {2026},
eprint = {2609.09921},
archivePrefix = {arXiv},
}
@article{galiano2022,
author = {Galiano, Gonzalo and Velasco, Juli{\'a}n},
title = {Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart},
journal = {Rev. R. Acad. Cienc. Exactas F{\'\i}s. Nat. Ser. A Mat. RACSAM},
volume = {116},
number = {3},
year = {2022},
pages = {93},
doi = {10.1007/s13398-022-01231-7},
}
@misc{aim-superlinear-skt-nonlocal-to-local-limit,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The local limit of superlinear multispecies nonlocal diffusion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/469-superlinear-skt-nonlocal-to-local-limit.md},
note = {Entry 469 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove convergence from regional nonlocal diffusion to the local Shigesada--Kawasaki--Teramoto system with superlinear transition rates. For any number of species and any fixed exponent greater than one, the given nonnegative strong nonlocal solutions admit a subsequence converging strongly in space-time on every finite time interval, under the symmetric coefficient and strict diagonal-dominance assumptions of the source problem. The limit …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/localization.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00838, "The local limit of superlinear multispecies nonlocal diffusion", clopen.solutions.