Localization limit for superlinear nonlocal multispecies systems

Prove the localization limit from the superlinear nonlocal multispecies population system with exponent s\ne 1 to the corresponding local SKT system, extending the known result for s=1 and two species to superlinear diffusion exponents and the general multispecies setting.

Background

The paper studies a nonlocal multispecies population system whose diffusion operator is intended to approximate the local Shigesada–Kawasaki–Teramoto (SKT) cross-diffusion system under a rescaling of the diffusion kernel. A localization result had previously been established only for the special case of linear diffusion potentials (s=1) with two species. The paper develops global existence and uniqueness for superlinear diffusion, but does not establish convergence of the nonlocal solutions to the local SKT solutions in this regime.

The authors report that one-dimensional numerical experiments strongly support the validity of the localization limit for superlinear exponents and suggest that the remaining obstacles are technical. Establishing this limit would connect the strong-solution theory for the nonlocal model with the available theory for the corresponding local SKT system.

References

The localization limit was proved for $s=1$ and $n=2$ in using duality estimates, whose extension to $s\neq 1$ is an open problem.

— Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules  (2609.09921 - Hirvonen et al., 9 Sep 2026) in Section 1, subsection “Model equations”