The nonlocal-to-local limit for superlinear SKT cross-diffusion

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 satisfies the local equation with its original initial data and homogeneous Neumann pressure flux. Both nonlinear pressures and reactions converge strongly in $L^1$. A power entropy gives compactness of the densities. A supplementary regularized logarithmic entropy controls weighted square-root differences, which yield an integrable pressure gradient and permit first-order testing up to the boundary.