Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart

We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada-Kawasaki-Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.

Automated summary

This study proves that under certain conditions, the nonlocal diffusion problem can be transformed into a problem with local diffusion by rescaling the kernel function. It also provides a new proof of existence for solutions to the local diffusion problem.