Analysis of a splitting-differentiation population model leading to cross-diffusion

We propose a proof of existence of solutions which is conceptually simpler than the previous proof in [1]. Indeed, our proof is based on a direct parabolic regularization of the problem, whereas previous proofs involved a change of unknowns rendering the problem to a parabolic-hyperbolic formulation. Moreover, we believe that our approach also gives a way to select a unique natural solution of the problem as a limit of vanishing viscosity solutions of the regularized parabolic problems and, therefore, it overcomes the possible non-uniqueness issue of the parabolic-hyperbolic formulation.