On a Splitting-Differentiation Process Leading to Cross-Diffusion
We generalize the dynamical system model proposed by Sánchez-Palencia for the splitting-differentiation process of populations to include spatial dependence. This gives rise to a family of cross-diffusion partial differential equations problems, among which we consider the segregation model proposed by Busenberg and Travis. For the one-dimensional case, we make a direct parabolic regularization of the problem to show the existence of solutions in the space of BV functions. Moreover, we introduce a Finite Element discretization of both our parabolic regularization and an alternative regularization previously proposed in the literature. Our numerical results suggest that our approach is more stable in the tricky regions where the solutions exhibit discontinuities.
- 15.Gambino, G., Lombardo, M.C., Sammartino, M.: Pattern formation driven by cross-diffusion in a 2D domain. Nonlinear Anal.: Real World Appl. 14, 1755–1779 (2013)Google Scholar
- 17.Sánchez-Palencia, E.: Competition of subspecies and structural stability. Survival of the best adapted or coexistence? In: International Congress on Nonlinear Models in Partial Differential Equations, Toledo, Spain (2011)Google Scholar