Abstract
We study a two-species cross-diffusion model that is inspired by a system of convectiondiffusion equations derived from an agent-based model on a two-dimensional discrete lattice. The latter model has been proposed to simulate gang territorial development through the use of graffiti markings. We find two energy functionals for the system that allow us to prove a weak-stability result and identify equilibrium solutions. We show that under the natural definition of weak solutions, obtained from the weak-stability result, the system does not allow segregated solutions. Moreover, we present a result on the long-term behavior of solutions in the case when the masses of the densities are smaller than a critical value. This result is complemented with numerical experiments.
| Original language | English |
|---|---|
| Pages (from-to) | 2139-2175 |
| Number of pages | 37 |
| Journal | Communications in Mathematical Sciences |
| Volume | 19 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - 2021 |
Bibliographical note
Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.
Keywords
- Cross-diffusion
- Entropy method
- Equilibrium solutions
- Gang dynamics
- Linear stability
- Long-time behaviour
- Numerical simulations
- Partial differential equations
- Weak stability