Stationary states of quadratic diffusion equations with long-range attraction

By Martin Burger, Marco Di Francesco, Marzena Franek. 2011

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We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient ow generated by the energy E, which is the sum of a quadratic free energy and the interaction energy. The interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diusivity constant is strictly smaller than the total mass of the interaction kernel. In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the Krein-Rutman theorem. The steady states are symmetric with respect to their center of mass x0, compactly supported on sets of the form (x0 L;x0 + L), C 2 on their support, strictly decreasing on (x0;x0+L). Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/0fcb0bd0c5f92f5146aeba8b9b73e8ccade4cf62

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In Applied Mathematics

  1. Razvan C. Fetecau
  2. Rustum Choksi
  3. Ihsan Topaloglu

In Modeling and Simulation

  1. Razvan C. Fetecau
  2. Rustum Choksi
  3. Ihsan Topaloglu

In Public Health, Environmental and Occupational Health

  1. Razvan C. Fetecau
  2. Rustum Choksi
  3. Ihsan Topaloglu

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