Stationary states of quadratic diffusion equations with long-range attraction

We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient flow 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 diffusivity 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 $x_0$, compactly supported on sets of the form $[x_0 − L, x_0 + L], C^2$ on their support, and strictly decreasing on $(x_0, x_0+L)$. Moreover, they are global minimizers of the energy functional $E$. The results are complemented by numerical simulations.

Keywords

nonlinear diffusion, nonlocal aggregation, population dynamics, stationary states, Krein-Rutman theorem

2010 Mathematics Subject Classification

35B33, 35B41, 35K65, 45C05, 45K05, 92D25, 92D50

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Published 14 May 2013