Global attractor for a nonlocal model for biological aggregation

We investigate the long term behavior in terms of global attractors, as time goes toinfinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the global attractor, and also discuss the convergence of any bounded solutions to steady states. In particular, under suitable assumptions on the parameters of the problem, we establish the convergence of the bounded solution $u(t)$ to a single steady state $u_*$, and the rate of convergence

$\parallel u(t) - u_* \parallel {L^P \Omega} \sim (1+t)^{- \rho}$, as $t \to \infty$,

for any $p \gt 1$, and some $\rho = \rho (u_*, p) \in (0,1)$. Finally, the existence of an exponential attractor is also demonstrated for sufficiently smooth kernels in the case of non-degenerate diffusion. Our analysis extends and complements the analysis from [17] and many other fundamental works.

Keywords

global existence, gradient structure, global attractor, convergence to steady states, exponential attractor, chemotaxis, biological aggregation

2010 Mathematics Subject Classification

35A01, 35A02, 35K55

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Published 7 February 2014