期刊
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
卷 20, 期 12, 页码 2267-2291出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202510004921
关键词
Non-local interaction equation; double-well potential; stability analysis; numerical simulation
资金
- King Abdullah University of Science and Technology (KAUST) [KUK-I1-007-43]
- Austria-France project [FR 05/2007, Amadeus 13785 UA]
- WPI, Wolfgang Pauli Institute, University of Vienna
In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential W-epsilon converging to a singular repulsive interaction potential W, the Dirac-type stationary states (rho) over bar (epsilon) approximate weakly a unique stationary state (rho) over bar is an element of L-infinity. We illustrate our results with numerical examples.
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