Equilibria of biological aggregations with nonlocal repulsive-attractive interactions

We consider the aggregation equation $\rho_{t}-\nabla\cdot(\rho\nabla K\ast\rho) =0$ in $\mathbb{R}^{n}$, where the interaction potential $K$ incorporates short-range Newtonian repulsion and long-range power-law attraction. We study the global well-posedness of solutions and investigate analytically and numerically the equilibrium solutions. We show that there exist unique equilibria supported on a ball of $\mathbb{R}^n$. By using the method of moving planes we prove that such equilibria are radially symmetric and monotone in the radial coordinate. We perform asymptotic studies for the limiting cases when the exponent of the power-law attraction approaches infinity and a Newtonian singularity, respectively. Numerical simulations suggest that equilibria studied here are global attractors for the dynamics of the aggregation model

Book Reviews

Reviews of the following books: The Long Argument: English Puritanism and the Shaping of New England Culture, 1570-1700 by Stephen Foster; The Salem Witch Crisis by Larry Gragg; A Home for Everyman: The Greek Revival and Maine Domestic Architecture by Joyce K. Bibber; The Gunpowder Mills of Maine by Maurice W. Hitten; In the Hands of Providence: Joshua L. Chamberlain And The American Civil War by Alice Rains Trulock; Hurricane Island: The Town That Disappeared by Eleanor Motley Richardson; Home Front On Penobscot Bay: Rockland During The War Years, 1940-1945by Paul G. Merriam, Thomas J. Molloy, and Theodore W. Sylvester, Jr.; The History of Broadcasting in Maine: The First Fifty Yearsby Ellie Thompson; New Compass Points by Roy P. Fairfiel