Nonlocal interaction equations: Stationary states and stability analysis. (English) Zbl 1265.35142

Raoul, Gaël

Differ. Integral Equ. 25, No. 5-6, 417-440 (2012).

The author is interested in the asymptotic behavior of the density \(\rho (t,x)\) satisfying the equation \(\rho _t=\nabla _x\cdot (\rho \nabla _x[W\ast \rho +V])\) for \(t>0\), \(x\in \mathbb {R}^d\). In the case of analytic potentials \(V,W,\) he proves that \(\rho (t,x)\) converges (in a weak sense) to steady-states of the above equation, which are necessarily finite sums of Dirac masses. Further, the author considers two cases of the potentials having attractive or repulsive singularities at \(x=0\). In the first case, every steady-state apart from Dirac masses is nonlinearly unstable, whereas in the second case the solution (of time dependent equation) is uniformly bounded in \(L^{\infty }(\mathbb {R})\).

Reviewer: Marie Kopáčková (Praha)

MSC:

35K55	Nonlinear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35B20	Perturbations in context of PDEs
35Q79	PDEs in connection with classical thermodynamics and heat transfer

Keywords:

interaction potential; external potential; Dirac mass; asymptotic behavior