93,237 research outputs found
Nonlocal interactions by repulsive-attractive potentials: radial ins/stability
In this paper, we investigate nonlocal interaction equations with
repulsive-attractive radial potentials. Such equations describe the evolution
of a continuum density of particles in which they repulse each other in the
short range and attract each other in the long range. We prove that under some
conditions on the potential, radially symmetric solutions converge
exponentially fast in some transport distance toward a spherical shell
stationary state. Otherwise we prove that it is not possible for a radially
symmetric solution to converge weakly toward the spherical shell stationary
state. We also investigate under which condition it is possible for a
non-radially symmetric solution to converge toward a singular stationary state
supported on a general hypersurface. Finally we provide a detailed analysis of
the specific case of the repulsive-attractive power law potential as well as
numerical results. We point out the the conditions of radial ins/stability are
sharp.Comment: 42 pages, 7 figure
Toward nonlinear stability of sources via a modified Burgers equation
Coherent structures are solutions to reaction-diffusion systems that are
time-periodic in an appropriate moving frame and spatially asymptotic at
to spatially periodic travelling waves. This paper is concerned
with sources which are coherent structures for which the group velocities in
the far field point away from the core. Sources actively select wave numbers
and therefore often organize the overall dynamics in a spatially extended
system. Determining their nonlinear stability properties is challenging as
localized perturbations may lead to a non-localized response even on the linear
level due to the outward transport. Using a modified Burgers equation as a
model problem that captures some of the essential features of coherent
structures, we show how this phenomenon can be analysed and nonlinear stability
be established in this simpler context.Comment: revised version with some typos fixe
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