In this paper we consider a one-dimensional nonlocal interaction
equation with quadratic porous-medium type diusion in which the interaction
kernels are attractive, nonnegative, and integrable on the real line. Earlier
results in the literature have shown existence of nontrivial steady states if the
L1 norm of the kernel G is larger than the diusion constant ". In this paper
we aim at showing that this equation exhibits a `multiple' behavior, in that
solutions can either converge to the nontrivial steady states or decay to zero for
large times. We prove the former situation holds in case the initial conditions
are concentrated enough and `close' to the steady state in the 1-Wasserstein
distance. Moreover, we prove that solutions decay to zero for large times
in the diusion-dominated regime "  kGkL1 . Finally, we show two partial
results suggesting that the large-time decay also holds in the complementary
regime " < kGkL1 for initial data with large enough second moment. We use
numerical simulations both to validate our local asymptotic stability result and
to support our conjecture on the large time decay.

Title

Kinetic and Related Models

Publisher

American Institute of Mathematical Sciences

Publisher Country

United States of America

Indexing

Scopus

Impact Factor

1.39

Publication Type

Online only

Volume

12

Year

2019

Pages

303-322