  We consider an aggregation-diffusion equation modelling particle interaction
with non-linear diffusion and non-local attractive interaction using a
homogeneous kernel (singular and non-singular) leading to variants of the
Keller-Segel model of chemotaxis. We analyse the fair-competition regime in
which both homogeneities scale the same with respect to dilations. Our analysis
here deals with the one-dimensional case and provides an almost complete
classification. In the singular kernel case and for critical interaction
strength, we prove uniqueness of stationary states via a variant of the
Hardy-Littlewood-Sobolev inequality. Using the same methods, we show uniqueness
of self-similar profiles in the sub-critical case by proving a new type of
functional inequality. Surprisingly, the same results hold true for any
interaction strength in the non-singular kernel case. Further, we investigate
the asymptotic behaviour of solutions, proving convergence to equilibrium in
Wasserstein distance in the critical singular kernel case, and convergence to
self-similarity for sub-critical interaction strength, both under a uniform
stability condition. Moreover, solutions converge to a unique self-similar
profile in the non-singular kernel case. Finally, we provide a numerical
overview for the asymptotic behaviour of solutions in the full parameter space
demonstrating the above results. We also discuss a number of phenomena
appearing in the numerical explorations for the diffusion-dominated and
attraction-dominated regimes.
