Aggregation--Diffusion Equations with Regular or Repulsive Interactions: Gaussian and Self-Similar Asymptotics

We study the large-time behavior of solutions to the aggregation--diffusion equation $$ u_t=Δu+\nabla\cdot\bigl(u\,\nabla K*u\bigr) \qquad \text{in } \mathbb{R}^d. $$ Our main results identify a transition, governed by the regularity and singularity of the interaction kernel, between Gaussian and no