David Gómez-Castro

We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential W(x) ∼\log |x| for |x| ≫1 appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the C^αregularity with a novel approach for this family of equations using modulus of continuity techniques.

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $\mathcal D^s,p (\mathbb R^n) D s , p ( R n ) and their embeddings, for s ∈(0,1] s ∈ ( 0 , 1 ] and p\ge 1 p ≥ 1 . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For s p < n s p < n or s = p = n = 1 s = p = n = 1 we show that \mathcal D^s,p( \mathbb R^n) D s , p ( R n ) is isomorphic to a suitable function space, whereas for s p \ge n$ s p ≥ n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.