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.

Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effects. Our parabolic system is the gradient flow of an energy functional, and in fact we show that the stationary states are minimizers of a relaxed energy. Here, we study radial solutions of an aggregation-diffusion model that combines nonlinear fast diffusion with a convection term driven by the gradient of a potential, both in balls and the whole space. We show that, depending on the exponent of fast diffusion and the potential, the steady state is given by the sum of an explicit integrable function, plus a Dirac delta at the origin containing the rest of the mass of the initial datum. Furthermore, it is a global minimizer of the relaxed energy. This splitting phenomenon is an uncommon example of blow-up in infinite time.

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.

\textcopyright 2018 Elsevier Ltd We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation (−∆)su+Vu=f posed in bounded domain Ω⊂Rnwith zero outside conditions. We consider general nonnegative potentials V∈Lloc1(Ω) and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration since it leads to so-called flat solutions. We comment on related literature.