The aggregation-diffusion equation with energy critical exponent

2023/02/19 by Bian, Shen · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.09490

Abstract

We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be m=(2d)/(d+2s) in such a way that the associated free energy is conformal invariant and there is a family of stationary solutions U(x)=c(\fracλλ²+|x-x₀|²)^(d+2s)/(2) for any constant c and some λ>0, x₀ ∈ \R^d. We analyze under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of dynamical solutions by virtue of stationary solutions. Precisely, solutions exist globally in time if the L^m norm of the initial data ‖u₀‖_L^m(\R^d) is less than the L^m norm of stationary solutions ‖U(x)‖_L^m(\R^d). Whereas there are blowing-up solutions for ‖u₀‖_L^m(\R^d)>‖U(x)‖_L^m(\R^d).

Cited by

• Sharp criteria for a degenerate diffusion-aggregation system with the intermediate exponent

Related

• The aggregation-diffusion equation with the intermediate exponent
• The Keller-Segel model with mass critical exponent
• Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models
• Asymptotic Behavior for Critical Patlak-Keller-Segel model and an Repulsive-Attractive Aggregation Equation
• Critical threshold for a two-species chemotaxis system with the energy critical exponent