Critical threshold for a two-species chemotaxis system with the energy critical exponent
Shen Bian
TL;DR
This paper analyzes a two-species chemotaxis system in $\mathbb{R}^d$ ($d\ge3$) with degenerate diffusion and nonlocal attraction on the energy-critical exponent $1/m_1+1/m_2=(d+2)/d$. The authors connect stationary states to extremals of the Hardy-Littlewood-Sobolev inequality via a variational structure of the free energy $F(u,v)$, and prove a sharp dichotomy: global existence if $\|u_0\|_{m_1}<\|U_s\|_{m_1}$ and $\|v_0\|_{m_2}<\|V_s\|_{m_2}$ (with $F(u_0,v_0)<F(U_s,V_s)$), and finite-time blow-up if both exceed these thresholds. They derive the blow-up via a second-moment method and establish global existence through uniform-in-time $L^p$ bounds and Moser iteration, all under the energy-critical scaling. The results illuminate the precise balance between diffusion and long-range attraction in multi-species aggregation and extend known single- and multi-species chemotaxis results to the critical, nonlocal setting.
Abstract
We consider a two-species chemotaxis model in $\R^d(d \ge 3)$ featuring nonlinear porous medium-type diffusion and nonlocal attractive power-law interaction. Here, the nonlinear diffusion is chosen to be $1/m_1+1/m_2=(d+2)/d$ in such a way that the associated free energy is conformal invariant, and there are radially symmetric, non-increasing and non-compactly supported stationary solutions $(U_s(x),V_s(x))$. We analyze the conditions on initial data $(u_0,v_0)$ under which attractive forces dominate over diffusion, and further classify the global existence and finite time blow-up of dynamical solutions by virtue of these stationary solutions. Specifically, the solution $(u,v)(x,t)$ exists globally in time if the initial data satisfy $\|u_0\|_{L^{m_1}(\R^d)}<\|U_s\|_{L^{m_1}(\R^d)}$ and $\|v_0\|_{L^{m_2}(\R^d)}<\|V_s\|_{L^{m_2}(\R^d)}$. In contrast, there are blowing-up solutions when $\|u_0\|_{L^{m_1}(\R^d)}>\|U_s\|_{L^{m_1}(\R^d)}$ and $\|v_0\|_{L^{m_2}(\R^d)}>\|V_s\|_{L^{m_2}(\R^d)}$.