Critical threshold for a two-species chemotaxis system with the energy critical exponent

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 $\|v_0\|_{L^{m_2}(\R^d)}>\|V_s\|_{L^{m_2}(\R^d)}$.