Multidimensional degenerate Keller-Segel system with critical diffusion exponent \(2n/(n+2)\) (Q2904738)

A degenerate diffusion Patlak-Keller-Segel system in \(n \geqslant 3\) dimension is considered. The main difference between the current work and many other recent studies on the same model is to study the diffusion exponent \(m = 2n/(n + 2) ,\) which is smaller than the usual exponent \(m^* {\text{ }} = {\text{ }}2 - 2/n\) used in other studies. With the exponent \(m = 2n/(n + 2),\) the associated free energy is conformal invariant, and there is a family of stationary solutions NEWLINE\[NEWLINEU_{\lambda ,x_0 } (x) = C\left( {\frac{\lambda } {\lambda ^2 + |x - x_0 |^2 }} \right)^{\frac{n + 2} {2}} NEWLINE\]NEWLINE \(\forall \lambda > 0,\,x_0 \in \mathbb{R}^n. \) For radially symmetric solutions it is proved that if the initial data are strictly below \(U_{\lambda ,0} (x)\) for some \(\lambda, \) then the solution vanishes in \(L_{loc}^1 \) as \(t \to \infty;\) if the initial data are strictly above \(U_{\lambda ,0} (x)\) for some \(\lambda,\) then the solution either blows up at a finite time or has a mass concentration at \(r = 0\) as time goes to infinity. For general initial data, it is proved that there is a global weak solution provided that the \(L^m\) norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. A finite time blow-up of the solution if the \(L^m\) norm for initial data is larger than the \(L^m\) norm of \(U_{\lambda ,x_0 } (x), \) which is constant independent of \( \lambda\) and \(x_o,\) and the free energy of initial data is smaller than that of \(U_{\lambda ,x_0 } (x)\) is proved.