Exact criterion for global existence and blow up to a degenerate Keller-Segel system
The life time of non-negative weak solutions to the Patlak-Keller-Segel (PKS) equation with nonlinear diffusion \[ \partial_t \rho = \mathrm{div} \left( \nabla \rho^m - \rho \nabla (E_n*\rho) \right)\;, \quad x\in\mathbb{R}^n\;, \;t>0\;, \] is studied when \(n\geq 3\) and \(m\in (2n/(n+2),2(n-1)/n)\). Here \(E_n\) denotes the Poisson kernel and the initial condition \(\rho_0\) is a non-negative function in \(L^1(\mathbb{R}^n)\cap L^m(\mathbb{R}^n)\). Two quantities play an important role in the dynamics of this equation, namely the mass \(\|\rho(t)\|_{L^1}\) of \(\rho\) at time \(t\) which is constant during time evolution \((\|\rho(t)\|_{L^1} = M_0:= \|\rho_0\|_{L^1})\) and the energy \[ \mathcal{F}(\rho(t)) := \int_{\mathbb{R}^n} \left[ \frac{\rho(t,x)^m}{m-1} - \frac{1}{2} \rho(t,x) (E_n*\rho)(t,x) \right]\, dx\;, \] which is a non-increasing function of time. Given \(M_0>0\) and \(m\in (2n/(n+2),2(n-1)/n)\), it is shown that there are two positive parameters \(\mathcal{F}^*\) and \(s^*\) depending only on \(n\), \(m\), and \(M_0\) such that the weak solution to the PKS equation with initial condition \(\rho_0\) exists globally if \(\mathcal{F}(\rho_0)<\mathcal{F}^*\) and \(\|\rho_0\|_{L^{2n/(n+2)}}<(s^*)^{(n-2)/2n(m-1)}\) while it blows up in finite time if \(\mathcal{F}(\rho_0)<\mathcal{F}^*\) and \(\|\rho_0\|_{L^{2n/(n+2)}}>(s^*)^{(n-2)/2n(m-1)}\). A complete classification of the behaviors of low energy solutions according to the value of the \(L^{2n/(n+2)}\)-norm of their initial condition is thus provided. The proof relies on the Hardy-Littlewood-Sobolev inequality.
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