Threshold of global behavior of solutions to a degenerate drift-diffusion system in between two critical exponents

Abstract

We consider large time behavior of weak solutions to degenerate drift-diffusion system related to Keller–Segel system. $$\begin{aligned} \left\{ \begin{array}{lll} &{}\partial _t \rho -\Delta \rho ^{\alpha } + \nabla \cdot (\rho \nabla \psi ) =0, &{}\quad t>0,\ x \in \mathbb {R}^n,\\ &{}-\Delta \psi = \rho ,&{}\quad t>0, \ x \in \mathbb {R}^n,\\ &{}\rho (0,x) = \rho _0(x), &{}\quad x \in \mathbb {R}^n,\\ \end{array} \right. \end{aligned}$$ where \(n\ge 3\) and \(\alpha >0\). There exist two critical diffusion exponents \(\alpha =\alpha ^*=2-\frac{2}{n}\) and \(\alpha =\alpha _*=2-\frac{4}{n+2}\) and for those cases, large time behavior of solutions is classified by the invariant norms of initial data. We consider the case of the intermediate exponent \(\alpha _{*}< \alpha < \alpha ^{*}\) and classify the global existence and finite time blow up of weak solutions by the combination of invariant norms of initial data. Besides we show that the threshold value which classifies the behavior of solutions is characterized by the best possible constant of the modified Hardy–Littlewood–Sobolev inequality: $$\begin{aligned} \int _{\mathbb {R}^n}f(-\Delta )^{-1}f dx \le C_{HLS,\alpha }\Vert f\Vert _1^{1-\sigma }\Vert f\Vert _{\alpha }^{1+\sigma }, \end{aligned}$$ where \(\sigma =\frac{\alpha }{\alpha -1}\frac{n-2}{n}-1\) and it is given by the radial stationary solution of the system. Here the result is continuous analogue of the known critical cases (Blanchet et al., Calc Var Partial Diff Equ 35:133–168, 2009 and Ogawa, Disc Contin Dyn Syst Ser S 4:875–886, 2011). Analogous result has been obtained in the theory of nonlinear Schrodinger equations. The global behavior of the weak solution is also given and the solution converges to the self-similar Barenbratt solution as time parameter goes to infinity.