Abstract:
A degenerate Keller-Segel system with diffusion exponent $2n/(n+2) < m < 2-\frac{2}{n}$ in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The exact criterion is closely related to the optimal constant in Haddy- Littlewood- Sobolev inequality. In the case of initial free energy less than a universal constant which depends on the inverse of total mass, there exists a constant such that if the $L^{2n/(n+2)}$ norm of initial data is less than this constant, then the weak solution exists globally; if the $L^{2n/(n+2)}$ norm of initial data is larger than the same constant, then the solution must blow-up in finite time. Our result shows that the total mass, which plays the deterministic role in two dimension case, might not be an appropriate criterion for existence and blow up discussion in multi-dimension, while the $L^{\frac{2n}{n+2}}$ norm of the initial data and the relation between initial free energy and initial mass are more important. Moreover, $L^{\infty}$-bound and uniqueness of solution to the degenerate Keller-Segel equations are given under the sharp condition on initial data. |