The Existence Of Weak Solution And L~∞ Estimate To A Degenerate Keller-Segel System

In this paper, we study the following multi-dimension degenerate Keller-Segel system with the nonlinear diffusion and the nonlinear concentration where n≥3,2<q≤2+n/2, the diffusion exponent m ∈ (q - (n+2)/4, q -n/2),ρo is a non-negative function and satisfies Here we prove the global existence of weak solutions and L∞-bound of the solutions. In particular, when the initial free energy and the L(n+2)/2n norm of the initial date are less than some constants, which depends on the initial of total mass, we prove the global existence by using the energy method and uniform estimates. Moreover, the L/∞-bound is obtained by bootstrap iterative method. This paper is divided into three parts. The first one is introduction and preliminary knowledge; The second part is the existence of the solution of the equations; Finally we give the uniform bound of L∞-norm.