Global Existence Of Weak Solutions To A Degenerate Parabolic-parabolic Keller-segel Equation

This article mainly studies the existence of weak solutions to a parabolic-parabolic Keller-Segel equations with degenerate diffusion under sharp initial conditions. In the literature [7], the authors gave the sharp initial criteria s* > 0 for a parabolic-elliptic degenerate Keller-Segel equations, where s* depends on the spatial dimension, system parameters and initial mass. In this paper, we prove that the parabolic-parabolic Keller-Segel equations with the diffusion exponent 2n/2+n< m < 2-2/n has a global weak solution under the same sharp initial conditions ???0??2n/Ln+2< s*. Here the sharp initial crite-ria is mainly from the best constant of the Sobolev inequality, that is different from the parabolic-elliptic model (The best initial criteria is from the best constant of the Hardy-Littlewood-Sobolev inequality). Detailedly, we first, construct a regularized prob-lem, which keeps the dissipation property of free energy; Then, give the uniform estimates for approximate solutions; Finally, use the Lions-Aubin lemma to give compactness argu-ment, and prove the existence of weak solutions.