Finite-time blowup in a Cauchy problem of parabolic-parabolic chemotaxis system

2018/06/01 Fri 15:30 - 17:30
溝口 紀子

This talk is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation in the whole plane. Although some results on blowup in the system were given in a disk, there have been no results in the whole plane except existence of a special solution constructed by Schweyer. In the case of a disk, solutions blow up in finite time if their initial energy is less than some specific value. On the other hand, the energy diverges to $-\infty$ as time goes to $+\infty$ for any forward selfsimilar solution in the plane. This implies that one cannot expect to get a criterion for finite-time blowup using energy. For a solution $(u,v)$, $u$ and $v$ denote density of cells and of chemical substance, respectively. Let $\tau$ be the coefficient of time derivative of $v$. I will talk that there exists $M(\tau)>0$ with $M(\tau) \to \infty$ as $\tau \to \infty$ such that all solutions $(u,v)$ with initial mass of $u$ larger than $M(\tau)$ blow up in finite time.