A quasi‐linear parabolic system of chemotaxis

We consider a quasi‐linear parabolic system with respect to unknown functions u and v on a bounded domain of n ‐dimensional Euclidean space. We assume that the diffusion coefficient of u is a positive smooth function A ( u ), and that the diffusion coefficient of v is a positive constant. If A ( u ) is a positive constant, the system is referred to as so‐called Keller‐Segel system. In the case where the domain is a bounded domain of two‐dimensional Euclidean space, it is shown that some solutions to Keller‐Segel system blow up in finite time. In three and more dimensional cases, it is shown that solutions to so‐called Nagai system blow up in finite time. Nagai system is introduced by Nagai. The diffusion coefficients of Nagai system are positive constants. In this paper, we describe that solutions to the quasi‐linear parabolic system exist globally in time, if the positive function A ( u ) rapidly increases with respect to u .