A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

作者：Mizukami Masaaki; Yokota Tomomi^*

来源：Mathematische Nachrichten, 2017, 290(16): 2648-2660.

DOI：10.1002/mana.201600394

• chemotaxis
• sensitivity function
• global existence
• boundedness

This paper deals with the Keller-Segel system
{u(t) = Delta u - del . u chi(v)del v), x is an element of Omega, t > 0,
v(t) = Delta v + u - v, x is an element of Omega, t > 0,
where Omega is a bounded domain in R-n with smooth boundary partial derivative Omega, n >= 2; chi is a nonnegative function satisfying chi(s) <= K(a + s)(-k) for some k >= 1 and a >= 0. In the case that k = 1 and a = 0, Fujie [2] established global existence of bounded solutions under the condition 0 < K < root 2/n. On the other hand, when k > 1, Winkler [14] asserted global existence of bounded solutions for arbitrary k > 0. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary K > 0. Moreover, the condition for K when k > 1 cannot connect with the condition when k = 1. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for chi and to build a mathematical bridge between the cases k = 1 and k > 1.

• 出版日期2017-11