A Time Asymptotics of a Solution to 1D Keller–Segel System on a Bounded Interval

In this paper, the time asymptotic behaviour of solutions (u, v) of a one-dimensional parabolic-parabolic type Keller–Segel system defined on the bounded interval $$I=(-\frac{\pi }{2}, \frac{\pi }{2})$$ , with the Neumann boundary condition, is considered. The system describes the phenomenon such that the cellular slime molds form an aggregation by the chemotaxis movement. It is shown that, there is $$\chi _0 >0$$ such that if $$0 \le \chi < \chi _0$$ , where $$\chi $$ reflects the sensitivity of the chemotactic response to the chemical substance, then a mild solution (u, v) of the system converges uniformly to a pair of positive constants $$(M, \frac{\alpha M}{\gamma })$$ , a stationary solution, where $$\alpha $$ and $$\gamma $$ are also the coefficients of the system, and M is a constant which only depends on $$u_0$$ , where $$u_0$$ is the initial data of u. Also, the convergence rates of the solution (u, v), and of $$\partial _x u$$ , $$\partial _x v$$ , $$\partial _{x}^2 v$$ to the value of the stationary solution are explicitly given.