Asymptotic convergence of solutions of Keller–Segel equations in network shaped domains

Abstract

The objective of this study is to examine the classical Keller–Segel equations in network shaped domains. First, the existence of unique strict global solutions of the equations is demonstrated by using the theory of strongly elliptic differential operators in network shaped domains and the theory of abstract parabolic semilinear evolution equations. Furthermore, by considering an angle condition as well as the Łojasiewicz–Simon inequality, it is proved that every global solution converges to a stationary solution, and the convergence is dominated by the values of a Lyapunov function along the global solution.