On the logarithmic Keller-Segel-Fisher/KPP system

Abstract

We consider the Cauchy problem of a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. We study the global well-posedness, long-time behavior, vanishing coefficient limit and decay rate of solutions in \begin{document}$ \mathbb{R} $\end{document} . By utilizing energy methods, we show that for any given classical initial datum which is a perturbation around a constant equilibrium state with finite energy (not small), there exists a unique global-in-time solution to the Cauchy problem, and the solution converges to the constant equilibrium state, as time goes to infinity. Under the same initial condition, it is shown that the solution with positive chemical diffusion coefficient converges to the solution with zero chemical diffusion coefficient, as the coefficient goes to zero. Furthermore, for a slightly smaller class of initial data, we identify the algebraic decay rates of the solution to the constant equilibrium state by employing time-weighted energy estimates.