On a chemotaxis model with singular sensitivity: Convergence rate towards spiky steady state

This paper is concerned with the stability analysis for a model of chemotaxis:$ \begin{equation*} \begin{aligned} u_t & = u_{xx} - \chi[u(\ln w)_x]_x, &(x, t)&\in \mathbb{R}_+ \times \mathbb{R}_+, \\ w_t & = \varepsilon w_{xx} - u^\gamma w^m, &(x, t)&\in \mathbb{R}_+ \times \mathbb{R}_+. \end{aligned} \end{equation*} $By utilizing spatially weighted energy method, it is shown that the non-trivial steady state associated with the model under certain parametric constraints and the boundary conditions:$ \begin{equation*} \begin{aligned} &u_x = \chi u(\ln w)_x, \ \ w = b>0, &&(x, t) \in \{0\}\times \mathbb{R}_+, \\ &(u, w) \to (0, 0), && x\to\infty, \ t\in \mathbb{R}_+, \end{aligned} \end{equation*} $is locally asymptotically stable. By exploring the idea of 'gaining temporal integrability through faster spatial decay', the explicit (algebraic) convergence rate, in terms of the system parameters, of general solutions of the initial-boundary value problem towards the steady state is identified.