Existence and stability of boundary spike layer solutions of an attractive chemotaxis model with singular sensitivity and nonlinear consumption rate of chemical stimuli

This paper is devoted to the study of the existence and stability of non-trivial steady state solutions to the following coupled system of PDEs on the half-line $R^{+}=(0,\infty )$: $u_t=u_{xx}-\chi {\left[u{(lnw)}_x\right]}_x,$$w_t=\varepsilon w_{xx}-u^{\gamma }{w}^m,$ which is a model of chemotaxis of Keller–Segel type. When $u$ is subject to the no-flux boundary condition, $w$ equals a positive value at the origin, and assuming the functions vanish at the far field, a unique steady state $(U,W)$ is constructed under suitable restrictions on the system parameters, which is capable of describing fundamental phenomena in chemotaxis, such as spatial aggregation. Moreover, the steady state is shown to be nonlinearly asymptotically stable if $(u_0-U)$ carries zero mass, $w_0\left(x\right)$ matches $W\left(x\right)$ at the far field, and the initial perturbation is sufficiently small in weighted Sobolev spaces.