Stability of solutions of chemotaxis equations in reinforced random walks (Q1849184)

The authors consider the following nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation: \[ \begin{aligned} \frac{\partial p}{\partial t}&= \operatorname {div}(\nabla p-p\chi (w) \nabla w), \quad x\in\Omega, \;t\geq 0,\\ \frac{\partial w}{\partial t}&= g(p, w), \quad x\in\Omega, \;t\geq 0,\end{aligned} \] with some boundary condition. Here \(\Omega\) is a domain in a Euclidean space, \(g\) and \(\chi\) are given scalar functions. This system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher, or lower, concentrations of the chemical substance, or by aggregating or dispersing. It is proved that stationary solutions of the system are asymptotically stable.