Traveling wave solutions of a singular Keller-Segel system with logistic source (Q2688407)

The author consider special solutions of the one-dimensional (partially dissipative) singular chemotaxis model with logistic source \[ u_t=(Du_x-\chi\frac{u}{c}c_x)_x+\mu u(1-u), \] \[ c_t=\varepsilon c_{xx}-uc, \] in both cases: viscous (\(\varepsilon>0\)) and inviscid (\(\varepsilon=0\)) in the second equation. The analysis of traveling waves is performed together with their stability properties. In particular, dependence on the parameter \(\varepsilon\searrow 0\) is studied, i.e. when the parabolic reaction-diffusion system degenerates into parabolic-ordinary differential equations system.