Existence of traveling wave solutions of parabolic–parabolic chemotaxis systems

The current paper is devoted to the study of traveling wave solutions of the following parabolic–parabolicchemotaxis system, ut=Δu−χ∇⋅(u∇v)+u(a−bu),x∈RNτvt=Δv−v+u,x∈RN,where u(x,t) represents the population density of a mobile species and v(x,t) represents the population density of a chemoattractant, and χ represents the chemotaxis sensitivity. In an earlier work (Rachidi et al., 2017) by the authors of the current paper, traveling wave solutions of the above chemotaxis system with τ=0 are studied. It is shown in Rachidi et al. (2017) that for every 00, there is0