Existence of traveling wave solutions of parabolic-parabolic chemotaxis systems

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 0<χc∗(χ) and ξ∈SN−1, the system has a traveling wave solution (u(x,t),v(x,t))=(U(x⋅ξ−ct;τ),V(x⋅ξ−ct;τ)) with speed c connecting the constant solutions (ab,ab) and (0,0). Moreover, limχ→0+c∗(χ)=2aif01.We prove in the current paper that for every τ>0, there is0<χτ∗c∗(χ,τ)≥2a satisfying that for every c∈(c∗(χ,τ),c∗∗(χ,τ)) and ξ∈SN−1, the system has a traveling wave solution (u(x,t),v(x,t))=(U(x⋅ξ−ct;τ),V(x⋅ξ−ct;τ)) with speed c connecting the constant solutions (ab,ab) and (0,0), and it does not have such traveling wave solutions of speed less than 2a. Moreover, limχ→0+c∗∗(χ,τ)=∞,limχ→0+c∗(χ,τ)=2aif0