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Existence of traveling wave solutions of parabolic–parabolic chemotaxis systems

2018; Elsevier BV; Volume: 42; Linguagem: Inglês

10.1016/j.nonrwa.2017.12.004

1878-5719

Rachidi B. Salako, Wenxian Shen,

Evolution and Genetic Dynamics

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 0<χ c∗(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)=2aif0 1.We prove in the current paper that for every τ>0, there is0<χτ∗<b2 such that for every 0<χ 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<a≤1+τa(1−τ)+1+τa(1−τ)++a(1−τ)+1+τaifa≥1+τa(1−τ)+, and limx→∞U(x;τ)e−μx=1,where μ is the only solution of the equation μ+aμ=c in the interval (0,min{a,1+τa(1−τ)+}). Furthermore, limτ→0+χτ∗=b2,limτ→0+c∗(χ;τ)=c∗(χ),limτ→0+c∗∗(χ;τ)=∞.