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, u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + u ( a − b u ) , x ∈ R N τ v t = Δ v − v + u , x ∈ R N , 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 < b 2 , there is c ∗ ( χ ) such that for every c > c ∗ ( χ ) and ξ ∈ S N − 1 , the system has a traveling wave solution ( u ( x , t ) , v ( x , t ) ) = ( U ( x ⋅ ξ − c t ; τ ) , V ( x ⋅ ξ − c t ; τ ) ) with speed c connecting the constant solutions ( a b , a b ) and ( 0 , 0 ) . Moreover, lim χ → 0 + c ∗ ( χ ) = 2 a if 0 < a ≤ 1 1 + a if a > 1 . We prove in the current paper that for every τ > 0 , there is 0 < χ τ ∗ < b 2 such that for every 0 < χ < χ τ ∗ , there exist two positive numbers c ∗ ∗ ( χ , τ ) > c ∗ ( χ , τ ) ≥ 2 a satisfying that for every c ∈ ( c ∗ ( χ , τ ) , c ∗ ∗ ( χ , τ ) ) and ξ ∈ S N − 1 , the system has a traveling wave solution ( u ( x , t ) , v ( x , t ) ) = ( U ( x ⋅ ξ − c t ; τ ) , V ( x ⋅ ξ − c t ; τ ) ) with speed c connecting the constant solutions ( a b , a b ) and ( 0 , 0 ) , and it does not have such traveling wave solutions of speed less than 2 a . Moreover, lim χ → 0 + c ∗ ∗ ( χ , τ ) = ∞ , lim χ → 0 + c ∗ ( χ , τ ) = 2 a if 0 < a ≤ 1 + τ a ( 1 − τ ) + 1 + τ a ( 1 − τ ) + + a ( 1 − τ ) + 1 + τ a if a ≥ 1 + τ a ( 1 − τ ) + , and lim x → ∞ U ( x ; τ ) e − μ x = 1 , where μ is the only solution of the equation μ + a μ = c in the interval ( 0 , min { a , 1 + τ a ( 1 − τ ) + } ) . Furthermore, lim τ → 0 + χ τ ∗ = b 2 , lim τ → 0 + c ∗ ( χ ; τ ) = c ∗ ( χ ) , lim τ → 0 + c ∗ ∗ ( χ ; τ ) = ∞ . [ABSTRACT FROM AUTHOR]