Boundedness in a two-species chemotaxis-consumption system with nonlinear diffusion and sensitivity. Issue 16 (10th December 2021) - Zhang, Jing; Hu, Xuegang
ABSTRACT: This paper deals with the following two-species chemotaxis system 1 u t = ∇ ⋅ ( D 1 ( u ) ∇ u ) − ∇ ⋅ ( S 1 ( u ) ∇ w ) + μ 1 u ( 1 − u − a 1 v ), x ∈ Ω, t > 0, v t = ∇ ⋅ ( D 2 ( v ) ∇ v ) − ∇ ⋅ ( S 2 ( v ) ∇ w ) + μ 2 v ( 1 − a 2 u − v ), x ∈ Ω, t > 0, w t = Δ w − ( u + v ) w, x ∈ Ω, t > 0 with homogeneous Neumann boundary conditions and suitable initial conditions, where Ω ⊂ R n ( n ≥ 1 ) is a bounded and smooth domain. Moreover, the parameters μ 1, μ 2 > 0, and a 1, a 2 > 0 . For i = 1, 2, we assume that D i ( s ) ≥ c D i ( s + 1 ) m i − 1 a n d S i ( s ) ≤ c S i ( s + 1 ) q i − 1 w i t h S i ( s ) ≥ 0 f o r a l l s ≥ 0, where c D i > 0, c S i > 0, m i ∈ R and q i ∈ R . Then if one of the following cases holds: q i < m i 2 + 2 − n n + 2 ; q i < m i 2 + 3 2 and μ i is sufficiently large, it is proved that the system possesses a unique global bounded classical solution.