Global boundedness in an attraction–repulsion Chemotaxis system with nonlinear productions and logistic source

Abstract

Abstract This paper deals with the attraction–repulsion chemotaxis system with nonlinear productions and logistic source, u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( Φ ( u ) ∇ v ) + ∇ ⋅ ( Ψ ( u ) ∇ w ) + f ( u ) , v t = Δ v + α u k − β v , τ w t = Δ w + γ u l − δ w , τ ∈ { 0 , 1 } , $$\begin{aligned}& u_{t} = \nabla \cdot \bigl( D(u) \nabla u \bigr) - \nabla \cdot \bigl( \Phi (u) \nabla v \bigr) + \nabla \cdot \bigl( \Psi (u) \nabla w \bigr) + f(u),\\& v_{t} = \Delta v+\alpha {{u}^{k}}-\beta v,\qquad \tau w_{t} = \Delta w+\gamma {{u}^{l}}-\delta w,\quad \tau \in \{0,1 \}, \end{aligned}$$ in a bounded domain Ω ⊂ R n $\Omega \subset {{\mathbb{R}}^{n}}$ ( n ≥ 1 $n \ge 1 $ ), subject to the homogeneous Neumann boundary conditions and initial conditions, where D , Φ , Ψ ∈ C 2 [ 0 , ∞ ) $D,\Phi ,\Psi \in {{C}^{2}}[0,\infty )$ are nonnegative with D ( s ) ≥ ( s + 1 ) p $D(s)\ge {{(s+1)}^{p}}$ for s ≥ 0 $s\ge 0$ , Φ ( s ) ≤ χ s q $\Phi (s)\le \chi {{s}^{q}}$ , ξ s g ≤ Ψ ( s ) ≤ ζ s j $\xi {{s}^{g}}\le \Psi (s) \le \zeta s^{j}$ , s ≥ s 0 $s\ge {{s}_{0}}$ , for s 0 > 1 ${{s}_{0}}>1$ , the logistic source satisfies f ( s ) ≤ s ( a − b s d ) $f(s)\le s(a-b{{s}^{d}})$ , s > 0 $s>0$ , f ( 0 ) ≥ 0 $f(0)\ge 0$ , and the nonlinear productions for the attraction and repulsion chemicals are described via α u k $\alpha {{u}^{k}}$ and γ u l $\gamma {{u}^{l}}$ , respectively. When k = l = 1 $k=l=1$ , it is known that this system possesses a globally bounded solution in some cases. However, there has been no work in the case k , l > 0 $k,l>0$ . This paper develops the global boundedness of the solution to the system in some cases and extends the global boundedness criteria established by Tian, He, and Zheng (2016) for the attraction–repulsion chemotaxis system.

Keywords

• Chemotaxis
• Attraction–repulsion
• Nonlinear productions
• Logistic source
• Boundedness
• Fully parabolic