Abstract
Abstract This paper is concerned with a chemotaxis system u t = Δ u m − ∇ ⋅ ( χ 1 ( w ) u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v n − ∇ ⋅ ( χ 2 ( w ) v ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , x ∈ Ω , t > 0 , w t = Δ w − ( α u + β v ) w , x ∈ Ω , t > 0 , \left\{\begin{array}{ll}{u}_{t}=\Delta {u}^{m}-\nabla \cdot \left({\chi }_{1}\left(w)u\nabla w)+{\mu }_{1}u\left(1-u-{a}_{1}v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta {v}^{n}-\nabla \cdot \left({\chi }_{2}\left(w)v\nabla w)+{\mu }_{2}v\left(1-{a}_{2}u-v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {w}_{t}=\Delta w-\left(\alpha u+\beta v)w,& x\in \Omega ,\hspace{0.33em}t\gt 0,\end{array}\right. under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 3 \Omega \subset {{\mathbb{R}}}^{3} with smooth boundary, where μ 1 , μ 2 > 0 {\mu }_{1},{\mu }_{2}\gt 0 , a 1 , a 2 > 0 {a}_{1},{a}_{2}\gt 0 , α , β > 0 \alpha ,\beta \gt 0 , and the chemotactic sensitivity function χ i ∈ C 1 ( [ 0 , ∞ ) ) {\chi }_{i}\in {C}^{1}({[}0,\infty )) , χ i ′ ≥ 0 {\chi }_{i}^{^{\prime} }\ge 0 . It is proved that for any large initial data, for any m , n > 1 m,n\gt 1 , the system admits a global weak solution, which is uniformly bounded.
Highlights
Chemotaxis refers to the effect of chemical substances in the environment on the movement of species
This can lead to strict directional movement or partial orientation and partial tumbling movement
Chemotaxis is an important means of cellular communication
Summary
Chemotaxis refers to the effect of chemical substances in the environment on the movement of species. When D(u) ≡ 1, in the three-dimensional case, Zheng et al proved that the system (2) with a logistic type source μu(1 − u) admits a unique global classical solution if the initial data of w are small in [5]. If N ≤ 2, if N ≥ 3, where λ0 is a positive constant which is corresponding to the maximal Sobolev regularity, for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. Global existence and boundedness 951 if χi (w) = χi are positive constants They considered asymptotic behavior of solutions to the system: when a1, a2 ∈ (0, 1), u(⋅, t) → 1 − a1 , v(⋅, t) → 1 − a2 , w → α(1 − a1) + β(1 − α2) in L∞(Ω) as t → ∞;.
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