Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source

Abstract

Abstract In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: { u t = Δ ( γ ( v ) u ) + μ u ( 1 − u α ) , x ∈ Ω , t > 0 , 0 = Δ v − v + u r , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω $$ \textstyle\begin{cases} u_{t}=\Delta {\bigl(\gamma (v)u\bigr)}+\mu u\bigl(1-u^{\alpha}\bigr), &x \in \Omega , t > 0, \\ 0=\Delta v-\ v+u^{r} , &x\in \Omega , t > 0, \\ u(x, 0) = u_{0}(x), &x\in \Omega \end{cases} $$ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 $\Omega \subset \mathbb{R}^{2}$ , where the motility function γ ( v ) $\gamma (v)$ satisfies γ ( v ) ∈ C 3 ( [ 0 , ∞ ) ) $\gamma (v)\in C^{3}([0,\infty ))$ with γ ( v ) > 0 $\gamma (v)>0$ , and | γ ′ ( v ) | 2 γ ( v ) $\frac{|\gamma '(v)|^{2}}{\gamma (v)}$ is bounded for all v > 0 $v > 0$ . The purpose of this paper is to prove that the model possesses globally bounded solutions. In addition, we show that all solutions ( u , v ) $(u, v)$ of the model will exponentially converge to the unique constant steady state ( 1 , 1 ) $(1, 1)$ as t → + ∞ $t\rightarrow +\infty $ when μ ≥ K 4 1 + r $\mu \geq \frac{K}{4^{1+r}}$ with K = max 0 < v ≤ ∞ | γ ′ ( v ) | 2 γ ( v ) $K=\max_{0< v\leq \infty} \frac{|\gamma '(v)|^{2}}{\gamma (v)}$ .

Keywords

• Global boundedness
• Logistic source
• Chemotaxis
• Signal-dependent motility
• Exponential decay