Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source

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• 作者：Xiao He^a ; ^b ; ^{xiaohe@mail.dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Sining Zheng^b ; ^{snzheng@dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chemotaxis ; Global existence ; Large time behavior ; Logistic source ; Convergence rate
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：436
• 期：2
• 页码：970-982
• 全文大小：357 K

文摘

We study the global attractors to the chemotaxis system with logistic source: u_t−Δu+χ∇⋅(u∇v)=au−bu²$u_t-\mathrm{\Delta }u+\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)=au-b{u}^2$, τv_t−Δv=−v+u$\tau v_t-\mathrm{\Delta }v=-v+u$ in Ω×R⁺$\mathrm{\Omega }\times {\mathbb{R}}^{+}$, subject to the homogeneous Neumann boundary conditions, where smooth bounded domain Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$, with 466488f83eadb4aff9f1cfa9" title="Click to view the MathML source">χ,b>0$\chi ,b>0$, 467" title="Click to view the MathML source">a∈R$a\in \mathbb{R}$, and τ∈{0,1}$\tau \in \{0,1\}$. For the parabolic–elliptic case with τ=0$\tau =0$ and N>3$N>3$, we obtain that the positive constant equilibrium $(\frac{a}{b},\frac{a}{b})$ is a global attractor if a>0$a>0$ and $b>\max \{\frac{N-2}{N}\chi ,\frac{\chi \sqrt{a}}{4}\}$. Under the assumption b06dd123424087918fd38326b0aa6c" title="Click to view the MathML source">N=3$N=3$, it is proved that for either the parabolic–elliptic case with τ=0$\tau =0$, a>0$a>0$, 46a2d">$b>\max \{\frac{\chi }{3},\frac{\chi \sqrt{a}}{4}\}$, or the parabolic–parabolic case with τ=1$\tau =1$, a>0$a>0$, $b>\frac{\chi \sqrt{a}}{4}$ large enough, the system admits the positive constant equilibrium $(\frac{a}{b},\frac{a}{b})$ as a global attractor, while the trivial equilibrium (0,0)$(0,0)$ is a global attractor if a≤0$a\le 0$ and 46b5bce1da71f2a20eb665d00021" title="Click to view the MathML source">b>0$b>0$. It is pointed out that here the convergence rates are established for all of them. The results of the paper mainly rely on parabolic regularity theory and Lyapunov functionals carefully constructed.