Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source

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• 作者：G. Viglialoro ^{giuseppe.viglialoro@unica.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear parabolic systems ; Chemotaxis ; Logistic source ; Boundedness
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：34
• 期：Complete
• 页码：520-535
• 全文大小：707 K

文摘

In this paper we study the chemotaxis-system defined in a convex smooth and bounded domain a1f52f380e355c9" title="Click to view the MathML source">Ω$\Omega$ of a1fa36d23816" title="Click to view the MathML source">R³${\mathbb{R}}^3$, with a160730a92ce3f3abbaeb1b6" title="Click to view the MathML source">χ>0$\chi >0$ and endowed with homogeneous Neumann boundary conditions. The source g$g$ behaves similarly to the logistic function and verifies 9f806d5d12fe346fe1cdd082c7124d4" title="Click to view the MathML source">g(s)≤a−bs^α$g\left(s\right)\le a-b{s}^{\alpha }$, for s≥0$s\ge 0$, with 9fbd3ec4d2c719278" title="Click to view the MathML source">a≥0$a\ge 0$, ae9ecabc7c0a" title="Click to view the MathML source">b>0$b>0$ and α>1$\alpha >1$. In line with Viglialoro (2016), where for a162e145d2084b404ce802">$\alpha \in (\frac{5}{3},2)$ the global existence of very weak solutions (u,v)$(u,v)$ to the system is shown for any nonnegative initial data 9fc1701418596e597a6f7">$(u_0,v_0)\in C^0\left(\bar{\Omega }\right)\times C^2\left(\bar{\Omega }\right)$ and under zero-flux boundary condition on a18152c972549dfe65472f47343f9172" title="Click to view the MathML source">v₀$v_0$, we prove that no chemotactic collapse for these solutions may present over time. More precisely, we establish that if the ratio $\frac{a}{b}$ does not exceed a certain value and for 9fb9f2b37691502">$\frac{9}{5}<p<\alpha <2$ the initial data are such that 9342d4" title="Click to view the MathML source">‖u₀‖_L^p(Ω)${\Vert u_0\Vert }_{L^p\left(\Omega \right)}$ and ‖∇v₀‖_L⁴(Ω)${\Vert \nabla v_0\Vert }_{L^4\left(\Omega \right)}$ are small enough, then (u,v)$(u,v)$ is uniformly-in-time bounded.