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 8121816301225&_mathId=si2.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=d9b8aa2e84d3b676fa1f52f380e355c9" title="Click to view the MathML source">Ω$\Omega$ of 8121816301225&_mathId=si3.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=8d9616131a586987a7b7a1fa36d23816" title="Click to view the MathML source">R³${\mathbb{R}}^3$, with 8121816301225&_mathId=si4.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=7d106e8ca160730a92ce3f3abbaeb1b6" title="Click to view the MathML source">χ>0$\chi >0$ and endowed with homogeneous Neumann boundary conditions. The source 8121816301225&_mathId=si5.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=bfb45a44b9e720dadf9275c9de4ea201" title="Click to view the MathML source">g$g$ behaves similarly to the logistic function and verifies 8121816301225&_mathId=si6.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=99f806d5d12fe346fe1cdd082c7124d4" title="Click to view the MathML source">g(s)≤a−bs^α$g\left(s\right)\le a-b{s}^{\alpha }$, for 8121816301225&_mathId=si7.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=07548ccbf123895e9dedaf5e86c8bf2e" title="Click to view the MathML source">s≥0$s\ge 0$, with 8121816301225&_mathId=si8.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=c8a5418049acc909fbd3ec4d2c719278" title="Click to view the MathML source">a≥0$a\ge 0$, 8121816301225&_mathId=si9.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=124ddd9da767694658f8ae9ecabc7c0a" title="Click to view the MathML source">b>0$b>0$ and 8121816301225&_mathId=si10.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=768f44717dfd1b7c521302687370734a" title="Click to view the MathML source">α>1$\alpha >1$. In line with Viglialoro (2016), where for 8121816301225&_mathId=si11.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=df8d322462a162e145d2084b404ce802">8121816301225-si11.gif">$\alpha \in (\frac{5}{3},2)$ the global existence of very weak solutions 8121816301225&_mathId=si12.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v)$(u,v)$ to the system is shown for any nonnegative initial data 8121816301225&_mathId=si13.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=0e5852bdf679fc1701418596e597a6f7">8121816301225-si13.gif">$(u_0,v_0)\in C^0\left(\bar{\Omega }\right)\times C^2\left(\bar{\Omega }\right)$ and under zero-flux boundary condition on 8121816301225&_mathId=si14.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=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 8121816301225&_mathId=si15.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=01c287b32702defd0b7a61fc0df50ec1">8121816301225-si15.gif">$\frac{a}{b}$ does not exceed a certain value and for 8121816301225&_mathId=si16.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=26c17d77bf05bbd049fb9f2b37691502">8121816301225-si16.gif">$\frac{9}{5}<p<\alpha <2$ the initial data are such that 8121816301225&_mathId=si17.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=28faa0bfc00b53884d83624a329342d4" title="Click to view the MathML source">‖u₀‖_L^p(Ω)${\Vert u_0\Vert }_{L^p\left(\Omega \right)}$ and 8121816301225&_mathId=si18.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=14e3e9e487fd9a712c0ac38223e97025" title="Click to view the MathML source">‖∇v₀‖_L⁴(Ω)${\Vert \nabla v_0\Vert }_{L^4\left(\Omega \right)}$ are small enough, then 8121816301225&_mathId=si12.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v)$(u,v)$ is uniformly-in-time bounded.