I
n this
pa
per we study the chemotaxis-system
defi
ned i
n a co
nvex smooth a
nd bou
nded domai
n pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si2.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=d9b8aa2e84d3b676fa1f52f380e355c9" title="Click to view the MathML source">Ωpan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> of
pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si3.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=8d9616131a586987a7b7a1fa36d23816" title="Click to view the MathML source">Rp>3p>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n>, with
pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si4.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=7d106e8ca160730a92ce3f3abbaeb1b6" title="Click to view the MathML source">χ>0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> a
nd e
ndowed with homoge
neous Neuma
nn bou
ndary co
nditio
ns. The source
pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si5.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=bfb45a44b9e720dadf9275c9de4ea201" title="Click to view the MathML source">gpan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> behaves similarly to the logistic fu
nctio
n a
nd verifies
pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si6.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=99f806d5d12fe346fe1cdd082c7124d4" title="Click to view the MathML source">g(s)≤a−bsp>αp>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n>, for
pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si7.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=07548ccbf123895e9dedaf5e86c8bf2e" title="Click to view the MathML source">s≥0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n>, with
pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si8.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=c8a5418049acc909fbd3ec4d2c719278" title="Click to view the MathML source">a≥0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n>,
pan id="mmlsi9" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si9.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=124ddd9da767694658f8ae9ecabc7c0a" title="Click to view the MathML source">b>0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> a
nd
pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si10.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=768f44717dfd1b7c521302687370734a" title="Click to view the MathML source">α>1pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n>. I
n li
ne with Viglialoro (2016), where for
pan id="mmlsi11" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si11.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=df8d322462a162e145d2084b404ce802">nlineImage" height="18" width="59" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816301225-si11.gif"><noscript>n:bottom" width="59" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816301225-si11.gif">noscript>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> the global existe
nce of very weak solutio
ns
pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si12.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> to the system is show
n for a
ny
no
nnegative i
nitial data
pan id="mmlsi13" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si13.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=0e5852bdf679fc1701418596e597a6f7">nlineImage" height="15" width="153" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816301225-si13.gif"><noscript>n:bottom" width="153" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816301225-si13.gif">noscript>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> a
nd u
nder zero-flux bou
ndary co
nditio
n o
n pan id="mmlsi14" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si14.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=a18152c972549dfe65472f47343f9172" title="Click to view the MathML source">v0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n>, we
prove that
no chemotactic colla
pse for these solutio
ns may
prese
nt over time. More
precisely, we establish that if the ratio
pan id="mmlsi15" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si15.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=01c287b32702defd0b7a61fc0df50ec1">nlineImage" height="15" width="8" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816301225-si15.gif"><noscript>n:bottom" width="8" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816301225-si15.gif">noscript>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> does
not exceed a certai
n value a
nd for
pan id="mmlsi16" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si16.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=26c17d77bf05bbd049fb9f2b37691502">nlineImage" height="17" width="89" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816301225-si16.gif"><noscript>n:bottom" width="89" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816301225-si16.gif">noscript>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> the i
nitial data are such that
pan id="mmlsi17" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si17.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=28faa0bfc00b53884d83624a329342d4" title="Click to view the MathML source">‖u0‖Lp>pp>(Ω)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> a
nd
pan id="mmlsi18" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si18.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=14e3e9e487fd9a712c0ac38223e97025" title="Click to view the MathML source">‖∇v0‖Lp>4p>(Ω)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> are small e
nough, the
n pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si12.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pa
n> is u
niformly-i
n-time bou
nded.