In thi
s paper we
study the chemotaxi
s-
sy
stem
ss="formula" id="fd000005">
defined in a convex
smooth and bounded domain <
span id="mml
si2" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si2.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=d9b8aa2e84d3b676fa1f52f380e355c9" title="Click to view the MathML
source">Ω
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> of <
span id="mml
si3" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si3.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=8d9616131a586987a7b7a1fa36d23816" title="Click to view the MathML
source">R<
sup>3
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, with <
span id="mml
si4" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si4.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=7d106e8ca160730a92ce3f3abbaeb1b6" title="Click to view the MathML
source">χ>0
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and endowed with homogeneou
s Neumann boundary condition
s. The
source <
span id="mml
si5" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si5.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=bfb45a44b9e720dadf9275c9de4ea201" title="Click to view the MathML
source">g
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> behave
s similarly to the logi
stic function and verifie
s <
span id="mml
si6" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si6.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=99f806d5d12fe346fe1cdd082c7124d4" title="Click to view the MathML
source">g(
s)≤a&minu
s;b
s<
sup>α
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, for <
span id="mml
si7" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si7.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=07548ccbf123895e9dedaf5e86c8bf2e" title="Click to view the MathML
source">
s≥0
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, with <
span id="mml
si8" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si8.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=c8a5418049acc909fbd3ec4d2c719278" title="Click to view the MathML
source">a≥0
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, <
span id="mml
si9" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si9.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=124ddd9da767694658f8ae9ecabc7c0a" title="Click to view the MathML
source">b>0
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and <
span id="mml
si10" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si10.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=768f44717dfd1b7c521302687370734a" title="Click to view the MathML
source">α>1
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. In line with Viglialoro (2016), where for <
span id="mml
si11" cla
ss="mathml
src">
source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si11.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=df8d322462a162e145d2084b404ce802">
ss="imgLazyJSB inlineImage" 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">script>
style="vertical-align: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">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> the global exi
stence of very weak
solution
s <
span id="mml
si12" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si12.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML
source">(u,v)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> to the
sy
stem i
s shown for any nonnegative initial data <
span id="mml
si13" cla
ss="mathml
src">
source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si13.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=0e5852bdf679fc1701418596e597a6f7">
ss="imgLazyJSB inlineImage" 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">script>
style="vertical-align: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">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and under zero-flux boundary condition on <
span id="mml
si14" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si14.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=a18152c972549dfe65472f47343f9172" title="Click to view the MathML
source">v<
sub>0
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, we prove that no chemotactic collap
se for the
se
solution
s may pre
sent over time. More preci
sely, we e
stabli
sh that if the ratio <
span id="mml
si15" cla
ss="mathml
src">
source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si15.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=01c287b32702defd0b7a61fc0df50ec1">
ss="imgLazyJSB inlineImage" 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">script>
style="vertical-align: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">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> doe
s not exceed a certain value and for <
span id="mml
si16" cla
ss="mathml
src">
source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si16.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=26c17d77bf05bbd049fb9f2b37691502">
ss="imgLazyJSB inlineImage" 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">script>
style="vertical-align: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">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> the initial data are
such that <
span id="mml
si17" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si17.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=28faa0bfc00b53884d83624a329342d4" title="Click to view the MathML
source">‖u<
sub>0
sub>‖<
sub>L<
sup>p
sup>(Ω)
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and <
span id="mml
si18" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si18.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=14e3e9e487fd9a712c0ac38223e97025" title="Click to view the MathML
source">‖∇v<
sub>0
sub>‖<
sub>L<
sup>4
sup>(Ω)
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> are
small enough, then <
span id="mml
si12" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_mathId=
si12.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML
source">(u,v)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> i
s uniformly-in-time bounded.