In thi
s p
aper we
study the chemot
axi
s-
sy
stem
ass="formul
a" id="fd000005">
ass="m
athml"><
sp
an id="mml
si1" cl
ass="m
athml
src"><
a title="View the M
athML
source" cl
ass="m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si1.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=323b8e5c1770bf7923f807689d71533d">
ass="imgL
azyJSB inlineIm
age" height="34" width="275"
alt="View the M
athML
source" title="View the M
athML
source"
src="/
sd/grey_pxl.gif" d
at
a-inlimgeid="1-
s2.0-S1468121816301
225-
si1.gif">
script>style="vertical-align:bottom" width="275" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816301225-si1.gif">script>
a><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si1.gif" overflow="scroll">{able class="cases">sub>utsub>=Δu−χ∇⋅(u∇v)+g(u)x∈Ω,t>0,sub>vtsub>=Δv−v+ux∈Ω,t>0,able>ath>
sp
an>
sp
an>
sp
an>
ass="temp"
src="/
sd/bl
ank.gif">
defined in
a convex
smooth
and bounded dom
ain <
sp
an id="mml
si2" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si2.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=d9b8
aa2e84d3b676f
a1f52f380e355c9" title="Click to view the M
athML
source">Ω
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si2.gif" overflow="scroll">Ωath>
sp
an>
sp
an>
sp
an> of <
sp
an id="mml
si3" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si3.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=8d9616131
a586987
a7b7
a1f
a36d23816" title="Click to view the M
athML
source">R<
sup>3
sup>
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si3.gif" overflow="scroll">sup>athvariant="double-struck">R3sup>ath>
sp
an>
sp
an>
sp
an>, with <
sp
an id="mml
si4" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si4.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=7d106e8c
a160730
a92ce3f3
abb
aeb1b6" title="Click to view the M
athML
source">χ>0
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si4.gif" overflow="scroll">χ>0ath>
sp
an>
sp
an>
sp
an>
and endowed with homogeneou
s Neum
ann bound
ary condition
s. The
source <
sp
an id="mml
si5" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si5.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=bfb45
a44b9e720d
adf9275c9de4e
a201" title="Click to view the M
athML
source">g
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si5.gif" overflow="scroll">gath>
sp
an>
sp
an>
sp
an> beh
ave
s simil
arly to the logi
stic function
and verifie
s <
sp
an id="mml
si6" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si6.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=99f806d5d12fe346fe1cdd082c7124d4" title="Click to view the M
athML
source">g(
s)≤
a&minu
s;b
s<
sup>α
sup>
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si6.gif" overflow="scroll">g(s)≤a−bsup>sαsup>ath>
sp
an>
sp
an>
sp
an>, for <
sp
an id="mml
si7" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si7.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=07548ccbf123895e9ded
af5e86c8bf2e" title="Click to view the M
athML
source">
s≥0
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si7.gif" overflow="scroll">s≥0ath>
sp
an>
sp
an>
sp
an>, with <
sp
an id="mml
si8" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si8.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=c8
a5418049
acc909fbd3ec4d2c719278" title="Click to view the M
athML
source">
a≥0
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si8.gif" overflow="scroll">a≥0ath>
sp
an>
sp
an>
sp
an>, <
sp
an id="mml
si9" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si9.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=124ddd9d
a767694658f8
ae9ec
abc7c0
a" title="Click to view the M
athML
source">b>0
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si9.gif" overflow="scroll">b>0ath>
sp
an>
sp
an>
sp
an>
and <
sp
an id="mml
si10" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si10.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=768f44717dfd1b7c521302687370734
a" title="Click to view the M
athML
source">α>1
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si10.gif" overflow="scroll">α>1ath>
sp
an>
sp
an>
sp
an>. In line with Vigli
aloro (2016), where for <
sp
an id="mml
si11" cl
ass="m
athml
src"><
a title="View the M
athML
source" cl
ass="m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si11.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=df8d322462
a162e145d2084b404ce802">
ass="imgL
azyJSB inlineIm
age" height="18" width="59"
alt="View the M
athML
source" title="View the M
athML
source"
src="/
sd/grey_pxl.gif" d
at
a-inlimgeid="1-
s2.0-S1468121816301
225-
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>
a><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si11.gif" overflow="scroll">α∈(ac>53ac>,2)ath>
sp
an>
sp
an>
sp
an> the glob
al exi
stence of very we
ak
solution
s <
sp
an id="mml
si12" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si12.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the M
athML
source">(u,v)
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si12.gif" overflow="scroll">(u,v)ath>
sp
an>
sp
an>
sp
an> to the
sy
stem i
s shown for
any nonneg
ative initi
al d
at
a <
sp
an id="mml
si13" cl
ass="m
athml
src"><
a title="View the M
athML
source" cl
ass="m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si13.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=0e5852bdf679fc1701418596e597
a6f7">
ass="imgL
azyJSB inlineIm
age" height="15" width="153"
alt="View the M
athML
source"
style="m
argin-top: -5px; vertic
al-
align: middle" title="View the M
athML
source"
src="/
sd/grey_pxl.gif" d
at
a-inlimgeid="1-
s2.0-S1468121816301
225-
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>
a><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si13.gif" overflow="scroll">(sub>u0sub>,sub>v0sub>)∈sup>C0sup>(accent="true">Ω̄)×sup>C2sup>(accent="true">Ω̄)ath>
sp
an>
sp
an>
sp
an>
and under zero-flux bound
ary condition on <
sp
an id="mml
si14" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si14.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=
a18152c972549dfe65472f47343f9172" title="Click to view the M
athML
source">v<
sub>0
sub>
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si14.gif" overflow="scroll">sub>v0sub>ath>
sp
an>
sp
an>
sp
an>, we prove th
at no chemot
actic coll
ap
se for the
se
solution
s m
ay pre
sent over time. More preci
sely, we e
st
abli
sh th
at if the r
atio <
sp
an id="mml
si15" cl
ass="m
athml
src"><
a title="View the M
athML
source" cl
ass="m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si15.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=01c287b32702defd0b7
a61fc0df50ec1">
ass="imgL
azyJSB inlineIm
age" height="15" width="8"
alt="View the M
athML
source"
style="m
argin-top: -5px; vertic
al-
align: middle" title="View the M
athML
source"
src="/
sd/grey_pxl.gif" d
at
a-inlimgeid="1-
s2.0-S1468121816301
225-
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>
a><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si15.gif" overflow="scroll">ac>abac>ath>
sp
an>
sp
an>
sp
an> doe
s not exceed
a cert
ain v
alue
and for <
sp
an id="mml
si16" cl
ass="m
athml
src"><
a title="View the M
athML
source" cl
ass="m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si16.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=26c17d77bf05bbd049fb9f2b37691502">
ass="imgL
azyJSB inlineIm
age" height="17" width="89"
alt="View the M
athML
source" title="View the M
athML
source"
src="/
sd/grey_pxl.gif" d
at
a-inlimgeid="1-
s2.0-S1468121816301
225-
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>
a><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si16.gif" overflow="scroll">ac>95ac><p<α<2ath>
sp
an>
sp
an>
sp
an> the initi
al d
at
a are
such th
at <
sp
an id="mml
si17" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si17.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=28f
aa0bfc00b53884d83624
a329342d4" title="Click to view the M
athML
source">‖u<
sub>0
sub>‖<
sub>L<
sup>p
sup>(Ω)
sub>
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si17.gif" overflow="scroll">sub>‖sub>u0sub>‖sup>Lpsup>(Ω)sub>ath>
sp
an>
sp
an>
sp
an>
and <
sp
an id="mml
si18" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si18.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=14e3e9e487fd9
a712c0
ac38223e97025" title="Click to view the M
athML
source">‖&n
abl
a;v<
sub>0
sub>‖<
sub>L<
sup>4
sup>(Ω)
sub>
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si18.gif" overflow="scroll">sub>‖∇sub>v0sub>‖sup>L4sup>(Ω)sub>ath>
sp
an>
sp
an>
sp
an>
are
sm
all enough, then <
sp
an id="mml
si12" cl
ass="m
athml
src"><
sp
an cl
ass="formul
atext
stixSupport m
athImg" d
at
a-m
athURL="/
science?_ob=M
athURL&_method=retrieve&_eid=1-
s2.0-S1468121816301
225&_m
athId=
si12.gif&_u
ser=111111111&_pii=S1468121816301
225&_rdoc=1&_i
ssn=14681218&md5=02cbdf8dbeb63e8e3e2b953f3867c909" title="Click to view the M
athML
source">(u,v)
sp
an><
sp
an cl
ass="m
athCont
ainer hidden"><
sp
an cl
ass="m
athCode">
ath altimg="si12.gif" overflow="scroll">(u,v)ath>
sp
an>
sp
an>
sp
an> i
s uniformly-in-time bounded.