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 <div class="formula" id="fd000005"><div class="mathml">n id="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301225&_mathId=si1.gif&_user=111111111&_pii=S1468121816301225&_rdoc=1&_issn=14681218&md5=323b8e5c1770bf7923f807689d71533d">nlineImage" height="34" width="275" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816301225-si1.gif"><noscript>der="0" 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">noscript>n class="mathContainer hidden">n class="mathCode">

{\begin{cases} d> u_{t} = Δ u &minus; χ &nabla; &sdot; (u &nabla; v) + g (u) & d> & d> x &isin; Ω, t > n>0 n>, & d> \\ d> v_{t} = Δ v &minus; v + u & d> & d> x &isin; Ω, t > n>0 n>, & d> \end{cases}

n>n>n>

d/blank.gif">div>div> defined in a convex smooth and bounded domain n id="mmlsi2" class="mathmlsrc">n 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">Ωn>n class="mathContainer hidden">n class="mathCode">

Ω

n>n>n> of n id="mmlsi3" class="mathmlsrc">n 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">R³n>n class="mathContainer hidden">n class="mathCode">

{nt="double-struck">R}^{n>3 n>}

n>n>n>, with n id="mmlsi4" class="mathmlsrc">n 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>0n>n class="mathContainer hidden">n class="mathCode">

χ > n>0 n>

n>n>n> and endowed with homogeneous Neumann boundary conditions. The source n id="mmlsi5" class="mathmlsrc">n 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">gn>n class="mathContainer hidden">n class="mathCode">

g

n>n>n> behaves similarly to the logistic function and verifies n id="mmlsi6" class="mathmlsrc">n 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−bs^αn>n class="mathContainer hidden">n class="mathCode">

g (s) \leq a &minus; b s^{α}

n>n>n>, for n id="mmlsi7" class="mathmlsrc">n 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≥0n>n class="mathContainer hidden">n class="mathCode">

s \geq n>0 n>

n>n>n>, with n id="mmlsi8" class="mathmlsrc">n 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≥0n>n class="mathContainer hidden">n class="mathCode">

a \geq n>0 n>

n>n>n>, n id="mmlsi9" class="mathmlsrc">n 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>0n>n class="mathContainer hidden">n class="mathCode">

b > n>0 n>

n>n>n> and n id="mmlsi10" class="mathmlsrc">n 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>1n>n class="mathContainer hidden">n class="mathCode">

α > n>1 n>

n>n>n>. In line with Viglialoro (2016), where for n id="mmlsi11" class="mathmlsrc">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">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>der="0" 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">noscript>n class="mathContainer hidden">n class="mathCode">

α &isin; (\frac{n>5 n>}{n>3 n>}, n>2 n>)

n>n>n> the global existence of very weak solutions n id="mmlsi12" class="mathmlsrc">n 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)n>n class="mathContainer hidden">n class="mathCode">

(u, v)

n>n>n> to the system is shown for any nonnegative initial data n id="mmlsi13" class="mathmlsrc">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">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>der="0" 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">noscript>n class="mathContainer hidden">n class="mathCode">

(u_{n>0 n>}, v_{n>0 n>}) &isin; C^{n>0 n>} (\overset{Ω}{nt="true">}) \times C^{n>2 n>} (\overset{Ω}{nt="true">})

n>n>n> and under zero-flux boundary condition on n id="mmlsi14" class="mathmlsrc">n 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">v₀n>n class="mathContainer hidden">n class="mathCode">

v_{n>0 n>}

n>n>n>, we prove that no chemotactic collapse for these solutions may present over time. More precisely, we establish that if the ratio n id="mmlsi15" class="mathmlsrc">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">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>der="0" 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">noscript>n class="mathContainer hidden">n class="mathCode">

\frac{a}{b}

n>n>n> does not exceed a certain value and for n id="mmlsi16" class="mathmlsrc">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">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>der="0" 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">noscript>n class="mathContainer hidden">n class="mathCode">

\frac{n>9 n>}{n>5 n>} < p < α < n>2 n>

n>n>n> the initial data are such that n id="mmlsi17" class="mathmlsrc">n 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‖_L^p(Ω)n>n class="mathContainer hidden">n class="mathCode">

{‖ u_{n>0 n>} ‖}_{L^{p} (Ω)}

n>n>n> and n id="mmlsi18" class="mathmlsrc">n 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‖_L⁴(Ω)n>n class="mathContainer hidden">n class="mathCode">

{‖ &nabla; v_{n>0 n>} ‖}_{L^{n>4 n>} (Ω)}

n>n>n> are small enough, then n id="mmlsi12" class="mathmlsrc">n 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)n>n class="mathContainer hidden">n class="mathCode">

(u, v)

n>n>n> is uniformly-in-time bounded.

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