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 mmlsi2" class="mathmlsrc">mulatext 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">ΩmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mi>Ωmi>math> of mmlsi3" class="mathmlsrc">mulatext 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">R³mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msup><mrow><mi mathvariant="double-struck">Rmi>mrow><mrow><mn>3mn>mrow>msup>math>, with mmlsi4" class="mathmlsrc">mulatext 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>0mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><mi>χmi><mo>>mo><mn>0mn>math> and endowed with homogeneous Neumann boundary conditions. The source mmlsi5" class="mathmlsrc">mulatext 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">gmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi>gmi>math> behaves similarly to the logistic function and verifies mmlsi6" class="mathmlsrc">mulatext 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&minus;bs^αmathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>gmi><mrow><mo>(mo><mi>smi><mo>)mo>mrow><mo>≤mo><mi>ami><mo>&minus;mo><mi>bmi><msup><mrow><mi>smi>mrow><mrow><mi>αmi>mrow>msup>math>, for mmlsi7" class="mathmlsrc">mulatext 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≥0mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mi>smi><mo>≥mo><mn>0mn>math>, with mmlsi8" class="mathmlsrc">mulatext 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≥0mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll"><mi>ami><mo>≥mo><mn>0mn>math>, mmlsi9" class="mathmlsrc">mulatext 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>0mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><mi>bmi><mo>>mo><mn>0mn>math> and mmlsi10" class="mathmlsrc">mulatext 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>1mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll"><mi>αmi><mo>>mo><mn>1mn>math>. In line with Viglialoro (2016), where for mmlsi11" class="mathmlsrc">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">mg class="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">mathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll"><mi>αmi><mo>∈mo><mrow><mo>(mo><mfrac><mrow><mn>5mn>mrow><mrow><mn>3mn>mrow>mfrac><mo>,mo><mn>2mn><mo>)mo>mrow>math> the global existence of very weak solutions mmlsi12" class="mathmlsrc">mulatext 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)mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll"><mrow><mo>(mo><mi>umi><mo>,mo><mi>vmi><mo>)mo>mrow>math> to the system is shown for any nonnegative initial data mmlsi13" class="mathmlsrc">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">mg class="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">mathContainer hidden">mathCode"><math altimg="si13.gif" overflow="scroll"><mrow><mo>(mo><msub><mrow><mi>umi>mrow><mrow><mn>0mn>mrow>msub><mo>,mo><msub><mrow><mi>vmi>mrow><mrow><mn>0mn>mrow>msub><mo>)mo>mrow><mo>∈mo><msup><mrow><mi>Cmi>mrow><mrow><mn>0mn>mrow>msup><mrow><mo>(mo><mover accent="true"><mrow><mi>Ωmi>mrow><mrow><mo>̄mo>mrow>mover><mo>)mo>mrow><mo>&times;mo><msup><mrow><mi>Cmi>mrow><mrow><mn>2mn>mrow>msup><mrow><mo>(mo><mover accent="true"><mrow><mi>Ωmi>mrow><mrow><mo>̄mo>mrow>mover><mo>)mo>mrow>math> and under zero-flux boundary condition on mmlsi14" class="mathmlsrc">mulatext 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">v₀mathContainer hidden">mathCode"><math altimg="si14.gif" overflow="scroll"><msub><mrow><mi>vmi>mrow><mrow><mn>0mn>mrow>msub>math>, we prove that no chemotactic collapse for these solutions may present over time. More precisely, we establish that if the ratio mmlsi15" class="mathmlsrc">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">mg class="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">mathContainer hidden">mathCode"><math altimg="si15.gif" overflow="scroll"><mfrac><mrow><mi>ami>mrow><mrow><mi>bmi>mrow>mfrac>math> does not exceed a certain value and for mmlsi16" class="mathmlsrc">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">mg class="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">mathContainer hidden">mathCode"><math altimg="si16.gif" overflow="scroll"><mfrac><mrow><mn>9mn>mrow><mrow><mn>5mn>mrow>mfrac><mo><mo><mi>pmi><mo><mo><mi>αmi><mo><mo><mn>2mn>math> the initial data are such that mmlsi17" class="mathmlsrc">mulatext 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)mathContainer hidden">mathCode"><math altimg="si17.gif" overflow="scroll"><msub><mrow><mrow><mo>‖mo><msub><mrow><mi>umi>mrow><mrow><mn>0mn>mrow>msub><mo>‖mo>mrow>mrow><mrow><msup><mrow><mi>Lmi>mrow><mrow><mi>pmi>mrow>msup><mrow><mo>(mo><mi>Ωmi><mo>)mo>mrow>mrow>msub>math> and mmlsi18" class="mathmlsrc">mulatext 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)mathContainer hidden">mathCode"><math altimg="si18.gif" overflow="scroll"><msub><mrow><mrow><mo>‖mo><mi>∇mi><msub><mrow><mi>vmi>mrow><mrow><mn>0mn>mrow>msub><mo>‖mo>mrow>mrow><mrow><msup><mrow><mi>Lmi>mrow><mrow><mn>4mn>mrow>msup><mrow><mo>(mo><mi>Ωmi><mo>)mo>mrow>mrow>msub>math> are small enough, then mmlsi12" class="mathmlsrc">mulatext 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)mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll"><mrow><mo>(mo><mi>umi><mo>,mo><mi>vmi><mo>)mo>mrow>math> is uniformly-in-time bounded.