Boundedness in a Keller-Segel system with external signal production

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• 作者：Tobias Black ^{tblack@math.upb.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chemotaxis ; Boundedness ; External signal production ; Critical mass
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：436-455
• 全文大小：501 K

文摘

We study the Neumann initial-boundary problem for the chemotaxis system<div class="formula" id="fm0010"><div class="mathml">d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=33faf6557e2b006e62d04d629553bb0d">dth="360" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304814-si1.gif">dden">de">d/blank.gif">div>div> in a smooth, bounded domain d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=d2b8905626058a3e0a20fbd876c8e623" title="Click to view the MathML source">Ω⊂Rⁿdden">de"> with d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=acad3082ef576480ec3dfa2c479bb257" title="Click to view the MathML source">n≥2dden">de">$n\ge 2$ and d="mmlsi378" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si378.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=cca8d49d761f0cb01428cc7ced1fd90e">dth="310" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304814-si378.gif">dden">de">$f\in {\mathrm{L}}^{\infty }([0,\infty );{\mathrm{L}}^{\frac{n}{2}+{\delta }_0}(\mathrm{\Omega }))\cap C^{\alpha }(\mathrm{\Omega }\times (0,\infty ))$ with some d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=95434e241337eb338ce91c58cb1c670c" title="Click to view the MathML source">α>0dden">de">$\alpha >0$ and d="mmlsi48" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si48.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=ddf0df3b39b11e19fab632730f971026" title="Click to view the MathML source">δ₀∈(0,1)dden">de">${\delta }_0\in (0,1)$. First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of d="mmlsi7" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si7.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=5e4ce9bad4c0dac797db03b1a3b1680d" title="Click to view the MathML source">n=2dden">de">$n=2$ and f   being constant in time, requiring the nonnegative initial data d="mmlsi8" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si8.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=d9fa814a0741900fc29a643346e9b188" title="Click to view the MathML source">u₀dden">de">$u_0$ to fulfill the property d="mmlsi22" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si22.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=560f82b355d91f41805f9218a30f75a2">dth="98" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304814-si22.gif">dden">de"> ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller&ndash;Segel model (coinciding with d="mmlsi10" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=3b03d4fc43ab8444e7f11a85d934ba18" title="Click to view the MathML source">f≡0dden">de">$f\equiv 0$ in the system above) to the case d="mmlsi11" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=1126bb8672db71f42d8fc95b701f3f62" title="Click to view the MathML source">f≢0dden">de">$f\not\equiv 0$. Under certain smallness conditions imposed on the initial data and f   we furthermore show that for more general space dimension d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304814&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304814&_rdoc=1&_issn=0022247X&md5=acad3082ef576480ec3dfa2c479bb257" title="Click to view the MathML source">n≥2dden">de">$n\ge 2$ and f not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller&ndash;Segel system to the present situation.