Singularity Formation in Chemotaxis--A Conjecture of Nagai

dc.contributor.author Levine, Howard
dc.contributor.author Renclawowicz, Joanna
dc.contributor.department Mathematics
dc.date 2018-02-18T00:30:56.000
dc.date.accessioned 2020-06-30T06:00:39Z
dc.date.available 2020-06-30T06:00:39Z
dc.date.copyright Thu Jan 01 00:00:00 UTC 2004
dc.date.issued 2004-01-01
dc.description.abstract <p>Consider the initial-boundary value problem for the system (<em>S</em>)<em>u<sub>t</sub></em> = <em>u<sub>xx</sub></em> - (<em>uv<sub>x</sub></em>)<em><sub>x</sub></em>, <em>v<sub>t</sub></em>= <em>u</em>- <em>av</em> on an interval [0,1] for <em>t</em> > 0, where <em>a</em> > 0 with <em>u<sub>x</sub></em>(0,<em>t</em>) = <em>u<sub>x</sub></em>(1,<em>t</em>)= 0. Suppose \mu, <em>v</em><sub>0</sub> are positive constants. The corresponding spatially homogeneous global solution <em>U</em>(<em>t</em>) = \mu, <em>V</em>(<em>t</em>) = \mu <em>a</em> + (<em>v</em><sub>0</sub> - \mu <em>a</em>)\exp(-<em>at</em>) is stable in the sense that if (\mu',<em>v</em><sub>0'</sub> ) are positive constants, the corresponding spatially homogeneous solution will be uniformly close to (<em>U</em>(\cdot),<em>V</em>(\cdot)).</p> <p>We consider, in sequence space, an approximate system (<em>S</em>') which is related to (<em>S</em>) in the following sense: The chemotactic term (<em>uv<sub>x</sub></em>)<em><sub>x</sub></em> is replaced by the inverse Fourier transform of the finite part of the convolution integral for the Fourier transform of (<em>uv<sub>x</sub></em>)<em><sub>x</sub></em>. (Here the finite part of the convolution on the line at a point <em>x</em> of two functions, <em>f</em>,<em>g</em>, is defined as $\int_0^x(f(y)g(y-x)\,dy$.) We prove the following: If \mu > <em>a</em>, then in every neighborhood of (\mu,<em>v</em><sub>0</sub> ) there are (spatially nonconstant) initial data for which the solution of problem (<em>S</em>') blows up in finite time in the sense that the solution must leave <em>L</em><sup>2</sup> (0,1)\times <em>H</em><sup>1</sup> (0,1) in finite time <em>T</em>. Moreover, the solution components <em>u</em>(\cdot,<em>t</em>),<em>v</em>(\cdot,<em>t</em>) each leave <em>L</em><sup>2</sup> (0,1).If \mu > <em>a</em>, then in every neighborhood of (\mu,<em>v</em><sub>0</sub> ) there are (spatially nonconstant) initial data for which the solution of problem (<em>S</em>) on (0,1) \times (0,<em>T</em><sub>max</sub> ) must blow up in finite time in the sense that the coefficients of the cosine series for (<em>u</em>,<em>v</em>) become unbounded in the sequence product space $\ell^1\times\ell^1_1$.</p> <p>A consequence of (2) states that in every neighborhood of (\mu,<em>v</em><sub>0</sub> ), there are solutions of (<em>S</em>) which, if they are sufficiently regular, will blow up in finite time. (Nagai and Nakaki [<em>Nonlinear Anal</em>., 58 (2004), pp. 657--681] showed that for the original system such solutions are unstable in the sense that if \mu > <em>a</em>, then in every neighborhood of (\mu,\mu <em>a</em>), there are spatially nonconstant solutions which blow up in finite <em>or</em> infinite time. They conjectured that the blow-up time must be finite.) Using a recent regularity result of Nagai and Nakaki, we prove this conjecture.</p>
dc.description.comments <p>This is an article from <em>SIAM Journal on Applied Mathematics</em> 65 (2004): 336, doi:<a href="http://dx.doi.org/10.1137/S0036139903431725">10.1137/S0036139903431725</a>. Posted with permission.</p>
dc.format.mimetype application/pdf
dc.identifier archive/lib.dr.iastate.edu/math_pubs/40/
dc.identifier.articleid 1041
dc.identifier.contextkey 9339860
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath math_pubs/40
dc.identifier.uri https://dr.lib.iastate.edu/handle/20.500.12876/54636
dc.language.iso en
dc.source.bitstream archive/lib.dr.iastate.edu/math_pubs/40/0-dsmComments_for_Levine__Howard_A_vita_November_2015.pdf|||Sat Jan 15 00:07:18 UTC 2022
dc.source.bitstream archive/lib.dr.iastate.edu/math_pubs/40/1-2004_Levine_ErratumSingularity.pdf|||Sat Jan 15 00:07:17 UTC 2022
dc.source.bitstream archive/lib.dr.iastate.edu/math_pubs/40/2-2004_Levine_ErratumSingularity.pdf|||Sat Jan 15 00:07:19 UTC 2022
dc.source.bitstream archive/lib.dr.iastate.edu/math_pubs/40/2004_Levine_ErratumSingularity.pdf|||Sat Jan 15 00:07:20 UTC 2022
dc.source.uri 10.1137/S0036139903431725
dc.subject.disciplines Applied Mathematics
dc.subject.disciplines Cell Biology
dc.subject.disciplines Partial Differential Equations
dc.subject.keywords chemotaxis
dc.subject.keywords finite time singularity formation
dc.subject.keywords Keller-Segel model
dc.supplemental.bitstream 2004_Levine_ErratumSingularity.pdf
dc.title Singularity Formation in Chemotaxis--A Conjecture of Nagai
dc.type article
dc.type.genre article
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48
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