## 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

Consider the initial-boundary value problem for the system (S)ut = uxx - (uvx)x, vt= u- av on an interval [0,1] for t > 0, where a > 0 with ux(0,t) = ux(1,t)= 0. Suppose \mu, v0 are positive constants. The corresponding spatially homogeneous global solution U(t) = \mu, V(t) = \mu a + (v0 - \mu a)\exp(-at) is stable in the sense that if (\mu',v0' ) are positive constants, the corresponding spatially homogeneous solution will be uniformly close to (U(\cdot),V(\cdot)).

We consider, in sequence space, an approximate system (S') which is related to (S) in the following sense: The chemotactic term (uvx)x is replaced by the inverse Fourier transform of the finite part of the convolution integral for the Fourier transform of (uvx)x. (Here the finite part of the convolution on the line at a point x of two functions, f,g, is defined as $\int_0^x(f(y)g(y-x)\,dy$.) We prove the following: If \mu > a, then in every neighborhood of (\mu,v0 ) there are (spatially nonconstant) initial data for which the solution of problem (S') blows up in finite time in the sense that the solution must leave L2 (0,1)\times H1 (0,1) in finite time T. Moreover, the solution components u(\cdot,t),v(\cdot,t) each leave L2 (0,1).If \mu > a, then in every neighborhood of (\mu,v0 ) there are (spatially nonconstant) initial data for which the solution of problem (S) on (0,1) \times (0,Tmax ) must blow up in finite time in the sense that the coefficients of the cosine series for (u,v) become unbounded in the sequence product space $\ell^1\times\ell^1_1$.

A consequence of (2) states that in every neighborhood of (\mu,v0 ), there are solutions of (S) which, if they are sufficiently regular, will blow up in finite time. (Nagai and Nakaki [Nonlinear Anal., 58 (2004), pp. 657--681] showed that for the original system such solutions are unstable in the sense that if \mu > a, then in every neighborhood of (\mu,\mu a), there are spatially nonconstant solutions which blow up in finite or 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.

This is an article from SIAM Journal on Applied Mathematics 65 (2004): 336, doi:10.1137/S0036139903431725. Posted with permission.

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
##### Original bundle
Now showing 1 - 4 of 4
No Thumbnail Available
Name:
Size:
210.08 KB
Format:
Description:
No Thumbnail Available
Name:
2004_Levine_ErratumSingularity.pdf
Size:
98.48 KB
Format:
Description:
No Thumbnail Available
Name:
2-2004_Levine_ErratumSingularity.pdf
Size:
98.48 KB
Format: