Campus Units
Mathematics
Document Type
Article
Publication Version
Published Version
Publication Date
2004
Journal or Book Title
SIAM Journal on Applied Mathematics
Volume
65
Issue
1
First Page
336
Last Page
360
DOI
10.1137/S0036139903431725
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.
Copyright Owner
Society for Industrial and Applied Mathematics
Copyright Date
2004
Language
en
File Format
application/pdf
Recommended Citation
Levine, Howard A. and Renclawowicz, Joanna, "Singularity Formation in Chemotaxis--A Conjecture of Nagai" (2004). Mathematics Publications. 40.
https://lib.dr.iastate.edu/math_pubs/40
Erratum
Comments
This is an article from SIAM Journal on Applied Mathematics 65 (2004): 336, doi:10.1137/S0036139903431725. Posted with permission.