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Singularity Formation in Chemotaxis--A Conjecture of Nagai

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Consider the initial-boundary value problem for the system (*S*)*u _{t}* =

*u*- (

_{xx}*uv*)

_{x}*,*

_{x}*v*=

_{t}*u*-

*av*on an interval [0,1] for

*t*> 0, where

*a*> 0 with

*u*(0,

_{x}*t*) =

*u*(1,

_{x}*t*)= 0. Suppose \mu,

*v*

_{0}are positive constants. The corresponding spatially homogeneous global solution

*U*(

*t*) = \mu,

*V*(

*t*) = \mu

*a*+ (

*v*

_{0}- \mu

*a*)\exp(-

*at*) is stable in the sense that if (\mu',

*v*

_{0'}) 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 (*uv _{x}*)

*is replaced by the inverse Fourier transform of the finite part of the convolution integral for the Fourier transform of (*

_{x}*uv*)

_{x}*. (Here the finite part of the convolution on the line at a point*

_{x}*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,

*v*

_{0}) 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

*L*

^{2}(0,1)\times

*H*

^{1}(0,1) in finite time

*T*. Moreover, the solution components

*u*(\cdot,

*t*),

*v*(\cdot,

*t*) each leave

*L*

^{2}(0,1).If \mu >

*a*, then in every neighborhood of (\mu,

*v*

_{0}) there are (spatially nonconstant) initial data for which the solution of problem (

*S*) on (0,1) \times (0,

*T*

_{max}) 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,*v*_{0} ), 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.

##### Comments

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