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.

Comments

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

Copyright Owner

Society for Industrial and Applied Mathematics

Language

en

File Format

application/pdf

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