Discrete and Continuum Multiscale Behaviour in Bacterial Communication - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 302

The limit t !1 is both physically meaningful and of mathematical interest
(specifically, in the associated continuum limit, blow up can occur at an arbitrary
real value of x, whereas in the discrete version the blow up point needs to be at an
integer i: the switch from continuous to discrete translation invariance in going
from the PDE to the differential-difference system has a number of important
implications for the proper understanding of such discrete problems, including for
the pinning phenomena noted later). We adopt the modified definition

x ¼ð i i c Þ= t, where i c 2 is to be chosen such that blow up in the continuum

limit occurs at x ¼ 0. Setting

R

u i ð t; t Þ u ð x ; t Þþ 1

t 2 v ð x ; t Þ

t !þ1

ð 17 Þ

as

implies

2 u
ox 2 þ u 2 ;

2 v 4 u

ox 4 ;

o u

ot ¼ o

o v

ot ¼ o ox 2 þ 2uv þ 1

o

ð 18 Þ

12

wherein the fourth derivative term is the sole remnant of discreteness thus far. The
blow-up behaviour of the first of ( 18 ) is well known and motivates the interme-
diate-asymptotic variables

1
ð t c t Þ f ð n ; s Þ;

1
ð t c t Þ 2 g ð n ; s Þ;

x
ð t c t Þ 2 ;

u ¼ v ¼ n ¼ s ¼ ln ð t c t Þ:

Introducing

on 2 n ¼ 0

j ð s Þ¼ o 2 f

with j ! 0ass !þ1 to be determined, it can be shown that
f 1 þ j ð s Þ 1 1

2 n 2 þ j 2 ð s Þ 5 þ 1 4 n 4

as j ! 0 ð 19 Þ

with

dj
ds ¼ 4j 2 ;

j ¼ 1 = 4 ð s þ s c Þ ð 20 Þ

arising as a solvability condition (cf., [ 3 , 6 ], for example). Since g ð n ; s Þ satisfies a
linear problem, we may superpose the 'complementary function' contribution

g T c ð 1 þ j ð s Þð 1 n 2 ÞÞ;

ð 21 Þ

corresponding to a shift of T c = t 2 in the blow up time t c , with the constant T c
depending on the initial data, and a 'particular integral'