Biomedical Engineering Reference
In-Depth Information
u
ð
x
;
t
Þ¼
U
ð
t
Þ
v
ð
x
;
t
Þ
to give
ot
ð
x
;
t
Þ¼
o
2
v
ox
2
ð
x
;
t
Þþ
U
ð
t
Þ
o
v
U
ð
t
Þ
ð
v
2
ð
x
;
t
T
Þ
v
ð
x
;
t
ÞÞ:
We conjecture, then, that the large time behaviour takes the quasi-steady form
f
¼
U
ð
t
Þ
U
ð
t
Þ
2
v
ð
x
;
t
Þ
V
ð
f
Þ;
x
as
t
!þ1
with
f
¼
O
ð
1
Þ
ð
11
Þ
whereby V
ð
f
Þ
satisfies the novel nonlinear functional differential equation
df
2
ð
f
Þþ
V
2
f
p
V
ð
f
Þ¼
0
;
d
2
V
V
¼
O
ð
e
j
f
j
Þ
2
as
j
f
j!1;
ð
12
Þ
where we have made use of (
10
), which implies in (
11
) that x
¼
O
ð
e
t ln 2
=
2T
Þ
, i.e.
that u is largest in an exponential narrow spike, and that the neglected terms are
exponentially smaller than those included. The dominant balance for larger x is
simply the heat equation
2
u
ox
2
ð
x
;
t
Þ
;
o
u
ot
ð
x
;
t
Þ
o
ð
13
Þ
u remains hyperexponentially large out to x
¼
O
ð
e
t ln 2
=
2T
Þ
, a caustic of the
resulting rays associated with the Liouville-Green approximation to (
12
) being
present in this exponentially large range of x. Confirmation of the applicability of
this postulated scenario of course requires in particular the existence of a non-
trivial solution of the boundary-value problem (
12
), which represents an open
problem; stability is also important: if a solution to (
12
) exists, it will be unstable
to modes associated with translations of t (or equivalently of U
0
) and of x, but
should not be to any others if it is to represent the generic large-time behaviour.
3 Spatio-Temporal Blow-Up Phenomena: The Effects
of Discreteness
The analog of the linear problem (
1
) now reads
du
i
dt
ð
t
Þ¼
t
2
ð
u
i
þ
1
ð
t
Þ
2u
i
ð
t
Þþ
u
i
1
ð
t
ÞÞ þ
u
i
ð
t
Þ; 1
\i\
þ1 ð
14
Þ
wherein the integers i are to be viewed as labelling distinct cells, the simplest
interpretation being that the signalling molecules (concentration u
i
ð
t
Þ
) can be
transported directly across the membranes of adjacent cells, the constant t
2
being a
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