Biomedical Engineering Reference
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to give
ot
þ
o
f
2
of
þ
e
Tof
=
ot
¼
0
;
ox
the large-time behaviour to which takes the form
f
¼
tF
ð
g
Þ;
g
¼
x
=
t
whereby (since F
ð
g
Þ
satisfies a first-order ODE of Clairaut's form) F is given
parametrically in terms of P
dF
=
dg by
F
gP
þ
e
T
ð
F
gP
Þ
þ
P
2
¼
0
;
g
Tge
T
ð
F
gP
Þ
þ
2P
¼
0
;
so that
F
k
þð
1
þ
kT
Þ
g
2
=
4
as
g
!
0
;
F
¼
g
2
=
4
þ
exponentially small terms
as
g
!þ1;
the former being consistent with (
8
) [i.e., in the terminology of matched asymp-
totic expansions, matching with (
8
)] and the latter implying that the purely dif-
fusive effects dominate the far field, contrasting with the zero-delay case (
2
)in
which
the
feedback
term
continues
to
generate
an
exponentially
growing
contribution.
We now turn to the more complicated nonlinear case
2
u
ox
2
ð
x
;
t
Þþ
u
2
ð
x
;
t
T
Þ1
\x\
þ1
o
u
ot
¼
o
(in this case we could rescale T to 1 but we forego the opportunity). That this case
is both significantly more involved and qualitatively distinct from the zero-delay
version (
3
) is already apparent from the spatially uniform solution, which we
denote U
ð
t
Þ
: applying the Liouville-Green method in the form
U
ð
t
Þ¼
e
U
ð
t
Þ
as
t
!þ1
gives
U
U
o
e
t ln 2
=
T
U
ð
t
Þ
2U
ð
t
T
Þ;
for some constant U
0
that depends on the initial data (implying strong sensitivity to
the initial state); constructing correction terms yields
U
ð
t
Þ
4 ln 2
T
U
0
e
t ln 2
=
T
e
U
0
e
t ln 2
=
T
as
t
!þ1;
ð
10
Þ
implying that blow up is in infinite, rather than in finite, time, albeit at a much
faster-than-exponential rate. In the spatially structured case we set
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