4
SIGURD ANGENENT AND JOOST HULSHOF
Since the convergence of
u(y, t)
to
U(y)
follows from more general theory, we
will concentrate here on proving the asymptotic formula for
Rep ( t).
1.2. Proof of lemma 1.1.
Both 'P and
rp(r, t)
=
n
/2 are solutions of (1.2),
so that their difference satisfies a linear parabolic equation to which we can apply
the Sturmian theorem: the number of zeroes of
r
r--+
tp( r, t) -
11"
/2 does not increase
with time. Since it starts out being 1, and since the boundary conditions in
tp
force
tp(r, t)-
n/2 to change sign at least once between
r
= 0 and
r
= 1, we conclude
that for each
t
0 the function
r ,__.. tp(r, t) - n
/2 must vanish exactly once.
2. Constructing Formal Solutions
In this section we recall how in
[BHK]
a formal solution for (1.2) is derived.
We consider the function
u(y, t)
=
tp(yR(t), t),
where
R(t)
will be an approximation of
Rq,(t).
This function satisfies
2
Uy f(u)
(2.1) R(t) Ut
=
Uyy
+--
-2-
+
RRtYUy,
y y
Since we expect
u(y, t)
--
U(y)
as
t /
oo, we write
u
=
U(y)
+
v(y, t).
Assuming
v
is small compared with U, at least for y
«
R(t)-
1
,
one obtains the
following equation for
v
28v f"(U;v)
2
(2.2)
R -
8
=
J\1
[v] -
2
v
+
RRtyUy
+
RRtYVy
t
y
Here
J\1
is the differential operator
(2.3)
J\1 _ (_?__)2
+
~__?___
_
f'(U(y)) _ (_?__)2
+
~__?___
_
2._
+
8
- 8y y 8y
y2 -
8y y 8y y2 (1
+
y2)2
0
Also, we use the following notation:
f(n)(u; v)
=
1n
1
(1- Tt-l f(n)(u +TV) dT,
so that the Taylor-Maclaurin formula with remainder can be written as
f "(u) f(n-l)(u) f(n)(u· v)
f(u
+
v)
=
f(u)
+
f'(u)v
+
--v2
+ ... +
vn-l
+ '
vn.
2! (n- 1)! n!
We set
1
v(y, t)
=
a(t)'l/h(y),
where
'l/Jl(y)
is a solution of
(2.4)
J\1~1
=
~o(Y) ~f
yU'(y)
=
1
!yy2
which satisfies
~ 1
(0)
=
0,
and a(t) is determined by the boundary condition at
r
=
1, i.e. at
y
=
R-
1.
Namely, we require
U(R-
1
)
+
a(t)~ 1
(R-1)
=
n,
i.e.
()
_ 2arctanR(t)
(2.5) at -
~ 1 (
1
/R(t))
·
10f
course one does not expect the method of separation of variables to yield solutions for
the nonlinear equation which
v
must satisfy. However, the method may produce functions which
are close to solutions. As the subsequent analysis in this paper shows, this is indeed the case here.
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