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|p0y + tp|2 + (tp0 + rpr + (n в€’ 1)z)2 + (rpП„ )2 + (t2 + r2)F (z)
ec = 2

is sometimes called the conformal energy of the solution z(t, y).
It is also sometimes convenient to set
nв€’1
ed = rp0pr + zp0
2

so that
П•dil в‰Ў (ed + te)dy (mod {dt}),
and also
П•inv в‰Ў (ted + 1 (t2 + |y|2 )e в€’ nв€’1 2
(mod {dt}).
4 z )dy
2

The fact that the integrals of these quantities are constant in t yields results
about growth of solutions.
In fact, in the analysis of wave equations that are perturbations of the con-
formally invariant equation (3.68), the most eп¬Ђective estimates pertain to the
quantities appearing in these conservation laws. One can think of the conser-
vation laws as holding for our вЂњп¬‚atвЂќ non-linear wave equation (3.68), and then
the estimates are their analogs in the вЂњcurvedвЂќ setting.

3.5.3 Energy in Three Space Dimensions
We conclude this section by discussing a few more properties that involve the
energy in three space dimensions.
The fact that the energy E(t) is constant implies in particular that Rn zt dy
t
is bounded with respect to t. This allows us to consider the evolution of the
spatial L2 -norm of a solution to z = z 3 as follows:

d2
z 2 dy 2
= 2 (z ztt dy + zt )dy
dt2 R3 R3
t t

(zв€†z в€’ z 4 + zt )dy
2
= 2
R3
t

2
+ z 4)dy + 2 2
в€’2
= (| y z| zt dy
R3 Rn
t t

в‰¤ 4E,

where the second equality follows from the diп¬Ђerential equation and the third
from GreenвЂ™s theorem (integration by parts). The conclusion is that ||z||2 2(Rn )
L t
grows at most quadratically,

z 2dy в‰¤ 2Et2 + C1 t + C2 , (3.77)
R3
t
128 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

z 2 dy = O(t2 ).
and in particular

The energy plays another interesting role in the equation

z = в€’z 3 (3.78)

Here, it is possible for the energy to be negative:

z4
12
+ || z||2) в€’
E= (z dy. (3.79)
2t 4
R3
t

We will prove that a solution to (3.78), with compactly supported initial data
z(0, y), zt (0, y) satisfying E < 0, must blow up in п¬Ѓnite time ([Lev74]). No-
tice that any non-trivial compactly supported initial data may be scaled up to
achieve E < 0, and may be scaled down to achieve E > 0.
The idea is to show that the quantity

def 12
I(t) = z dy
2
R3
t

becomes unbounded as t T for some п¬Ѓnite time T > 0. We start by computing
its derivatives

I (t) = zzt dy,

2
I (t) = zt dy + zztt dy

2
| z|2dy + z 4 dy.
zt dy в€’
=

The last step uses GreenвЂ™s theorem, requiring the solution to have compact y-
support for each t в‰Ґ 0. To dispose of the z 4 term, we add 4E to each side
using (3.79):
2
| z|2dy.
I (t) + 4E = 3 zt dy +

We can discard from the right-hand side the positive gradient term, and from
the left-hand side the negative energy term, to obtain

2
I (t) > 3 zt dy.

To obtain a second-order diп¬Ђerential inequality for I, we multiply the last in-
equality by I(t) to obtain

3
z 2dy 2
I(t)I (t) > zt dy
2
(t)2 .
3
в‰Ґ I
2
3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 129

The last step follows from the Cauchy-Schwarz inequality, and says that I(t)в€’1/2
has negative second derivative. We would like to use this to conclude that I в€’1/2
vanishes for some T > 0 (which would imply that I blows up), but for this we
would need to know that (I в€’1/2 ) (0) < 0, or equivalently I (0) > 0, which may
not hold.
To rectify this, we shift I to

J(t) = I(t) в€’ 1 E(t + П„ )2 ,
2

with П„ > 0 chosen so that J (0) > 0. We now mimic the previous reasoning to
show that (J в€’1/2) (t) < 0. We have

zzt dy в€’ E(t + П„ ),
J (t) =

2
zztt dy в€’ E
J (t) = zt dy +

2
|| z||2dy в€’ 5E
=3 zt dy +

2
zt dy в€’ E .
>3

From this we obtain
2
J(t)J (t) в€’ 3 J (t)2 > 3
z 2 в€’ E(t + П„ )2 2
zt в€’ E в€’ zzt в€’ E(t + П„ ) ,
2 2

which is positive, again by the Cauchy-Schwarz inequality. This means that
(J в€’1/2 ) (t) < 0. Along with J в€’1/2(0) > 0 and (J в€’1/2 ) (0) < 0, this implies
that for some T > 0, J в€’1/2(T ) = 0, so J(t) blows up.

We conclude by noting that the qualitative behavior of solutions of z =
f(z) depends quite sensitively on the choice of non-linear term f(z). In contrast
to the results for z = В±z 3 described above, we have for the equation

z = в€’z 2 (n = 3),

that every solution must blow up in п¬Ѓnite time ([Joh79]). We will outline the
proof in case the initial data are compactly supported and satisfy

u(0, t)dy > 0, ut(0, t)dy > 0.

Note that replacing z by в€’z gives the equation z = z 2 , which therefore has
the same behavior.
This proof is fairly similar to the previous one; we will derive diп¬Ђerential
inequalities for
def
J(t) = z dy
Rn
t
130 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

which imply that this quantity blows up. We use integration by parts to obtain

(в€†z + z 2)dy = z2,
J (t) = ztt dy =

and using HВЁlderвЂ™s inequality on Supp z вЉ‚ {|y| в‰¤ R0 + t} in the form ||z||L1 в‰¤
o
||z||L2 ||1||L2, this gives
2
(R0 + t)в€’3 в‰Ґ C(1 + t)в€’3 J(t)2 .
J (t) в‰Ґ C z dy (3.80)

This is the п¬Ѓrst ingredient.
Next, we use the fact that if z0 (y, t) is the free solution to the homogeneous
wave equation z0 = 0, with the same initial data as our z, then

z(y, t) в‰Ґ z0 (y, t)

for t в‰Ґ 0; this follows from a certain explicit integral expression for the solution.
Note that if we set J0(t) = Rn z0 , then it follows from the equation alone that
t
J0 (t) = 0, and by the hypotheses on our initial data we have J0 (t) = C0 + C1 t
with C0, C1 > 0. Another property of the free solution is that its support at
time t lies in the annulus At = {t в€’ R0 в‰¤ |y| в‰¤ t + R0}. Now

C0 + C 1 t в‰¤ z dy
At
в‰¤ ||z||L1(A)
в‰¤ ||z||L2(A) ||1||L2(A)
2
2
в‰¤ C(1 + t) z dy .

This gives
1/2
C0 + C 1 t
2
z dy в‰Ґ
J (t) = ,
C(1 + t)
and in particular, J > 0. With the assumptions on the initial data, this gives

J (t) >0
в‰Ґ C(1 + t)2.
J(t)

We can use (3.80, 3.81, 3.81) to conclude that J must blow up at some п¬Ѓnite
time. This follows by writing

C(1 + t)в€’3 J 3/2J 1/2
в‰Ґ
J (t)
C(1 + t)в€’2 J 3/2.
в‰Ґ

Multiply by J and integrate to obtain

J (t) в‰Ґ C(1 + t)в€’1 J(t)5/4 .
3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 131

Integrating again, we have
в€’4
J(t) в‰Ґ J(0)в€’1/4 в€’ 1 C ln(1 + t) ,
4

and because J(0) > 0 and C > 0, J(t) must blow up in п¬Ѓnite time.
132 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS
Chapter 4

Additional Topics

4.1 The Second Variation
In this section, we will discuss the second variation of the Lagrangian func-
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