314 Exact Solutions and Invariant Subspaces
this theory was driven by various versions of so-called affine Bernstein conjectures,
including Chern’s and Calabi’s conjecture. A typical result says that, in the class of
complete hypersurfaces, M must be an elliptic paraboloid [384] (cf. the J¨orgens–
Calabi–Pogorelov results for the inhomogeneousM-A equation detD
2
u = 1inIR
N
;
see Remarks). The corresponding fourth-order parabolic flows
u
t
= F
4
[u] + (lower-order terms)
can be locally well-defined on the classes of convex functions. In particular, a related
modified equation
u
t
=
detD
2
u
+ (lower-order terms) in IR
2
× IR
+
(for simplicity, is the Laplacian in IR
N
) can be restricted to the subspace (6.127),
and the DS determines the blow-up or global evolution of convex surfaces in IR
2
.
Example 6.66 (mth-order equations) Consider a general mth-order fully nonlin-
ear equation of the M-A-type in IR
2
× IR
+
(m ≥ 2)
u
t
=
(|µ|=|ν|=m)
a
µ,ν
D
µ
x
uD
ν
x
u + (lower-order terms), (6.131)
where µ and ν are multi-indices, and the matrix #a
µ,ν
# satisfies a positivity-type as-
sumption for local existence(see Remarks). This equation admitsa finite-dimensional
restriction on the polynomial subspace W = L{x
α
y
β
, 0 ≤ α + β ≤ 2m}, with a
typical blow-up dynamics of convex solutions driven by a quadratic DS.
Example 6.67 (Third-order equation)Wefinish our list of M-A-typemodelswith
a related third-order PDE, written in the following evolution-lookingform:
u
ttt
= F[u] ≡ (u
txx
)
2
− u
ttx
u
xxx
. (6.132)
This is known as one of the associativity equations in 2D field theory [157]. On the
other hand, it is a parameterized form of the M-A-type equation
v
xxx
v
yyy
− v
xxy
v
xyy
= 1,
as the compatibility condition for a PDE system (a reduction of the Gauss–Codazzi
equations) governing hypersurfaces M
2
⊂ A
3
with a flat centroaffine metric, where
x and y are the asymptotic coordinates on M
2
, [182]. An abundance of explicit so-
lutions of (6.132) is available in [157]; see also the table in [182, p. 41].
The quadratic operator F in (6.132)admitsthe 5D subspace (nowa module) W
5
=
L{1, x , x
2
, x
3
, x
4
}, so that the PDE restricted to W
5
with solutions
u(x , t) = C
1
(t) + C
2
(t)x + C
3
(t)x
2
+ C
4
(t)x
3
+ C
5
(t)x
4
(6.133)
is equivalent to the following fifteenth-order DS:
C
1
=−6C
4
C
2
+ 4(C
3
)
2
,
C
2
=−24C
5
C
2
− 12C
4
C
3
+ 24C
3
C
4
,
C
3
=−48C
5
C
3
− 18C
4
C
4
+ 36(C
4
)
2
+ 48C
3
C
5
,
C
4
=−72C
5
C
4
− 24C
4
C
5
+ 144C
4
C
5
,
C
5
=−96C
5
C
5
+ 144(C
5
)
2
.
© 2007 by Taylor & Francis Group, LLC