Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
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9 Invariant Subspaces for Discrete Operators 443
finite-timeextinctionphenomena;see [235]for details.The pressure functionu = v
σ
solves
u
t
= F
0
[u] ≡ uu +
1
σ
|∇u|
2
+ σδ.
Operator F preserves the (N+1)-dimensional subspace
W
N+1
= L
1, x
2
1
, ..., x
2
N
. (9.60)
The same subspace is admitted by the discrete approximation of F
0
in the PDE
u
t
= F
h
[u] ≡ u
h
u +
1
σ
|∇
h
u|
2
+ σδ,
with difference operators from (9.46). This implies that, for solutions (9.44),
C
0
= 2C
0
M + σδ, M =
N
i=1
C
i
,
C
k
= 2C
k
M +
4
σ
C
2
k
, k = 1, 2, ..., N.
(9.61)
Due to the subspace (9.60), the DS (9.61) coincides with the DS in the differential
case.
Example 9.19 (Blow-up for exponential PDEs) Consider the quasilinear heatequa-
tion with exponential nonlinearities,
v
t
= e
v
+ e
v
+ ae
−v
+ b.
Setting u = e
v
yields the quadratic PDE
u
t
= F[u] ≡ uu + u
2
+ a + bu. (9.62)
F admits the subspace (see Proposition 6.9)
W
2
= L{1, f (x)}, (9.63)
where f is a solution of the linear elliptic equation
f + f = 0inIR
N
. (9.64)
Therefore, substituting u(x, t) = C
1
(t) + C
2
(t) f (x ) leads to the DS
C
1
= C
2
1
+ bC
1
+ a,
C
2
= C
1
C
2
+ bC
2
,
(9.65)
that can be solved explicitly. The corresponding discretized equation
u
t
= F
h
[u] ≡ u
h
u + u
2
+ a + bu
possesses the same solutionson the subspace(9.63),providedthat f solves the linear
discrete equation
h
f + f = 0. The discrete system for expansion coefficients
coincides with (9.65).
Example 9.20 (On quasilinear hyperbolic PDEs) Consider the evolution PDE
u
tt
= F[u] ≡ uu + u
2
+ a + bu. (9.66)
Fixing an arbitrary finite number n of linearly independent solutions f
1
, ..., f
n
of
(9.64) (n = 2ifN = 1) yields the subspace
W
n+1
= L{1, f
1
, ..., f
n
}. (9.67)
© 2007 by Taylor & Francis Group, LLC
444 Exact Solutions and Invariant Subspaces
Plugging u(x , t) = C
0
(t) + C
1
(t) f
1
+ ... + C
n
(t) f
n
gives the DS
C
0
= C
2
0
+ bC
0
+ a,
C
k
= C
0
C
k
+ bC
k
,
for k = 1, ..., n. Similar solutions exist for the corresponding discrete equation.
For the parabolic problem (9.62), last ODEs of the DS are
C
k
= C
k
(b +C
0
) ⇒
C
k
(t)
C
l
(t)
= constant
for all k, l = 1, ..., n. Hence, the solutions on the extended subspace (9.67) coincide
with those on the 2D subspace (9.63), where f is a linear combination of f
1
, ..., f
n
.
This is not the case for the hyperbolic PDE (9.66).
Example 9.21 (Fast diffusion) Consider the PDE
v
t
=∇·
v
−
4
N+2
∇v
+ bv
N+6
N+2
+ cv
in IR
N
× IR
+
with N ≥ 2. The pressure variable u = v
−
4
N+2
satisfies
u
t
= F
0
[u] −
4
N+2
b −
4
N+2
cu, where F
0
[u] = uu −
N+2
4
|∇u|
2
.
In Example 6.25, we have shown that F admits W
3
= L{1, |x|
2
, |x |
4
}, and have
constructed the corresponding solutions. Consider now the discretized radially sym-
metric operator
F
h
[u] = u
u
rr
+
N−1
r
u
r
−
N+2
4
(u
r
)
2
, where r =|x |, (9.68)
that is defined in the space of functions on the uniformgrid X
+
h
= X
h
∩{r > 0}, with
the standard approximation of the derivatives u
r
and u
rr
. The subspace of functions
restricted to X
+
h
is then invariant under the operator (9.68). Hence,
u = C
1
+ C
2
r
2
+ C
3
r
4
⇒ (9.69)
F
h
[u] = 2NC
1
C
2
+ 4(N −1)C
1
C
3
h + 2C
1
C
3
h
2
+
4(N + 2)C
1
C
3
+ (N −2)C
2
2
− 12C
2
C
3
h
+
2C
2
C
3
− 4(N +2)C
2
3
h
2
r
2
+
2NC
2
C
3
− 4(N +5)C
2
3
h + 2C
2
3
h
2
r
4
∈ W
3
.
Therefore, the discrete equation admits solutions (9.69), which, for h,τ = 0, coin-
cide with the correspondingdifferential ones.
9.3.3 Five-dimensional invariant subspaces for N = 1
1. Polynomial subspaces. We begin with the 1D quasilinear heat equation from
Example 1.14,
v
t
=
v
−
4
3
v
x
x
+ av
7
3
+ bv.
The pressure function u = v
−
4
3
satisfies the quadratic PDE
u
t
= F[u] ≡ uu
xx
−
3
4
(u
x
)
2
−
4
3
a −
4
3
bu. (9.70)
© 2007 by Taylor & Francis Group, LLC
9 Invariant Subspaces for Discrete Operators 445
F is known to preserve the 5D subspace, W
5
= L{1, x , x
2
, x
3
, x
4
}, so that (9.70)
possesses 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
, (9.71)
C
1
= 2C
1
C
3
−
3
4
C
2
2
−
4
3
a −
4
3
bC
1
≡
1
,
C
2
= 6C
1
C
4
− C
2
C
3
−
4
3
bC
2
≡
2
,
C
3
= 12C
1
C
5
+
3
2
C
2
C
4
− C
2
3
−
4
3
bC
3
≡
3
,
C
4
= 6C
2
C
5
− C
3
C
4
−
4
3
bC
4
≡
4
,
C
5
= 2C
3
C
5
−
3
4
C
2
4
−
4
3
bC
5
≡
5
.
(9.72)
For the corresponding discretized equation (9.70), the same subspace is obtained.
Proposition 9.22
The discrete operator
F
in
(9.70)
admits
W
5
.
Proof. Substituting (9.71) restricted to X
h
into (9.70) and using formulae
(x )
x
= 1,(x
2
)
x
= 2x,(x
3
)
x
= 3x
2
+ h
2
,(x
4
)
x
= 4x
3
+ 4xh
2
, (9.73)
we find that (9.70) has exact solutions (9.71), where the coefficients solve the DS
C
1
=
1
+ (2C
1
C
5
−
3
2
C
2
C
4
)h
2
−
3
4
C
2
4
h
4
,
C
2
=
2
− (4C
2
C
5
+ 3C
3
C
4
)h
2
− 6C
4
C
5
h
4
,
C
3
=
3
− (10C
3
C
5
+
9
2
C
2
4
)h
2
− 12C
2
5
h
4
,
C
4
=
4
− 22C
4
C
5
h
2
,
C
5
=
5
− 22C
2
5
h
2
.
(9.74)
Passing to the limit as h,τ→ 0 in (9.74) yields the DS (9.72).
Example 9.23 (Extended polynomial subspaces) All the polynomial subspaces
remain to exist for discrete higher-order operators. For instance, Proposition 3.2 es-
tablished that the fourth-order quadratic thin film equation (TFE)
u
t
= F[u] ≡−uu
xxxx
+ βu
x
u
xxx
,
where the basic subspace is known to be W
5
= L{1, x , x
2
, x
3
, x
4
}, and admits solu-
tions on the extended subspaces
W
6
= L{1, x , x
2
, x
3
, x
4
, x
5
}, if β =
2
5
,
W
7
= L{1, x , x
2
, x
3
, x
4
, x
5
, x
6
}, if β =
1
2
.
The same subspaces can beused for the discretized operator, F[u], with any standard
approximation of the derivatives, and lead to similar discrete DSs.
2. Trigonometric subspaces. Next consider the quasilinear heat equation with ab-
sorption from Example 1.15,
v
t
=
v
−
4
3
v
x
x
− v
−
1
3
.
The pressure transformation u = v
−
4
3
reduces it to
u
t
= F
0
[u] ≡ uu
xx
−
3
4
(u
x
)
2
+
4
3
u
2
, (9.75)
© 2007 by Taylor & Francis Group, LLC
446 Exact Solutions and Invariant Subspaces
where the quadratic operator F admits
W
5
= L
1, cos(λx), sin(λx), cos(
λx
2
), sin(
λx
2
)
, with λ=
4
√
3
. (9.76)
Hence, (9.75) has solutions
u= C
1
+ C
2
cos(λx)+ C
3
sin(λx)+ C
4
cos(
λx
2
)+ C
5
sin(
λx
2
). (9.77)
The DS is given in Example 1.15.
Unlike in the case of polynomial subspaces, for trigonometric ones, the invariance
of the nonlinear operator is no longer true for the discretization taken in the same
form as in (9.75). For preserving the subspace in the discrete case, we consider a
slightly different approximation of the operator,
u
t
= F
h
[u]≡ uu
xx
+α(u
x
)
2
+βu
2
, (9.78)
where the parametersα andβ depend on h, and conclude as follows:
Proposition 9.24
Operator
F
h
in
(9.78)
admits subspace
(9.76)
if the constants
α,β
and
λ
satisfy the system
m
1+α cos
2
(
λh
2
)
= β,
m
1 + 4cos
2
(
λh
4
)[1 + α cos(
λh
2
)]
= β
(9.79)
(
here
m =
4
h
2
sin
2
(
λh
2
)
as in
(9.49))
.
In the case of the quasilinear heat operator (cf. (9.75) withσ=−
4
3
),
F
h
[u] = uu
xx
+
1
σ
(u
x
)
2
− σ u
2
,
i.e., we still fix α =
1
σ
and β =−σ with some σ = σ(h), (9.79) gives the following
asymptotic values of the parameters σ and λ for which the 5D subspace (9.76) exists:
as h → 0,
σ =−
4
3
+
68
27
h
2
+ O(h
4
) and λ
2
=
16
3
+
32
27
h
2
+ O(h
4
).
For h = 0, we obtain the differential case σ =−
4
3
, λ =
4
√
3
, and subspace (9.76).
Finally, let us present the discrete DS for the coefficients of the solution (9.77) of
the discrete equation (9.78). Under the hypotheses of Proposition 9.24, this is
C
1
= βC
2
1
+ γ
1
C
2
2
+ C
2
3
+ γ
2
C
2
4
+ C
2
5
,
C
2
= γ
3
C
1
C
2
+ γ
4
C
2
4
− C
2
5
,
C
3
= γ
3
C
1
C
3
+ 2γ
4
C
4
C
5
,
C
4
= γ
5
C
1
C
4
+ γ
6
(C
2
C
4
+ C
3
C
5
),
C
5
= γ
5
C
1
C
5
− γ
6
(C
2
C
5
− C
3
C
4
),
where the coefficient {γ
i
} are
γ
1
= β − 4ρ
1
,γ
2,4
=
1
2
(β ± αρ
1
− 4ρ
2
), γ
3
= 2(β − 2ρ
1
),
γ
5
= 2(βh
2
− 2ρ
2
), γ
6
=
2
h
2
α sin(
λh
2
) sin(λh),
(9.80)
with ρ
1
=
1
h
2
sin
2
(
λh
2
) =
m
4
and ρ
2
=
1
h
2
sin
2
(
λh
4
).Forh = 0, coefficients (9.80)
determine those in the differential system.
© 2007 by Taylor & Francis Group, LLC
9 Invariant Subspaces for Discrete Operators 447
Example 9.25 (The TFE) It follows from Proposition 3.16 that the TFE
u
t
= F
0
[u] ≡−uu
xxxx
+
15
22
u
x
u
xxx
+
56
11
u
2
admits solutions on W
5
= L{1, cos x, sin x , cos2x, sin2x }. In order to keep this
subspace invariant in the discretized case, the operator is chosen to contain two pa-
rameters, depending on h,
u
t
= F
h
[u] ≡−uu
xxxx
+ βu
x
u
xxx
+ δu
2
,
where β →
15
22
and δ →
56
11
as h → 0.The correspondingdiscrete DS also converges
to the continuous one.
9.3.4 Exact solutions on partially invariant subspaces
Example 9.26 (Quasilinear heat equations) We takesolutionsfrom Example8.13,
and consider the PDE
v
t
= (ϕ(v))
xx
− (av + b)
1 +
2a
ϕ
(v)
G(v)
, where G(v) =
v
ϕ
(z) dz
az+b
.
Then the function u = G(v) solves
u
t
= F[u] ≡ (u)(u
xx
− 1) + a[(u
x
)
2
− 2u], (9.81)
with (u) = ϕ
(G
−1
(u)). Consider (9.81) on the subspace W
2
= L{1, x
2
} which is
not invariant under F. Setting u = C
1
+ C
2
x
2
∈ W
2
yields
F[u] = (u)(2C
2
− 1) + a
2C
2
(2C
2
− 1)x
2
− 2C
1
,
so that F[u] ∈ W
2
for all u ∈ W
2
. We have shown that, for F, W
2
is partially
invariant and there exists the invariant set (an affine subspace in W
2
),
M =
u = C
1
+ C
2
x
2
: C
2
=
1
2
, such that F[M] ⊆ W
2
.
On M, (9.81) reduces to an overdeterminedDS which is consistent. Substituting into
(9.81) yields the solution
u(x , t) = C
1
(t) +
1
2
x
2
∈ M, where C
1
=−2aC
1
. (9.82)
The same analysis applies to the discrete equation (9.81), since the discrete oper-
ator F admits the set M. Therefore, there exists the solution (9.82) on M, with the
coefficient C
1
(t) satisfying the discrete linear equation C
1
=−2aC
1
in T
τ
.
Theresults on partially invariantpolynomialsubspacesare easy to extendto higher-
order PDEs similar to those in Example 8.7 and Proposition 8.8. In the next PDE
associated with Example 8.44, the conclusion for the discrete equation is not so
straightforward, since we deal with trigonometric subspaces for which the consis-
tency of overdetermined DSs is not easy to check.
Example 9.27 (Semilinear parabolic model) Consider the PDE
u
t
= F
1
[u] + F
2
[u] ≡
u
xx
−
1
2
(u
x
)
2
u
+
2
u
+ 2u
+
1
2
(u
x
)
2
+ 2u
2
+ 2
. (9.83)
© 2007 by Taylor & Francis Group, LLC
448 Exact Solutions and Invariant Subspaces
Forconvenience,we briefly repeat the argumentfrom Example 8.44.Subspace W
2
=
L{1, sin 2x} is invariant under F
2
and, for any
u(x , t) = C
1
(t) + C
2
(t) sin 2x, (9.84)
F
2
[u] = 2
C
2
1
+ C
2
2
+ 2 + 4C
1
C
2
sin2x ∈ W
2
, but W
2
is not invariant under F
1
,
where
F
1
[u] =
2(C
2
1
−C
2
2
+1)
C
1
+C
2
sin2x
for u ∈ W
2
. (9.85)
There exists the set M on W
2
(a hyperbolain the variables {C
1
, C
2
}),
M =
u = C
1
+ C
2
sin2x : C
2
1
− C
2
2
+ 1 = 0
(9.86)
on which F
1
[u] = 0. Plugging (9.84) into (9.83) yields the overdetermined DS
C
1
= 2
C
2
1
+ C
2
2
+ 2,
C
2
= 4C
1
C
2
,
C
2
1
− C
2
2
+ 1 = 0,
(9.87)
which yields the solution of (9.83), u(x, t) = tan4t +
1
cos4t
sin2x ∈ M.
Consider the corresponding discretized equation
u
t
= F
h
[u] ≡ F
1h
[u] + F
2h
[u] (9.88)
on the subspace W
h
2
= L{1, sin(λx)}.OperatorF
2h
is as shown in (9.83), and
F
1h
[u] = αu
xx
−
1
2u
(u
x
)
2
+
2
u
+ 2u,
where α(h) → 1(andλ(h) → 2) as h → 0.
Proposition 9.28 (i) W
h
2
is invariant under
F
2h
iff
1
h
sin(λh) = 2. (9.89)
(ii)
If, in addition,
α = cos
2
(
λh
2
),
then
F
h
[M] ⊆ W
h
2
.
Since F
1h
[u] = 0onM and, as in the differential case,
F
2h
[u] = 2
C
2
1
+ C
2
2
+ 2 + 4C
1
C
2
sin(λx),
(9.88) on M is equivalent to the overdetermined discrete system (9.87), where the
continuous derivatives (·)
are replaced by discrete ones. It can be shown that, unlike
in the differential case, for both the implicit and explicit finite-difference schemes,
the discrete dynamical system does not have a solution satisfying C
2
1
− C
2
2
+ 1 = 0
on T
τ
, so our result is negative. On the other hand, if the time is not discretized,
system (9.87) gives the exact solutions of u
t
= F
h
[u]. Similarly, we can analyze
invariant sets and consistency of the DSs for discretized higher-order PDEs studied
in Examples 8.43–8.45.
9.4 Invariant properties of moving mesh operators and applications
The movingmesh methods(MMMs)are knownto be an efficient approachto solving
the nonlinear evolution PDEs where the solutions exhibit essentially non-stationary
© 2007 by Taylor & Francis Group, LLC
9 Invariant Subspaces for Discrete Operators 449
and singular behavior, such as blow-up, extinction, or quenching. We will describe
some basics of MMM theory and refer to [293, 292, 99] for general principles and
applications.
We apply the results on invariant subspaces for discrete operators to some MMMs
for typical parabolic or hyperbolic PDEs. We show that, for a class of nonlinear 1D
evolution equations,the MMMs may preservelinearinvariant subspaces of nonlinear
differential operators under a special approximation of the spatial gradient operator.
The corresponding discrete evolution on such subspaces may coincide with that for
fixed meshes, and even with the continuous ones, regardless of the fast deformation
of the moving mesh.
9.4.1 Introduction: MMMs and invariant subspaces
We deal with a general nonlinear evolution PDE
u
t
= F[u], x ∈ S ⊆ IR , t > 0, (9.90)
where F[u] = F
u, u
x
, ..., D
k
x
u
, F ∈ C
∞
is an kth-order ordinary differential
operator, and S is an interval. Assume that, being endowed with suitable boundary
conditions on the lateral boundary of Q = S × IR
+
(if S = IR ), the equation (9.90)
defines a smooth flow for bounded initial data u
0
.ForS = IR , we mean the Cauchy
problem with a given initial function u
0
(x ).
The MMMs approximate (9.90) on a moving mesh (MM) ={x = X
i
(t), i ∈
I, t > 0}⊂Q, I ={1, 2, ..., K }, with aprioriunknown curves x = X
i
(t), depend-
ing on the solution u(x , t) under consideration. Using identity
d
dt
u(X
i
, t) = u
t
(X
i
, t) + u
x
(X
i
, t)X
i
,
the differential equation for U ={U
i
(t) = u(X
i
(t), t)} reads
U
i
− u
x
(X
i
, t)X
i
= F[u(X
i
, t)], t > 0; i ∈ I.
Discretizing this equation yields a dynamical system,
U
i
− [D
h
U]
i
X
i
= F
h
[U]fort > 0, (9.91)
where F
h
is a suitable approximation of the nonlinear operator F,andD
h
approxi-
mates the gradient operator D
x
=
d
dx
. The evolution equations for the moving mesh
are added to (9.91), and these, all together, give a finite-dimensional DS for the
functions{U
i
(t), i ∈ I }. Such approaches make it possible to eventually concentrate
the MM close to crucial singularities of the solutions. Giving such advantages in
computing the singular behavior, the extra linear discrete gradient operator D
h
that
appears in (9.91) may strongly change the discrete PDE.
Evenfor semilinear second-orderparabolic PDEs, equation (9.91)no longer obeys
the Maximum Principle (MP). On the contrary, if X
i
(t) ≡ 0foralli, i.e., the mesh is
fixed (stationary), the MP holds, provided that the approximation F
h
preserves such
a positivity property. Therefore, moving meshes may destroy the positivity or mono-
tonicityfeaturesof solutionsthat are expectedto be inheritedby the discreteequation
from the originalparabolic PDE. Mathematical analysis of MMMs is harder than for
© 2007 by Taylor & Francis Group, LLC
450 Exact Solutions and Invariant Subspaces
the corresponding explicit or implicit difference schemes with stationary meshes.
This includes the questions of convergence of MMMs and their asymptotic behav-
ior. Comparison of solutions of MMMs for parabolic PDEs becomes a complicated
procedure, because the solutions are defined on the distinct MMs, depending on so-
lutions that are unknown apriori.
We will study the principal question of invariant linear subspaces generated by
the MM operators (MMOs). It was shown in the previous section how to choose a
linear subspace W that is invariant under a discrete nonlinear operator F
h
. In con-
trast, for typical DSs (9.91) occurring for the MMMs, extra work should be done.
We will need to prove the necessary and sufficient condition for the gradient approx-
imation D
h
to preserve the invariance of a given subspace W . This guarantees that
the invariant MMO admits the same subspace and, moreover, the discrete dynamics
on W coincides with the differential one, regardless of the structure of the MM.
For parabolic PDEs, we establish the necessary and sufficient conditions for the
positivity of the solutions (the weak MP), and also prove comparison of solutions
of MMMs. As a general conclusion, for typical semilinear parabolic PDEs, these
important properties remain valid if the MM does not move too fast.
We will also consider invariant MMOs for higher-order evolution PDEs,
D
m
t
u = F[u] ≡ F
u, u
t
, ..., D
m−1
t
u, u
x
, ..., D
k
x
u
, (9.92)
where m ≥ 2andm ≥ k. In particular, we consider the second-order hyperbolic
PDEs
u
tt
= F(u, u
x
, u
xx
).
9.4.2 Invariant subspaces for differential and discrete operators
For convenience, we list below a number of already known differential and corre-
sponding discrete quadratic operators preserving invariant subspaces.
1. Invariant subspaces of differential operators. Let us state the main invariance
hypothesis on the nonlinear operator F.
Hypothesis (Inv). The operator F admits a 2D subspace W
2
= L{1, f (x )},where f
is a given smooth function, i.e.,
F[W
2
] ⊆ W
2
. (9.93)
Under hypothesis(Inv), the PDE (9.90) on W
2
is equivalentto a nonlinearsecond-
order DS. In view of (9.93), for any u = C
1
+ C
2
f ∈ W
2
,
F[C
1
+ C
2
f ] =
1
(C
1
, C
2
) +
2
(C
1
, C
2
) f ∈ W
2
. (9.94)
Substituting (9.94) into (9.90) yields the DS
C
1
=
1
(C
1
, C
2
),
C
2
=
2
(C
1
, C
2
).
(9.95)
The following basic examples of operators and invariant subspaces are taken from
Sections 1.3–1.5 and 3.1.
© 2007 by Taylor & Francis Group, LLC
9 Invariant Subspaces for Discrete Operators 451
(i) Quadratic operators with 2D subspaces:
F
1
[u] = u
xx
+ (u
x
)
2
+ u, W
2
= L{1, x
2
}, (9.96)
F
2
[u] = u
xx
+ (u
x
)
2
+ u
2
, W
2
= L{1, cos x }, (9.97)
F
3
[u] = uu
xx
+
1
σ
(u
x
)
2
+ αu + β, W
2
= L{1, x
2
}, (9.98)
F
4
[u] = uu
xx
+
1
σ
(u
x
)
2
+ σ u
2
, W
2
= L{1, cos(λx )},λ=
σ
√
σ +1
, (9.99)
F
5
[u] =−u
xxxx
− (u
x
)
2
+ u
2
, W
2
= L{1, cosh x}, (9.100)
F
6
[u] =−uu
xxxx
+ αu
x
u
xxx
+ β(u
xx
)
2
+ γ u + δ, W
2
= L{1, x
4
}. (9.101)
The operators F
1
– F
4
correspond to the semilinear and quasilinear second-order
parabolic PDEs of reaction-diffusion type, where F
3
and F
4
contain the differential
operatoruu
xx
+
1
σ
(u
x
)
2
of the porousmedium(σ>0) or fast diffusion(σ<0) type.
The operator F
5
reproducesa fourth-order parabolic equation, which is a Kuramoto–
Sivashinsky-type PDE with the extra lower-order source term u
2
(see Section 3.8),
while F
6
occurs in TFE theory (Section 3.1). PDEs
u
tt
= F
3
[u] + γ u
xxxx
+ δ u
xx
and u
tt
= F
4
[u] + γ u
xxxx
+ δ u
xx
areBoussinesq-typeequationsfrom waterwave theory(Section5.3).We nextdiscuss
their exact solutions in greater detail.
Example 9.29 (PME with source) The quasilinear heat equation with a source,
u
t
= F
4
[u] ≡ uu
xx
+
1
σ
(u
x
)
2
+ σ u
2
,σ>0, (9.102)
plays a key role in the study of localization of blow-up in combustion, [509, Ch. 4].
The quadratic operator F
4
admits subspace W
2
given in (9.99), so that this PDE has
the solutions
u(x , t) = C
1
(t) + C
2
(t) cos(λx) ∈ W
2
, (9.103)
C
1
= σ C
2
1
+
σ
σ +1
C
2
2
,
C
2
=
σ(σ+2)
(σ +1)
C
1
C
2
.
(9.104)
Such solutions describe localization and blow-up phenomena. In particular, the Stur-
mian intersection comparison with such solutions establishes [509, p. 249] that the
interface s(t) of any weak solution u(x , t) ≥ 0 of (9.102) with the connected com-
pact support satisfies the localization estimate
|s(t) − s(0)|≤
1
2
L
S
= π
√
σ +1
σ
for any t ∈ [0, T )
(L
S
is the fundamental length), where T is the blow-up time of the solution. If
C
1
(t) ≡ C
2
(t) in (9.104), we obtain the ODE C
1
= α C
2
2
with α =
σ(σ+2)
σ +1
. Hence,
C
1
(t) = [α(T − t)]
−1
, and this gives the separable solution
u(x , t) =
1
α(T −t)
2cos
2
(
λx
2
) ≥ 0. (9.105)
Therefore, subspace W
2
contains the 1D manifold (with the parameter T ∈ IR )of
similarity solutions (9.105). It is important that these solutions are asymptotically
stable; see the stability analysis of blow-up dynamics in Example 3.17.
© 2007 by Taylor & Francis Group, LLC
452 Exact Solutions and Invariant Subspaces
Example 9.30 (Quasilinear hyperbolic equation)ThePDE
u
tt
= F
4
[u] ≡ (uu
x
)
x
+ u
2
,
which is hyperbolicin the positivity domain {u > 0}, admits solutions
u(x , t) = C
1
(t) + C
2
(t) cos
x
√
2
∈ W
2
= L
1, cos
x
√
2
,
C
1
= C
1
+
1
2
C
2
2
,
C
2
=
3
2
C
1
C
2
.
For C
1
= C
2
,wefind the ODE C
1
=
3
2
C
2
1
, which is integrated in quadratures
with the solutions given by elliptic functions. Taking C
1
(t) = 4(T − t)
−2
gives the
separable blow-up solution
u(x , t) =
4
(T −t )
2
1 + cos
x
√
2
.
Its asymptotic stability on W
2
can be checked, as done in Example 5.1.
(ii) Five and more-dimensional subspaces exist for special quadratic operators:
F
7
[u] = uu
xx
−
3
4
(u
x
)
2
, with W
5
= L{1, x , x
2
, x
3
, x
4
}
(notice that (9.101) also admits W
5
), and
F
8
[u] = uu
xx
−
3
4
(u
x
)
2
+
4
3
u
2
,
with W
5
givenin (9.76).Notethat F
7
also admits 2D subspaces L{1, x
2
}and L{1, x
4
}.
Operator
F
9
[u] = uu
xx
−
2
3
(u
x
)
2
admits W
2
= L{1, x
3
}, which is extended to the 4D W
4
= L{1, x , x
2
, x
3
};see
Example 1.13. Fourth-order thin film operators from Section 3.1 are the origin of
many invariant subspaces. For instance, in the previous section we have used
F
10
[u] =−uu
xxxx
+
2
5
u
x
u
xxx
,
admitting W
6
= L{1, x , x
2
, x
3
, x
4
, x
5
}and the 2D restrictionsL{1, x
5
}and L{1, x
4
}.
Operator
F
11
[u] =−uu
xxxx
+
1
2
u
x
u
xxx
admits W
7
= L{1, x, x
2
, x
3
, x
4
, x
5
, x
6
},aswellasL{1, x
6
} and L{1, x
4
}.
2. Discrete operators. In Section 9.3, we haveshown that invariantsubspaces can be
preservedand are stable with respect to standard discretizations of operators on fixed
meshes. In other words, the corresponding discrete operators, F
kh
, with sufficiently
small steps of uniform discretization h > 0 can admit subspaces with the basis
functions being O(h)-perturbations of those for differential operators. For the 2D
subspace, W
2
= L{1, f (x)}, this implies that
F
h
[C
1
+ C
2
f ] ≡
1h
(C
1
, C
2
) +
2h
(C
1
, C
2
) f ∈ W
2
, (9.106)
so that the discrete equation on a fixed mesh,
U
t
= F
h
[U],
© 2007 by Taylor & Francis Group, LLC