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 433

linearly independent solutions v

1,i

= 2

and v

2,i

= (−2)

. By Proposition 9.3, there

exist two linearly independent ﬁrst integrals of (9.15),

[u]

= 2



i+1



and I

[u]

= (−2)



i+1

−



It follows from Theorem 9.1 that the family of all the ﬁrst-order nonlinear operators

admitting the 2D subspace (9.16) is given by

F[u]

= A

[u]

, I

[u]

−i

+ A

[u]

, I

[u]

)(−2)

−i

where A

, A

: IR

→ IR are arbitrary functions. Due to Corollary 9.2, for any

g ∈ W

, equation F[u] = g is equivalent to a system of two algebraic equations.

For applications, it is important to derive all the autonomous operators admitting

, i.e., those that do not depend on the independent variable i ∈ Z. Denoting

i+1

) = a and (−2)

i+1

−

) = b, we infer that such operators,

which are invariant under translation i → i + 1, satisfy

(2a, −2b) − 2A

(a, b)]2

−i−1

+[A

(2a, −2b) + 2A

(a, b)](−2)

−i−1

= 0

for all i ∈ Z and a, b ∈ IR . In view of the linear independence of the functions,

(2a, −2b) = 2A

(a, b) and A

(2a, −2b) =−2A

(a, b) (9.17)

for all a, b ∈ IR . Thus the autonomous operators have the form

F[u]

= A



i+1

, u

i+1

−



+ A



i+1

, u

i+1

−



where A

and A

are arbitrary functions satisfying (9.17). For instance, the operators

[u]

i+1

)

i+1

−

)

and F

[u]

i+1

)

i+1

−

)

admit W

. A more general operator preserving W

is given by

F[u]

= C

[u]

+ C

[u]

L[u]

where C

and C

are constants, and

L is any linear operator admitting W

9.2 On the forward problem of invariant subspaces

As in the differential case in Chapter 2, the general forward Problem I of invariant

subspaces: given a nonlinear ﬁnite-order difference operator, determine all the linear

subspaces that it admits, remains

OPEN.

In this section, as a key example, we illustrate the forward problemfor the simplest

quadratic ﬁrst-order operator

F[u]

= (u

i+1

− u

)

. (9.18)

The forward problem for the corresponding differential Hamilton–Jacobi operator

[u] = (u

)

((9.18) is its difference approximation[

i+1

−u

)]

with h = 1) has been solved;

434 Exact Solutions and Invariant Subspaces

see Section 2.2. To be precise, up to translations in x , the operator (9.18) admits: (i)

the 1D subspaces W

= L{x

} and W

= L{1}, (ii) 2D subspaces W

= L{1, x } and

= L{1, x

}, given by the ODEs

= 0and u

and (iii) a single 3D subspace W

= L{1, x, x

} (u

xxx

= 0).

For convenience, step by step, we give a classiﬁcation of the invariant subspaces

for the discrete operator (9.18) with n ≤ 2, n = 3, and n ≥ 4 respectively.

Case I: n ≤ 2.

Theorem 9.5

Operator

(9.18)

admits

with

n ≤ 2

given by the linear equation

with constant coefﬁcients

L[u]

≡ u

i+1

− Bu

− Cu

i−1

= 0 (B, C ∈ IR ), (9.19)

in the following four cases only:

(i) B = C = 0 (

equation

i+1

= 0)

={0}

;

(ii) B = 1, C = 0 (u

i+1

− u

= 0)

= L{1}

;

(iii) B = 0, C = 1 (u

i+1

− u

i−1

= 0)

= L{1,(−1)

}

;

(iv) B = 2, C =−1 (u

i+1

− 2u

+ u

i−1

= 0)

= L{1, i }

The subspace in (iii) containing 2-periodic functions is not available in the classi-

ﬁcation of the differential operators.

Proof. Consider the invariance criterion L[F[u]] = 0ifL[u] = 0, or

i+2

− u

i+1

)

− B(u

i+1

− u

)

− C(u

− u

i−1

)

≡ 0

on solutions of (9.19). Using (9.19) to eliminate u

i+1

and u

i+2

, and taking into ac-

count that u

, u

i−1

,andu

i−1

are linearly independent, we arrive at the system



+ C − B)

= B(B − 1)

+ C,

C(B

+ C − B)(B − 1) = C[B(B − 1) − 1],

(B − 1)

= C(BC + 1).

(9.20)

Consider two cases:

1. Case C = 0. The ﬁrst equation yields B(B − 1)

= 0, so B = 0or1.

2. Case C = 0. Then, substituting C from the second equation (9.20), C =−B(B −

2) −

B−1

(B = 1), yields the following two equations for B:





B −

B−1



− B(B − 1)

=−B(B − 2) −

B−1

−[(B − 1)

− B]



B(B − 2) +

B−1



= 1,

that are reduced to B

(B − 2)(B

− 4B + 5) = 0andB

(B − 2)

(B − 1) = 0, so

either B = 2, C =−1, or B = 0, C = 1.

Case II: n = 3. A similar result holds.

Theorem 9.6

Operator

(9.18)

admits a 3D subspace

given by

L[u]

≡ u

i+3

− Bu

i+2

− Cu

i+1

− Du

= 0

iff

9 Invariant Subspaces for Discrete Operators 435

(i) B = 3, C =−3, D = 1

= L{1, i, i

}

;

(ii) B = C = 0, D = 1

= L



1, cos

2πi

, sin

2πi



;and

(iii) B = C = 1, D =−1

= L{1, i,(−1)

}

Case III: n ≥ 4. Finally, consider linear subspaces given by equations

L[u]

≡ u

i+n

−



n−1

k=0

i+k

= 0, n ≥ 4 (a

∈ IR ). (9.21)

Theorem 9.7

The only subspaces

with

n ≥ 4

invariant under

(9.18)

are

(i) u

i+n

= u

i+n−1

+ u

i+1

− u

= L



1, i, e

2πiI/(n−1)

, ..., e

2πiI(n−2)/(n−1)



;and

(ii) u

i+n

= u

= L



1, e

2πiI/n

, ..., e

2πiI(n−1)/n



where

=−1

A real-valued representation of these subspaces is straightforward.

Proof. Consider the invariance criterion: for any solutions of (9.21),

i+n+1

− u

i+n

)



n−1

k=0

i+k+1

− u

i+k

)

. (9.22)

We exclude the variables

i+n+1



n−1

k=0

i+k+1

= a

n−1

i+n



n−1

k=1

k−1

i+k



n−1

k=1

n−1

+ a

k−1

i+k

+ a

n−1

and u

i+n

given by (9.21), so that

i+n+1

− u

i+n



n−1

k=1

[(a

n−1

− 1)a

+ a

k−1

i+k

+ (a

n−1

− 1)a

i+n

− u

i+n−1

= (a

n−1

− 1)u

i+n−1



n−2

k=1

i+k

+ a

For convenience, setting

β = a

n−1

− 1, (9.23)

and substituting the representations given above into (9.22) yields





n−1

k=1

(βa

+ a

k−1

i+k



+ 2βa



n−1

k=1

(βa

+ a

k−1

i+k

+β

= a

n−1



βu

i+n−1



n−2

k=1

i+k



+2a



βu

i+n−1



n−2

k=1

i+k



+ a



n−2

i+n−1

− u

i+n−2

)

+ a

n−3

i+n−2

− u

i+n−3

)

+... + a

i+2

− u

i+1

)

+ a

i+1

− u

)

(9.24)

Equating the coefﬁcients of the factors u

in both sides for m = 0, 1, and 2, respec-

tively, we obtain the following. The equation corresponding to u

has the form





n−1

k=1

(βa

+ a

k−1

i+k



= a

n−1



βu

i+n−1



n−2

k=1

i+k



n−2

i+n−1

− u

i+n−2

)

+ ... + a

i+2

− u

i+1

)

+ a

i+1

(9.25)

The equation for u



n−1

k=1

(βa

+ a

k−1

i+k

= a



n−1



βu

i+n−1



n−2

k=1

i+k



− u

i+1



(9.26)

Since a

= 0, analyzing the coefﬁcients in (9.26) we obtain:

(i) for u

i+n−1

β[(β − 1)a

n−1

+ a

n−2

] = 0; (9.27)

436 Exact Solutions and Invariant Subspaces

(ii) for u

i+k

with 2 ≤ k ≤ n − 2,

β(βa

+ a

k−1

) = a

n−1

; (9.28)

and (iii) for u

i+1

β(βa

+ a

) = a

n−1

− 1. (9.29)

Finally, considering the coefﬁcients of u

in (9.24) yields

= a

n−1

+ 1. (9.30)

I. If β = 0, i.e., a

n−1

= 1, the system (9.27)–(9.30)takes the form

= 0for2≤ k ≤ n − 2, a

= 1, a

=−1

(where (9.25) becomes the identity), and we arrive at the case (i) of the theorem.

The subspace is determined from the characteristic equation λ

= λ

n−1

+ λ − 1,

so that λ = 1orλ

n−1

= 1, i.e., denoting I

=−1, we have λ

= e

2πkI/(n−1)

for

k = 1, 2, ..., n − 2, and λ

n−1

= λ

= 1.

II. Let β = 0, i.e., a

n−1

= 1. Then the system (9.27)–(9.30) reads

(β − 1)a

n−1

+ a

n−2

= 0,(β

− a

n−1

+ βa

k−1

= 0, k = 2, ..., n − 2,

(β

− a

n−1

+ βa

=−1,(β

− a

n−1

= 1.

Denoting

α = β

− a

n−1

= β

− β − 1, (9.31)

we obtain the solution

(α = 0), a

=−

α+β

, a



−



k−1

, (9.32)

where k = 2, ..., n − 2anda

n−2

= 1 −β

,sincea

n−1

= 1 +β by (9.23).

Consider (9.25). The coefﬁcient of u

i+1

is (βa

)

= a

n−1

, which

yields, by (9.23) and (9.32), [β(α+β)−α]

= (1+β)(α +β)

−βα

. Substituting

α from (9.31) yields β

(β +1)(β − 2)(β

− β + 1) = 0. If β =−1, then α = 1by

(9.31) that gives the subspace in (ii).

If β = 2, then α = 1anda

n−1

= 1 + β = 3, a

= 1, a

=−3and

=−3(−2)

k−1

for k = 2, ..., n − 2. Then the last equation in (9.32) implies

that (−2)

n−3

= 1 holds, which results in n = 3, contradicting the assumption. This

completes the proof.

It is obvious that if u

i+n

= u

as in (ii), then

F[u]

i+n

= F[u]

(9.33)

for any operator F that does not depend explicitly on i. The linear subspace W

given

in (ii) is invariant under such an arbitrary operator.

On the maximal dimension of invariant subspaces. In the differential case, the or-

der p ofa nonlinear ordinarydifferentialoperator F is knownto satisfy the following

estimate (Theorem 2.8)

n ≤ 2p + 1, (9.34)

where n is the maximal dimension of an invariant subspace. If n > 2p + 1, F is a

9 Invariant Subspaces for Discrete Operators 437

linear operator. It follows from Theorem 9.7 that this is not the case for the discrete

operators, where the maximal dimension n can be arbitrarily large, even for the ﬁrst-

order quadratic operator (9.18). Actually, this again emphasizes the obviousfact that

the set of difference operators is wider than that of the differential ones. Each pth-

order differential operator

[u] = F(u, u

, u

, ..., D

can be obtained from a one-parameter family of the difference operators

[u] = F(u, u

, u

, ..., D

u),

where the arguments are discrete approximations of the derivatives, for instance,



i+1

− u

i−1



, u



i−1

− 2u

+ u

i+1



, ... . (9.35)

This is understood in the sense of a formal limit, F

= lim F

as h → 0. Without

fear of confusion, we will use the same notation for derivatives in continuous and

discrete cases.

Example 9.8 (5D subspace) The discrete second-order quadratic operator

[u] = uu

−

)

with the derivative approximations (9.35), admits the same 5D subspace as the dif-

ferential operator F

[u] (Example 1.14)

= L{1, x , x

, x

Both continuous and discrete parabolic equations, u

= F

[u]andu

= F

[u]on

, reduce to the ﬁfth-order DS for the coefﬁcients {C

(t)} of the solution

u(x , t) =



k=0

(t)x

Similarly, the hyperbolic second-order equations, u

= F

[u]andu

= F

[u]on

, reduce to tenth-order DSs.

9.3 Invariant subspaces for ﬁnite-difference operators

We perform a more detailed analysis of linear subspaces that are invariant under op-

erators, corresponding to ﬁnite-difference approximations of differential operators,

with polynomial nonlinearities. As a result, a certain structural stability of invari-

ant subspaces and sets of nonlinear differential operators of reaction-diffusion type,

with respect to their spatial discretization, is established, and lower-dimensional re-

ductions of the ﬁnite-difference solutions on the invariant subspaces are constructed.

We study invariant properties and exact solutions of ﬁnite-difference schemes for

nonlinear evolution PDEs

= F[u]inIR

× IR

, (9.36)

where F is an ordinary differential (elliptic) operator with polynomial-like nonlin-

438 Exact Solutions and Invariant Subspaces

earities. We illustrate the results using the quasilinear parabolic equations from non-

linear diffusion and combustion theory with operators

F[u] ≡ ϕ(u)u

+ ψ(u)(u

)

+ f (u), (9.37)

where ϕ, ψ,and f are given sufﬁciently smooth functions, and ϕ ≥ 0 (the parabol-

icity condition).

One of the basic tools of solving the PDE (9.36) numerically is the method of

ﬁnite differences. Consider ﬁrst the case of discretization in both the spatial and

time-variables. Using a given uniform time-grid, T

={nτ : n = 0, 1, ...} with the

ﬁxed, sufﬁciently small step τ>0,andaninﬁnite uniform space-grid, X

={x =

kh : k = 0, ±1, ...} with the ﬁxed small step h > 0, we replace (9.36), (9.37) by the

following implicit ﬁnite-difference equation

= F

[u] ≡ ϕ(u)u

+ ψ(u)(u

)

+ f (u) (9.38)

for (x, t) ∈ Q

hτ

= X

× T

. We again keep the same notation, u

, for discrete and

continuous derivatives. As usual, u = u(x, t) belongs to the space of grid functions

on Q

hτ

, and the derivatives denote



u(x , t + τ)− u(x, t)





u(x +h, t) − u(x − h, t)





u(x +h, t) + u(x − h, t) − 2u(x, t)



(9.39)

so that F

[u] = F[u] with the derivatives replaced by those given in (9.39). We

consider the Cauchy problem on X

with prescribed initial data, so we do not need

boundary conditions. The PDE is then replaced by a discrete, inﬁnite-dimensional

DS. In the case of the spatial discretization only (the method of lines), there occurs

the continuoustime-derivativeon the left-handside of (9.38) that is a continuous DS.

In the construction of the ﬁnite-difference approximation of the problem, the el-

liptic differential operator F

: C

(IR ) → C(IR ) is replaced by the corresponding

ﬁnite-difference operator F

: V → V ,whereV ={u : X

→ IR } denotes the

space of grid functions on X

. The type of discretization depends on the properties

of the differential model, which it is necessary to preserve. Namely, these may be

conservation laws (conservation of the mass, or of the ﬁrst moment, corresponding

to the diffusion operator, or other higher-order moments), preservation of a ﬁxed

symmetry or scaling invariance of the equation, etc. In what follows, a standard and,

sometimes, non-divergent type of discretization is chosen in order to preserve the

invariant subspace property of the discretized nonlinear operator and the PDE.

We will establish that the ﬁnite-difference approximation may preserve the prop-

erty of the nonlinear operator F to admit a subspace. We show that, in some cases, if

F admits a ﬁnite-dimensional linear subspace W , such that F[W ] ⊆ W , the approx-

imating operator F

can also have a subspace W

⊂ V (i.e., F

] ⊆ W

)ofthe

same dimension and with similar basis functions. In the discretized problem, there

is also a reduction of the dimension, which gives solutions on the lower-dimensional

subspaces. The same effect is proved to be true for the partially discretized equation

(9.38)with the continuousderivativeu

, so that (9.38)is a systemof nonlinearODEs.

In the discrete case, the space V is inﬁnite-dimensional (as well as in the differential

9 Invariant Subspaces for Discrete Operators 439

case), so by the exact solutions we mean those that can be expressed in terms of a

ﬁnite-dimensional discrete or continuous DS.

In some cases, discrete operators may admit invariant sets on linear subspaces

for which the exact solutions are obtained from overdetermined DSs. We show that

reductions to lower-order systems via invariant subspaces and sets exist under the

standard approximations of the differential operator F on the uniform constant grid,

provided that F admits an invariantsubspace. Since for parabolic problems, the stan-

dard comparison theorem upon initial data is true for the implicit schemes such as

(9.38), [509, Ch. 7] (or for the corresponding DSs approximation), particular solu-

tions on subspaces or on sets can be used to estimate more general solutions. This

makes it possible to describe the asymptotic behavior and attractors for equations

(9.38).

9.3.1 First examples

We begin with a special example, exhibiting some common features of the group-

invariant and invariant subspace solutions.

Example 9.9 (Semilinear heat equation) Consider the parabolic PDE

= v

+ v lnv,

which admits a four-parameter Lie group of symmetries with operators [10, p. 135]

∂

∂t

, X

∂

∂x

, X

= 2e

∂

∂x

− e

∂

∂v

, X

= e

∂

∂v

and, therefore, has a wide class of invariant solutions. There is a simple invariant

subspace interpretation of those solutions generated by the symmetry ε

+ X

with a constant ε

, [10, p. 140]. The pressure-like function u = lnv satisﬁes

= F

[u] ≡ u

+ (u

)

+ u. (9.40)

The quadratic operator F admits a simple subspace W

= L{1, x

},sothat,for

u(x , t) = C

(t) + C

(t)x

, (9.41)

the following holds: F

[u] = C

+2C



+4C



∈ W

. Plugging (9.41) into

(9.40) yields a nonlinear DS





= C

+ 2C



= C

+ 4C

(9.42)

This is easily solved explicitly and gives exact solutions; see precise expressions

in [10, p. 140]. The standard ﬁnite-difference scheme actually given by (9.40) with

discrete derivatives also admits solutions (9.41), with a similar action of the operator

and the same DS.

Example 9.10 The corresponding N-dimensional PDE

= v + v lnv

440 Exact Solutions and Invariant Subspaces

by the same transformation v = e

is reduced to

= F

[u] ≡ u +|∇u|

+ u in IR

× IR

. (9.43)

This quadratic PDE possesses the solutions

u(x , t) = C

(t) +



k=1

(t)x

(9.44)

on the subspace W

N+1

= L{1, x

, ..., x

} that is easily seen to be also admitted by

the operator of the corresponding scheme on a uniform grid X

= F

[u] ≡ 

u +|∇

+ u, (9.45)

with the standard discrete approximations



u =



k=1

and |∇



k=1

)

. (9.46)

Substituting (9.44) into (9.45) yields the system





= C

+ 2



(i)



= C

+ 4C

, k = 1, ..., N.

The DS for solutions (9.44) of the parabolic PDE (9.43) is the same with the differ-

ence derivatives

Remark 9.11 (Hyperbolic PDEs) Invariantreductions to lower-order systems exist

for discretized hyperbolic equations

= F

[u]

with the quadratic operators from (9.40) or (9.45), and for other operators to be

considered. Observe that, in the case of higher-order time-derivatives, the set of so-

lutions on the 3D subspace W

= L{1, x, x

} contains new nontrivial solutions (for

the parabolic case (9.40) such a generalization reveals nothing new).

Example 9.12 (Semilinear heat equation with blow-up) Consider the semilinear

PDE from Example 1.11,

= v

+ v ln

Setting v = e

yields the PDE possessing blow-up solutions,

= F[u] ≡ u

+ (u

)

+ u

, (9.47)

with the quadratic operator F preserving the subspace W

= L{1, cos x }.

Let us state an analogy of the invariant property for equation (9.47), where all the

derivatives are discrete.

Proposition 9.13

The discrete operator

(9.47)

admits

= L{1, cos(λx)},

where

λ =

sin

−1

h (0 < h ≤ 1). (9.48)

Proof. Using simple formulae

(cos(λx))

=−l sinλx and (cos(λx))

=−m cosλx , (9.49)

9 Invariant Subspaces for Discrete Operators 441

where l =

sin(λh) and m =

sin

(

λh

),weﬁnd that, for any u = C

cos(λx) ∈ W

F[u] = (mC

+ 2C

) cos(λx ) +



cos

(λx ) + l

sin

(λx )



+ C

Then F[u] ∈ W

implies l

= 1, so, without loss of generality, we set l = 1, which

yields λ =

sin

−1

h, while m = 2/(1 +

√

1 − h

Hence, there exist the following solutions of (9.47) in the discrete case:

u(x , t) = C

(t) + C

(t) cos(λx), (9.50)





= C

+ C



= mC

+ 2C

If h → 0, then λ, m → 1 and, more precisely, λ(h) = 1 +

+ O(h

), so that

subspace (9.48) coincides with W

= L{1, cos x } at h = 0.

Remark 9.14 (KS-type equations) Similarsolutionson W

existfor the discretized

fourth-order Kuramoto–Sivashinskytype equation

=−u

xxxx

+ F[u].

Example 9.15 (The PME) Consider the PME with lower-order terms

= (v

)

+ v

σ +1

+ av

1−σ

+ bv,

where σ = 0, −1anda and b are given constants. The pressure function u = v

solves

= F[u] ≡ uu

)

+ σ u

+ σ a + σ bu. (9.51)

The quadratic operator F is known to admit the 2D subspaces (see Section 1.4)



L{1, cos(λx)}, if σ>−1,

L{1, cosh(λx )}, if σ<−1,

where λ = σ/

√

|σ + 1|. Consider, for instance, the case of σ>−1, where there

exist solutions (9.50), with the DS





= σ C

σ +1

+ σ bC

+ σ a,



σ(σ+2)

σ +1

+ σ bC

(9.52)

Let (9.51) now be a standard ﬁnite-difference scheme. Similarly to the previous

example, we establish the following:

Proposition 9.16

The discrete operator

(9.51)

has the subspace

= L{1, cos(λx)},

+ mσ − σ

= 0, (9.53)

where

and

are as given in

(9.49)

For σ>−1, on the subspace (9.53), i.e., for functions (9.50) restricted to X

,the

PDE (9.51) is equivalent to the following DS:





= σ C

+ σ bC

+ σ a,



= (2σ − m)C

+ σ bC

(9.54)

442 Exact Solutions and Invariant Subspaces

Setting k = cos(λh), let us write down the algebraic equation in (9.53) in the form

+ 2σ k + σ

− 2σ − 1 = 0.

It is easy to clarify for which σ and h this equation possesses roots k ∈ [−1, 1].

Avoiding a detailed analysis, we indicate, for example, that, for σ ∈



−

1+h

, 0



∪





, there exists the root k =−σ +



(σ + 1)

− σ

, and, for the correspond-

ing λ =

cos

−1

k, one obtains that

λ(h) =

√

σ +1

(σ +4)

24(σ +1)

5/2

+ O(h

) for small h > 0. (9.55)

This means convergence as h → 0 to the solution (9.50), with the DS (9.52), of the

PDE (9.51).There are other branches of λ(h) that do not have a differential analogy.

Subspaces of the hyperbolic cosh(λx) function are studied in a similar fashion.

Example 9.17 (Quasilinear heat equations)Letϕ(v) be an arbitrarysmoothmono-

tone function with ϕ

−1

inverse. Consider the PDE from Example 8.3,

= (ϕ(v))

aϕ(v)+b



(v)

+ [aϕ(v) + b]c.

Setting u = aϕ(v) + b yields

= F[u] ≡ (u)(u

+ acu) + au, (9.56)

where (u) = ϕ



(ϕ

−1

(

(u −b))).OperatorF was shown to admit the 2D subspace

= L{ρ

(x ), ρ

(x )}, (9.57)

where ρ

and ρ

are arbitrary linearly independent solutions of ρ



+acρ = 0. There-

fore, (9.56) possesses exact solutions

u(x , t) = C

(t)ρ

(x ) + C

(t)ρ

(x ), where C



= aC

, C



= aC

. (9.58)

It is easy to see that the same construction applies to the discrete equation (9.56),

and there exist exact solutions (9.58) with ρ satisfying the same linear difference

equation. Finally, we obtain a discrete linear DS.

9.3.2 N-dimensional quasilinear operators

Let us present further examples of subspaces for discretized nonlinear operators in

. We borrow some samples of such differential operators from Section 6.1 and

Proposition 6.1 therein.

Example 9.18 (Non-symmetric blow-up and extinction) Consider positive solu-

tions of the PME with an extra lower-order term

=∇·(v

∇v) + δv

1−σ

in IR

× IR

, (9.59)

where σ = 0isaﬁxed constant. Here, δ = 1 corresponds to the source term, gen-

erating for σ<0 non-symmetric blow-up solutions. For δ =−1, (9.59) becomes a

quasilinear heat equation with absorption that, for σ>0, describes non-symmetric