Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics

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3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 113

self-adjoint operators are difﬁcult. Let us determine the point spectrum of the special

extension. Notice ﬁrst that solutions of the eigenvalue equation

Bψ = λψ

exhibit the following generic exponential growth near the inﬁnite singular point:

ψ(η) ∼ e

µη

as η →+∞, with µ = (2a

)

−



± i

√



Therefore, a proper functional setting for B involves the weighted space L

with

the exponential weight ρ(η) = e

−bη

, where b ∈



(2a

)

−1/3



is a constant. It

turns out that B possesses the real point spectrum in spaces of functions that are

sufﬁciently smooth at η = 0 (e.g., such as H

(B) =



=−

+ 2, k = 0, 1, 2, ...



and each eigenfunction ψ

(η) is a kth-degree polynomial

(η) = b



2k(k−1)(k−2)

k−1

+ ...



where b

are some normalization constants. In a proper setting, the resolvent of

B is compact in L

(see Remarks for extra details), so the whole spectrum σ(B)

can be made discrete. These polynomials are complete and closed in L

(IR );see

Kolmogorov–Fomin [352, p. 431]. So, studying (3.51), eigenfunction expansions

can be used. To this end, let us introduce the adjoint operator

∗

v =−a

(ηv



)





η +

with the same spectrum and ﬁnd the bi-orthogonal(in L

(IR ))set{ψ

∗

}of eigenfunc-

tions satisfying, after normalization, (ψ

,ψ

∗

)=δ

. Then, in L

, we can look for

solutions of (3.51) in the form of

w(η,τ) =



(k≥0)

(τ )ψ

(η), where c

(τ ) =(w(·,τ),ψ

∗

and study the DS for the expansioncoefﬁcients {c

(τ )} that describes the asymptotic

behavior. It follows that the turning patterns satisfying the rescaled equation (3.51)

correspond to the evolution on the stable and center manifolds that are tangent to

the corresponding subspaces of B, i.e., as τ →∞, w(η, τ) ∼ e

(η), with

∈ σ

(B) for some k ≥ 4, so λ

≤ 0. In particular, according to the scaling (3.50),

taking k = 4 with λ

= 0 gives the asymptotic behavior for the ﬁrst turning pattern

(most probably, generic)

Y (y, t) ∼ e

−2τ

(η) + ... , where ψ

(η) = b



+ 32a



is the corresponding eigenfunction. This determines the behavior on sets deﬁned

as y = O(

√

− t) to be matched with the behavior (3.47) on smaller sets in the

variable ζ in (3.46), i.e., for y = O(t

− t), to get a global structure of this ﬁrst

turning pattern. A rigorous justiﬁcation of such asymptotic and matching analysis is

OPEN for higher-order TFEs. The exact solutions remain the only tool to test such a

curious singularity formation phenomenon.

114 Exact Solutions and Invariant Subspaces

u(x , t)

t = 1

(t)

−

(t)

t > 1

t < 1

Figure 3.2 The exact solution (3.29) with A

, E

< 0: (i) quenching at t = 1, and (ii)

two interfaces appear for t > 1.

3.2.6 Quenching patterns

Take A

again,and now ﬁxanarbitraryE

≥ 0 in (3.29), giving the single point

quenching, where the strictly positive solution ﬁrst touches the singularity zero level

{u = 0}at t = 1, as shown in Figure 3.2. If E

< 0 for the asymptotic description of

quenching,we still can use the same rescaled variables(3.39)to observeconvergence

(3.41), where the self-similar proﬁle is now strictly positive,

g(ξ) = 1 +|E

|ξ

The asymptotic quenching phenomenon is described by the same rescaled equation

(3.44) with, plausibly, a generic stabilization to the similarity proﬁle; proof is an

OPEN PROBLEM. After quenching, for t > 1, two interfaces appear with the non-

Lipschitz behavior

(t) =±

√

t − 1 + ... as t → 1

, (3.53)

so that a smooth ﬂow, corresponding to the uniformly parabolic TFE for t < 1, is re-

placed by the FBP for the degenerate equation for t > 1, with a quenching transition

at t = 1

−

. In general, the questions of solution extensions beyond singular quench-

ing remain

OPEN. For similar second-orderequations,such as (3.36), this determines

extensions of order-preserving semigroups for various types of singularities, [226,

Sect. 6.2].

A different quenching pattern occurs for E

= 0 in (3.29), and a distinct spatial

rescaled variable η is necessary for describing the limit t → 1

−

u(x , t) → 1 +

120

, where η =

(1−t )

1/4

. (3.54)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 115

Then we introduce another rescaled function, v(η,τ) = e

u(x , t),thatisdeﬁned

according to (3.54) and satisﬁes the following equation:

=−

η + v − 1 − e

−τ

(vv

ηηη

)

. (3.55)

Unlike the rescaled parabolic PDE (3.44), the thin ﬁlm operator now reduces to a

fourth-orderperturbationthat is exponentiallysmallas τ →∞.Thisgivesasingular

perturbation problem for the linear autonomous Hamilton–Jacobi equation

=−

η + v − 1. (3.56)

The convergence as τ →∞of some classes of solutions of (3.55) and (3.56) rep-

resents an interesting

OPEN PROBLEM; though such behavior is not expected to be

generic, unlike the above similarity with E

< 0. For PMEs with absorption (3.36)

for which, in (3.55),the fourth-orderoperatoris replaced by the diffusionone (vv

)

such behavior is proved to be stable; see results in [245, Ch. 5].

After quenching at t = 1, the interfaces also exhibit a different behavior (cf.

(3.53))

(t) =±



120(t − 1)



+ ... as t → 1

We expect that it is not generic and corresponds to very ﬂat initial data u(x, 1

−

)

created by such a peculiar unstable quenching.

3.2.7 On extinction patterns with zero contact angle, the Cauchy problem

In the case of the standard FBP having conditions as in (3.14), it seems that the

present TFE with absorption does not possess exact solutions on invariant subspaces

(for other parameters,such solutionsdo exist;see Example 3.10 below).On the other

hand, self-similar solutions of the form (3.42) may be introduced, where g(ξ) is an

even positivesolution of the ODE (3.43) on some interval ξ ∈ (0,ξ

) with conditions

g(ξ

) = g



(ξ

) = (gg



)(ξ

) = 0, and g



(0) = g



(0) = 0, (3.57)

correspondingto thesymmetry attheoriginξ = 0. The behaviorclose to theinterface

at ξ = ξ

g(ξ) = (ξ

− ξ)

√

2|ln(ξ

− ξ)|+C + ... (C ∈ IR ).

Existence (or nonexistence) of such g(ξ ) leads to two-parameter, {ξ

, C}, shooting

to satisfy two symmetry conditions at ξ = 0andremainsan

OPEN PROBLEM.

In the Cauchy problem for such degenerate PDEs to be discussed systematically

later on, solutions typically are of changing sign, so the PDE and the similarity ODE

are modiﬁed as follows:

=−(|u|u

xxx

)

− signu ⇒ −(|g|g



)



−



ξ + g − sign g = 0. (3.58)

Then, (3.57) are also valid at interfaces, but the solutions g(ξ ) are smoother and are

oscillatory with a different behavior (corresponding to (|g|g



)



+ sign g = 0)

g(ξ) = (ξ

− ξ)

ϕ(s), s = ln(ξ

− ξ), as ξ → ξ

−

where the oscillatory component ϕ(s) is a periodic solution of a nonlinear ODE.

116 Exact Solutions and Invariant Subspaces

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

g(ξ)

(a) six proﬁles

5 5.5 6 6.5

−12

−10

−8

−6

−4

−2

x 10

−4

g(ξ)

(b) oscillations, enlarged

Figure 3.3 Similarity extinction proﬁles satisfying the ODE (3.58) with the conditions (3.57);

six proﬁles (a) and oscillations near the interface for the second proﬁle (b).

Details will be presented in Section 3.7. Figure 3.3(a) shows six similarity proﬁles

for ξ>0. In (b), these numerical results reveal the oscillatory character of solutions

near interfaces, an intriguing part of our future analysis.

Example 3.6 (TFE with cubic operator) Take the unique operator from Proposi-

tion 3.5(ii) and consider the TFE with constant absorption

=−



xxx

−

)



− 1.

Taking the subspace of even fourth degree polynomials yields solutions

u(x , t) = C

(t) + C

(t)x

+ C

(t)x









= 3C

− 24C

− 1,



=−9(C

+ 2C



= 30(4C

− C

This DS is more difﬁcult and explicit solutions do not exist. A simpler system occurs

by setting C

(t) ≡ 0, which corresponds to the subspace W

= L{1, x

}, but it is

not clear whether the resulting extinction behavior is generic. The asymptotics of

extinction as t → T

−

is easy to detect, since, on any bounded orbit of the above DS,

(t) = T − t + O



(T − t)



while C

2,3

(t) remainalmostconstantfor t ≈ T . This givesthe necessaryasymptotics

of the extinction (or quenching for positive solutions) behavior. The corresponding

rescaled equations can be formulated in a similar fashion. Other features of the FBPs

can also be described, though some of the computations are not explicit or easy.

3.2.8 Quartic operators: applications to extinction and blow-up

As a new application, consider the fourth-order TFE with source or absorption

=−



xxx



+ g(v). (3.59)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 117

Introducing the new independent variable

v = u

, where µ =

, (3.60)

splits the thin ﬁlm operator into ﬁve primitive monomials of the algebraic homoge-

nuity four,

=−



xxxx

+ (4µ − 1)u

xxx

+3(µ − 1)u

)

+ 3(µ − 1)(2µ − 1)u(u

)

+µ(µ − 1)(µ − 2)(u

)



1−µ

g(u

(3.61)

Assume that the last zero-order term is a linear function, i.e.,

1−µ

g(u

) = a + bu. (3.62)

Then polynomial subspaces will depend on the differentialoperatorin (3.61). Firstly,

this admits the trivial subspace of linear functions W

= L{1, x}, which does not

provide us with interesting solutions. In particular, these are traveling waves that are

unbounded as x →∞. We are interested in solutions of typical bell-shaped forms

localized on a bounded interval in x. Secondly, extensions of W

are possible in the

following cases:

Proposition 3.7

The operatorgivenin

(3.61)

and

(3.62)

preserves

= L{1, x , x

}

iff

n = 3

n = 6

,or

n =−2

Proof. Plugging u = C

x +C

∈ W

into the quartic operator yields that the

terms on L{x

, x

} vanish iff

4(µ − 1)[3 + 6(2µ − 1) + 4µ(µ − 2)] = 0,

which yields either µ = 1or4µ

+ 4µ − 3 = 0, i.e., µ =

or µ =−

Example 3.8 (Extinction behavior) Let us choose a =−1andb = 0 in (3.62),

giving a constant absorption term in (3.61).

Case n = 3(µ = 1). This is quite simple, since the resulting PDE

= F

[u] − 1 ≡−u

xxxx

− 3u

xxx

− 1 (3.63)

does not contain lower differential terms, so that, being restricted to W

, it is equiv-

alent to the linear ODE

=−1. (3.64)

Curiously, this trivial evolution on W

describes the actual general phenomenon of

asymptoticdegeneracyof the PDEs near extinction.To show this, consider nonnega-

tive and even in x (symmetric)solutionson W

= L{1, x

}. Noticingthat the general

solution of (3.64) is

u(x , t) = h(x) − t,

where h(x ) is arbitrary(initial data), we choose the parabolicproﬁle h(x ) = T −dx

with positive constants T and d. Next, bearing in mind the FBP, take the positive part

to obtain the following pattern near the extinction time T :

u(x , t) =



T − t − dx



≡ (T − t)g(ξ) = (T − t)



1 − d ξ



, (3.65)

118 Exact Solutions and Invariant Subspaces

where ξ denotes the spatial rescaled variable x/

√

T − t. For general even bell-

shaped solutions of (3.63), this asymptotic behavior on W

suggests introducing the

rescaled solution

u(x , t) = (T − t)w(ξ, τ ), τ =−ln(T − t), (3.66)

satisfying a typical singular (exponentially) perturbed equation,

=−

ξ +w −1 +e

−2τ

[w]. (3.67)

Such PDEs occurred before for a number of the TFEs with absorption. Therefore,

we expect that, as τ →∞, bounded orbits {w(·,τ)} converge to stationary proﬁles

g, solving the linear ODE

−



ξ + g − 1 = 0,

that gives precisely the parabolic proﬁles g(ξ ) in (3.65). In this sense, solutions on

the invariant subspace can detect the correct generic asymptotic behavior of extinc-

tion. As usual, a rigorous passage to the limit in (3.67) represents a difﬁcult

OPEN

PROBLEM

Case n = 6(µ =

). In this case, the diffusion-absorption equation takes the form

=−



xxxx

+ u

xxx

−

)



− 1.

We have the following exact symmetric solutions on W

u(x , t) =



(t) + C

(t)x



, (3.68)





= 6C

− 1,



= 12C

We cannot solve the system explicitly and will compute the generic asymptotic be-

havior of orbits near the extinction time, as t → T

−

. This yields C

(t) = T −t +...

and C

(t) →−d < 0, so that the behavior (3.65) remains valid asymptotically.

Using the same rescaled variables (3.66) makes it possible to formulate the corre-

sponding singular perturbed problem (3.67) for which the exact solutions on W

are

expected to describe the generic asymptotic behavior. This is an

OPEN PROBLEM.

Example 3.9 (Blow-up) Case n =−2(µ =−

). In (3.59), we take

g(v) =−

−

bv,

so, again choosing a =−1andb = 0 yields the TFE with source

=−



xxx



. (3.69)

The reaction term ∼ v

5/3

is superlinear for v  1, so that blow-up is guaranteed for

solutions of (3.69) with sufﬁciently large positive initial data v

(x ).

Setting v = u

−3/2

yields quartic nonlinearities in the PDE

=−



xxxx

− 7u

xxx

−

)

+30u(u

)

−

105

)



− 1.

(3.70)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 119

v(x , t)

v(x , T

−

)

ﬁnal-time proﬁle

< t

< T

Figure 3.4 Single point blow-up described by positive exact solutions of (3.69) on W

The blow-up behavior for (3.69) is equivalent to the quenching for (3.70). We look

for nonnegativesolutions (3.68) and obtain the DS for expansion coefﬁcients





= 30C

− 1,



=−180C

Let T > 0betheﬁnite quenching time of a ﬁxed bounded orbit. Then the DS gives

as t → T

−

the behavior C

(t) = T − t + ... and C

(t) → d > 0. This yields the

blow-up behavior on W

,ast → T

−

v(x , t) = u

−

(x , t) ≈ (T − t)

−



1 + d ξ



−

,ξ=

√

T −t

. (3.71)

These exact solutions are strictly positive,so that these are classical smooth solutions

of the TFE (3.69) on IR × (0, T ).

The blow-up evolution is shown in Figure 3.4. By (3.71), there occurs the single

point blow-up as x = 0 only. Letting t → T

−

yields the ﬁnal-time proﬁle

v(x , T

−

) = d

−

|x|

−3

(1 + o(1)) for x ≈ 0.

As shown in the previous example, these exact solutions asymptotically converge as

t → T

−

to the sufﬁciently smooth proﬁles given by the ODE u

=−1. This implies

that the blow-up patterns (3.71) are described as t → T

−

by the ODE

This is easily seen from the equations (3.69),written for the rescaledfunction v(x, t) =

(T −t)

−3/2

w(ξ, τ) with the same spatial rescaling ξ = x/

√

T − t. The equation for

120 Exact Solutions and Invariant Subspaces

w contains a singular exponentially small perturbation as τ →∞,

=−

ξ −

w +

− e

−2τ



−2

ξξξ



The phenomenon of such an asymptotic degeneracy of these TFEs near blow-up is

OPEN PROBLEM.

3.3 Exact solutions with zero contact angle

Let us return to exact solutions satisfying the more physically meaningful zero con-

tact angleconditionin (3.14).We again beginby studyingthe extinctionphenomenon

for the TFE with absorption, which is as a manifestation of an “evaporation” thin ﬁlm

phenomenon; see Section 3.2.

Example 3.10 (Singularities and interfaces) We consider the TFE with a strong

non-Lipschitz absorption term,

=−(vv

xxx

)

−

√

v, (3.72)

and now impose the zero contact angle condition, so, at each interface,

v = v

= vv

xxx

= 0atx = s(t). (3.73)

Setting v = u

yields an equation with the cubic operator,

=−

u(u

)

xxxx

− u

)

xxx

−

≡ F

[u] −

, (3.74)

exhibiting the following invariant property.

Proposition 3.11

Operator

(3.74)

admits the subspace

= L{1, x , x

}

Taking u = C

+ C

x + C

yields

[u] =−12(C

+ C

− 60C

x − 60C

∈ W

. (3.75)

For simplicity, setting C

(t) ≡ 0 (meaning even and symmetric in x patterns) gives

the following solutions of the original PDE (3.72) on W

= L{1, x

v(x , t) = u

(x , t) ≡



(t) + C

(t)x



, (3.76)

which, clearly, satisfy all three free-boundary conditions (3.73). In this case, (3.75)

implies





=−12C

−



=−60C

(3.77)

The last ODE is solved,

(t) =±

√

120

√

for t > 0, (3.78)

and, from the ﬁrst,

(t) = A

−

t, where A

= constant. (3.79)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 121

Quenching and dynamic interface equation. Consider ﬁrst the plus sign in (3.78),

where, for convenience, setting A

in (3.79) yields

u(x , t) =



−



1 − t



√

120

√



. (3.80)

This solution is strictly positive for all t ∈ (0, 1), and vanishes for the ﬁrst time at

t = 1 at the point x = 0 that describesthe quenching phenomenon. After quenching,

for t > 1, two interfaces x =±s(t) appear, where

s(t) = R(t) ≡



√

120



− 1



. (3.81)

At the initial moment of time t = 1

, the interface is not Lipschitz continuous,

s(t) = (30)

√

t − 1 + ... ,

andthis poses a problemof a post quenchingbehaviorof the extensionof the solution

after this evolution singularity occurred at t = 1.

In order to derive the regularity interface equation, we use the same elementary

formulae (3.31) and calculate u

from (3.74) for the solution (3.80). We obtain the

following two-term expression:



= S[u] ≡ 6u

at x = s(t) for t > 1. (3.82)

Recall that, by construction, our solution already satisﬁes the necessary three free-

boundary conditions (3.73). Let us detect other interface equations by the time-

parameterization t =

)

that follows from (3.80). Then (3.81) yields



= R



(t) ≡ R





)



. (3.83)

This equation is again of the second order, but is different from (3.82). Equating the

right-hand sides of (3.82) and (3.83) yields a stationary second-order Neumann-type

free-boundary condition

= R





)



at x = s(t) for t > 1.

Towards local well-posedness of the FBP. Let us present some arguments justify-

ing local existence/uniqueness of the FBP. We use the von Mises transformation by

introducing the dependent variable

X = X (u, t), so X (u(x, t), t) ≡ x, (3.84)

which is assumed to be well-deﬁned in a neighborhood of the interface posed now

at u = 0, at least for sufﬁciently small t > 0. Initial data at t = 0aretakeninthe

bell-shaped form u

(x ) = [C

(0)+C

(0)x

]

, where C

(0)>0andC

(0)<0are

given constants, being initial data for the DS (3.77). This determines the unique so-

lution on the invariantsubspace with the known propertiesof extinction and singular

interfaces. In terms of the new function (3.84), initial data are smooth,

(u) =



(0)−u

(0)|

for small u ≥ 0. (3.85)

Assuming that X (u, t) is strictly monotone decreasing at least for small u > 0, and

122 Exact Solutions and Invariant Subspaces

calculating derivatives yields

=−

, u

=−

)

, u

xxx

=−

uuu

)

3(X

)

, ... .

On substitution into (3.74), the following PDE for X is derived:

= H (X) ≡−u



uuuu

)

−

10X

uuu

)

15(X

)



−u



uuu

)

−

21(X

)



−

)

(3.86)

Consider the principal fourth-order term in (3.86). Firstly, it follows that we need

to deal with solutions satisfying X

(0, t) = 0(andﬁnite) at the origin to exclude

gradient singularities, so it we check that the initial function (3.85) satisﬁes this

inequality of transversality. Secondly, (3.86) is degenerated at the boundary point

u = 0, and the quadratic rate of degeneracy, O(u

), in the higher-order term makes

it possible to pose some standard boundary conditions at u = 0.

The actual construction of the solution is as follows. The following decomposition

of the solution is needed:

X =

X +

X, where

X ∈ Span{1, u} and

X = X −

X. (3.87)

Projecting equation (3.86) onto

= Span{1, u} yields the ODE–PDE system:



H(X),

(3.88)

where

H and

H denote projections of H onto the correspondingsubspaces. The ﬁrst

equation is a two-dimensional dynamical system, and the second equation is then a

parabolic PDE with the principal degenerate operator

B =−u

. (3.89)

This linear operator is symmetric in the weighted space L

((0,δ)), with a small

δ>0andρ = u

−2

. By classical theory of symmetric ordinary differential operators

(see Naimark [432]), estimating the solutions of the eigenvalue equation

Bψ =±iψ for u ≈ 0

gives four types of asymptotics ψ

(u) = O(1), ψ

(u) = O(u) (both contain loga-

rithmic factors in higher-order terms), ψ

(u) = O(u

),andψ

(u) = O(u

),where

3,4

∈ L

and ψ

1,2

∈ L

. Therefore, the deﬁciency indices of B are (2,2), so that,

any pair of self-adjoint Dirichlet boundary conditions at u = 1 gives a self-adjoint

extension. We need to take the unique Friedrichs self-adjoint extension of B with a

discrete spectrum, compact resolvent, and a complete, closed eigenfunction set. As

usual, this extension is induced by the Dirichlet conditions at the singular endpoint

u = 0, i.e., w(0) = w



(0) = 0 for functions from the domain D(B). According

to the representation (3.87), the parabolic problem for solutions

X is well-posed,

provided that the transversality condition X

(0, t) = 0, as well as suitable (free-

boundary) conditions at u = 0, hold. Actually, the second equation in (3.88), after

dividing by u

, reduces to the PDE with the non-degenerate principal operator D

and the solution

X ∈ D(B) can be constructed by eigenfunction expansion, where