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 153

0 1 2 3 4 5 6 7

−5

−4

−3

−2

−1

x 10

−4

φ(s)

−φ

(a) n =

,λ= 1

0 1 2 3 4 5 6 7

−6

−4

−2

x 10

−6

φ(s)

(b) n =

,λ=−1

Figure 3.17 Solutions of the ODE (3.191) with n =

in the stabilization case λ = 1 (a) and

in the oscillatory case λ =−1(b).

The operator [v] is monotone in the topology of the Hilbert space H

−4

(IR ).Ex-

istence and uniqueness of weak continuous solutions are based on general theory of

tions, it is not difﬁcult to check that, for a specialclass of sufﬁciently regular (strong)

weak solutions {u(x, t)}, there exists a limit as n → 0

, establishing a homotopic

(non-singular) connection with the linear bi-harmonic PDE (3.2) in IR × IR

.The

uniform convergence of strong solutions as n → 0

is an important property that

will be used later on. For (3.188), the TWs satisfy

| f |

(4)

= λf



for y > 0, f (0) = 0,

so that, instead of the change (3.186),

f (y) = y

ϕ(s), s = ln y, with γ =

, where (3.190)

(4)

+ 2(2γ − 3)ϕ



+ (6γ

− 18γ + 11)ϕ



+ 2(2γ

− 9γ

+11γ − 3)ϕ



+ γ(γ − 1)(γ − 2)(γ − 3)ϕ =

|ϕ|

(ϕ



+ γϕ).

(3.191)

This ODE has two equilibria

±ϕ

=±



(γ −1)(γ −2)(γ −3)



. (3.192)

They exist in the parameter ranges n ∈





and n > 3ifλ<0, and for n ∈ (0, 1)

and n ∈



, 3



if λ>0.

Figure 3.17 shows different behavior of solutions in the case of n =

for λ>0

(convergenceto the positive constant equilibria (3.192); no traces of stable or unsta-

ble periodic patterns were observed) and for λ<0 (oscillatory). Periodic patterns

ϕ(s) persist for larger n, including n = 1.1; see Figure 3.18. Further increasing n,a

few oscillations appear followed by fast divergence of the solution ϕ(s), according

to the linear unstable exponential bundle.

Subcritical bifurcation. Numerically, the constant equilibrium ϕ

of (3.191) be-

comes asymptotically stable for n > n

,where

= 1.287...

monotone operators; see Lions [396, Ch. 2]. Writing (3.189) in the sense of distribu-

154 Exact Solutions and Invariant Subspaces

0 5 10 15

−0.1

−0.05

0.05

0.1

φ(s)

(a) n = 0.9

0 10 20 30 40 50 60 70

−2

φ(s)

n=1.1

n=1.12755

(b) n = 1.1and1.12755

Figure 3.18 Convergence to stable periodic solutions of the ODE (3.191), λ =−1, with

various Cauchy data at s = 0forn = 0.9 (a) and n = 1.1, n = 1.12755 (b).

0 5 10 15 20 25 30

−500

500

1000

φ(s)

n=1.287

n=1.2

Figure 3.19 Behavior as n → n

−

of solutions of the ODE (3.191), with λ =−1.

and as n → n

−

, the stable periodic solution becomes nonexistent,as shown in Figure

3.19. Therefore, for n > n

, the TW proﬁles are no longer oscillatory near such

interfaces (only a ﬁnite number of zeros is available). Such non-oscillatory behavior

becomes more and dominant for larger n >

, so that, possibly, the Cauchy problem

can be posed in the class of nonnegative solutions; an

OPEN PROBLEM.

Extra scaling for small n > 0. As usual, to see the oscillatory patterns for small

n > 0, an extra scaling in (3.191) with λ =−1 is performed by setting

ϕ(s) = Aψ(η), η =

, where a =

and A =





This leads to the simpliﬁed equation with a linear binomial operator,

(4)

+ 4ψ



+ 6ψ



+ 4ψ



+ ψ =−

|ψ|

(ψ



+ ψ), (3.193)

where we omit terms of the order O(n). In Figure 3.20, a stable periodic behavior

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 155

0 10 20 30 40 50

−1.5

−1

−0.5

0.5

1.5

x 10

−3

ψ(η)

(a) n = 0.3

0 10 20 30 40 50 60 70

−5

−4

−3

−2

−1

x 10

−5

ψ(η)

(b) n = 0.2

Figure 3.20 Stable periodic behavior for (3.193) for n = 0.3 (a) and n = 0.2(b).

of the ODE (3.193) is shown. For n = 0.2, the periodic solution is very small,

max|ϕ|∼10

−22

Comparison with the linear PDE. In view of the continuity at n = 0

, consider the

PDE (3.2) that admits TWs with the proﬁles satisfying

λf = f



⇒ f (y) = e

µy

, where µ

= λ.

We are interested in the behavior as y →−∞which is the left-hand “interface”

for solutions of the linear equation. Then, taking λ>0 gives the only possible non-

oscillatory exponential behavior

f (y) ∼ e

1/3

as y →−∞.

On the contrary, for λ<0, oscillatory patterns are obtained,

f (y) ∼ e

|λ|

1/3

cos



√

|λ|

y + A



, where A

= constant. (3.194)

This is in qualitative agreementwith the properties of the nonlinear ODE (3.191) for

n ∈ (0, 1]. It is curious that the period of linear oscillations in (3.194),



n=0

4π

√

= 7.26... (λ =−1),

is comparable with the nonlinear ones in Figure 3.17(b), for n =

, T



≈ 1.3.

The fundamental solution of the bi-harmonic equation (3.2),

b(x, t) = t

−

F(ζ ), ζ =

1/4

correspondsto the case λ>0aty =+∞. The rescaled kernel F is the unique radial

solution of the ODE problem

−F

(4)

ζ F



F = 0inIR ,



F = 1.

Then F = F(|ζ |) has exponential decay and oscillates as |ζ |→∞; general es-

timates on fundamental solutions can be found in Eidelman [164, p. 115]. More

precisely, using standard asymptotic expansion yields the following behavior of the

kernel as ζ →+∞:

F(ζ ) ∼ ζ

−

aζ

4/3

, with complex a satisfying a





, Rea < 0.

156 Exact Solutions and Invariant Subspaces

There exist two complex conjugate roots for exponentially decaying proﬁles

=−

−

(1 ± i

√

3) ≡−c

± ic

This yields the following oscillatory behavior as ζ →+∞:

F(ζ ) ∼ ζ

−

−c

4/3



cos





+ A

sin





, (3.195)

where A

and A

are constants. The algebraic factorζ

−1/3

is obtained by the WKBJ-

type technique. This is a periodic behavior with a single fundamental frequency. We

have seen numericallythat such a stableperiodicstructureis inheritedin the behavior

(3.190) for all n ∈ (0, n

Example 3.38 (Finite propagation and changing sign behavior in higher-order

semilinear models)Aswehaveseen,ﬁnite propagation and oscillatory solutions of

the CP can beobtained in any semilinear parabolic models by adding a non-Lipschitz

absorption-like term, e.g., (see [227] for properties and ε-regularization)

= F[u] ≡ (−1)

m+1

u −|u|

p−1

u, with p ∈ (−1, 1). (3.196)

For TWs, there occurs the ODE

−λf



= F[ f ]fory > 0, f (0) = 0,

which, close to the ﬁnite interface, becomes asymptotically stationary,

(−1)

m+1

(2m)

−|f |

p−1

f = 0 ⇒ f (y) = y

ϕ(ln y), (3.197)

where γ =

1−p

> 0. For anym ≥ 2, such ODEs have oscillatory solutions; see Sec-

tion 3.7. Note that (3.196) also describes the phenomenon of ﬁnite-time extinction

proved by energy estimates based on Saint–Venant’s principle; see [524] and survey

in [240]. The range p ≤−1 corresponds to nonexistence, [227].

Similarly, equations of Cahn–Hilliard type with mass conservation

= F[u] ≡ (−1)

m+1

u − D



|u|

p−1



for 1 ≤ k < m,

can also admit ﬁnite propagation. Solutions of changing sign (for k ≤ m − 2) are

given by (3.197) with 2m → 2(m −k). Depending on m and k, theperiodic behavior

of ϕ(s) can be stable or unstable, as in the examples discussed above.

3.9 Quasilinear pseudo-parabolic models: the magma equation

A typical pseudo-parabolic model to be studied is





+ (lower-order terms)(u ≥ 0), (3.198)

where n ∈ IR is a ﬁxed exponent. Such PDEs arise in mathematical modeling of the

segregation and migration of magma in the mantle of the Earth, [519, 420]. The full

model, called the magma equation,is



)

− 1





, (3.199)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 157

with two parameters n and l. Setting l = 0 gives the principal operator in (3.198).

Such PDEs belong to the class of pseudo-parabolic equations that may enjoy some

properties of standard parabolic ﬂows. Examples include the linear analogy

− u

xxt

= u

and the regularized PBBM equation (see Example 4.45)

− u

xxt

= uu

+ εu

(ε > 0).

The advantage of such pseudo-parabolic PDEs is that applying the inverse operator

(I − D

)

−1

with positive integrable kernel yields non-local equations with compact

operators, which can be studied by nonlinear integral operators theory.

Standard theory does not apply to quasilinear PDEs, such as (3.198), which, in

addition, are degenerate at the singular set {u = 0}. Mathematical theory of such de-

generate PDEs with nonnegative solutions (or, possibly, solutions of changing sign)

is unavailable. As usual, we will use solutions on invariant subspaces for formal de-

tecting some evolution singularities.

Example 3.39 (Quenching, interfaces, and the Cauchy problem) Consider the

PDE (3.198) for n = 1 with a constant absorption,

= F[u] ≡ (uu

)

− 1,

bearing in mind the phenomenon of ﬁnite-time extinction. F is associated with sub-

space (module) W

= L{1, x

}, so there exist exact solutions

u(x , t) = C

(t) + C

(t)x





= 2(C

)



− 1,



= 12C



We take, e.g., C

(any constantC

ﬁts), and the ﬁrst equation is then integrated,

u(x , t) =

(T − t) +

, (3.200)

where T > 0istheﬁnite quenching time. For t < T , the solution is strictly positive.

At t = T

−

, the solution touches the singularity level {u = 0}. We readily derive the

asymptotic extinction behavior as t → T ,

u(x , t) = (T − t)g

(ξ), ξ =

√

T −t

, where g

(ξ) =

. (3.201)

Hence, (3.201) belongs to the class of self-similar solutions u

(x , t) = (T − t)g(ξ),

where g (and g

) solves the following third-order ODE:







ξ − g





−



ξ + g − 1 = 0.

These solutions on W

explicitly describe the quenching behavior for this degen-

erate quasilinear PDE. A rigorous stability analysis of such quenching is

OPEN (it

is likely that it is not generic or robust at all). In the rescaled sense, this assumes,

introducing the rescaled function u(x, t) = (T −t)v(ξ, τ ), with τ =−ln(T −t), to

study the third-order rescaled PDE





ξ −v + v



ξξ

−

ξ +v − 1.

158 Exact Solutions and Invariant Subspaces

This means, passing to the limit τ →∞, that, for a class of positive, symmetric, and,

say, convex initial data (recall the parabolic shapes,

T +

, of initial functions

in W

), the convergence takes place,

v(ξ,τ) → g

(ξ) as τ →∞ (i.e., as t → T

−

Interface propagation. For t > T , the problem admits a suitable setting in the

class of nonnegative solutions in an FBP framework. It follows from (3.200) that the

right-hand interface is located at

s(t) = 6



√

t − T for t > T .

For such solutions u

, so that, at u = 0 in (3.200), −

(t − T ) + 3(u

)

= 0.

Plugging this into the expression for s



yields the dynamic interface equation



√

t−T

≡

at x = s(t). (3.202)

On the Cauchy problem: oscillatory and non-oscillatory patterns. Let us for-

mally detect the maximal regularity that can be attributed to the CP. To keep pseudo-

parabolicity and extinction properties, consider the signed equation for n = 1,

= (|u|u

)

− signu. (3.203)

Substituting the TWs u(x, t) = f (y), with y = x − λt, yields the ODE −λ f



−λ(| f | f



)



− sign f. Studying the right-hand interface, and hence, using the reﬂec-

tion y → −y, we keep the leading two terms on the right-hand side,

λ(| f |f



)



− sign f = 0fory > 0, f (0) = 0. (3.204)

It follows that the behavior near the interface at y = 0isgivenby

f (y) = y

ϕ(s), with s = ln y,

where the oscillatory component ϕ(s) solves a third-order ODE. Such ODEs were

studied earlier in the TFE applications; see (3.166). Therefore, ϕ(s) is oscillatory

and changes sign for λ<0, and is non-oscillatory for λ>0. Such TW properties

are expected to make sense in the CP for (3.203). Its well-posedness is

OPEN.Note

that the same ODE (3.204) occurs for the signed nonlinear dispersion equation (see

Section 4.3 for details)

=±(|u|u

)

− signu.

Secondly, for the original model (3.198) without strong absorption



|u|



(n > 0),

the same TW analysis leads to the ODE (| f |



)



= f for y > 0, which always

admits the strictly positive (non-oscillatory) solution

f (y) =



2(n+2)



The regularity at the interface y = 0 increases without bound as n → 0

when

approaching the linear PDE.

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 159

Example 3.40 (Sixth-order PDE) Considera higher-orderpseudo-parabolicmodel,

such as



|u|



xxxxxx

. (3.205)

The TW proﬁles satisfy

(| f |



)

(5)

= f for y > 0, f (0) = 0.

For n ∈ (0, 1), there exists a strictly positive, non-oscillatory solution f (y) = c

where c

> 0 is a constant.

Example 3.41 (Single point blow-up) Consider next a pseudo-parabolic PDE with

an extra reaction term,





+ v

2−n

, (3.206)

where, for n < 0, the source term is superlinear for v  1andmaycreateﬁnite-time

blow-up singularities. In order to deal with polynomial subspaces, set

v = u

, with µ =

to get a cubic PDE of the form

= u

xxt

+ 2(µ + 1)uu

+(µ + 1)



+ µ(u

)



≡ F

[u] +

(3.207)

Proposition 3.42 F

(3.207)

admits subspace

= L{1, x , x

}

iff

2µ

+ 7µ + 6 = 0 ⇐⇒ µ =−2,

µ =−

. (3.208)

Substituting into (3.207) u(x, t) = C

(t) +C

(t)x

yields that (3.208) annuls the

term on L{x

}.Thisgives





= 2(µ + 1)C



+ 2C



= 2(µ + 3)C



+ 2(µ + 1)(2µ + 1)C



(3.209)

Thus, such solutions exist in two cases, where (3.206) is



−1



+ v

for n =−1 (µ =−1);



−



+ v

for n =−

(µ =−

For both PDEs, we detect from the DS (3.209) the generic (on W

) extinction behav-

ior for the function u(x, t), which is equivalent to blow-up for v(x, t). For smooth

bounded orbits of the DS, the ﬁrst equation near extinction time T reads C



−

|µ|

+ ... , and the second equation yields C

(t) → d > 0, from which we ob-

tain the asymptotic pattern

u(x , t) = (T − t)

|µ|

+ dx

+ ... = (T − t)



|µ|

+ d ξ



+ ... as t → T

−

with the spatial rescaled variable ξ = x/

√

T − t. In terms of the original solution

v = u

2/n

of the initial PDE (3.206), this yields the following single point blow-up

behavior as t → T

−

for n =−1orn =−

w(ξ, τ) ≡ (T − t)

−

v(x , t) → g(ξ) ≡



|µ|

+ d ξ



(3.210)

160 Exact Solutions and Invariant Subspaces

uniformly in ξ ∈ IR . As happened before to other blow-up models, such behavior

corresponds to a delicate case of a singular perturbed ﬁrst-order Hamilton–Jacobi

equation. To reveal this singular limit, we write down the perturbed PDE for the

rescaled function w(ξ, τ) as follows:

=−

ξ +

w + w

2−n

+ e

−τ





−

w +



ξξ

. (3.211)

This is an exponentialperturbationof theHamilton–Jacobiequationw

=−

ξ+

w + w

2−n

that possesses g(ξ) in (3.210) as a stationary solution. There is a large

amount of mathematicalliterature devoted to inﬁnite-dimensionalsingular perturbed

DSs associated with blow-upand extinctionphenomenafor reaction-diffusionPDEs;

see [245, Ch. 5, 9–11]. These well-developed techniques do not apply to the per-

turbed PDE (3.211) for which establishing uniform boundedness and compactness

of the rescaled orbits in suitable metrics are also

OPEN.

Remarks and comments on the literature

In many occasions we put references concerning speciﬁc models, equations, and applications

alongside corresponding examples. Other references are given below.

§3.1.Earlier references on the derivation of the fourth-order TFE can be found in [263, 531],

where the ﬁrst analysis of some self-similar solutions was performed for n = 1. Source-

type (ZKB) similarity solutions for arbitrary n were studied in [48] for N = 1 and [185]

for the equation in IR

. More information on similarity and other solutions can be found

in [46, 45, 78]; see also a discussion of the TFE in the afterword of Barenblatt [25]. Thin

ﬁlm equations admit nonnegative solutions constructed by special parabolic approximations

of the degenerate nonlinear coefﬁcients; see the pioneering paper [44], various extensions in

[264, 167, 376, 576], and the references therein. For estimates of not necessarily nonnegative

solutions in IR

, see [265] and the bibliography therein.

The family (3.7) of generalized TFEs was studied in [345], where further references and

physical motivation can be found. The equation (3.3) was derived and studied in [155]; see

[529] for a parallel development. Equation (3.8) with m = n = 3 has been used to describe

bubble motion in a capillary tube and the Rayleigh–Taylor instability in a thin ﬁlm [279]. For

more information on the modeling and physics of thin liquid ﬁlms, we refer to survey papers

[430, 451, 34]; see also references in [431]. See [53]–[55], [293] for more general PDEs, such

as (3.9). The doubly nonlinear equation

=−(u

xxx

)

describes, for n = l +3, the surface tension-driven spreading of a power-law ﬂuid and, for n =

1, a power-law ﬂuidinaHele–Shaw cell; see [345], survey [451], and mathematics in [14].

The exponent l is determined by rheological characteristics of the liquid, so l = 0 corresponds

to a Newtonian liquid, while l = 0 appears for “power-law” (Ostwald–de Waele) liquids,

called shear-thinning if l > 0. For such liquids, a typical sample relation of the viscosity η

and the shear rate ˙γ is of the form η ∼˙γ

−l/(l+1)

Concerning the Benney equation (3.10) [39] and models of falling liquid ﬁlms, see [452].

Notice earlier experimental ﬁndings of P.L. and S.P. Kapitza [315] in 1949 related to travel-

ing waves in such models. On Marangoni instability in thin ﬁlm models (3.12), see [450].

Semilinear Cahn–Hilliard equations were introduced in [92]; see [438] and references therein.

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 161

The sixth-order thin ﬁlm equation (3.13) was introduced in [338, 339] in the case of n = 3,

and describes the spreading of a thin viscous ﬂuid under the driving force of an elastica or

light plate. In addition, [338, 339] treat a more general form of this equation (now allowing

for a reaction at the usual solid interface), which is shown to arise in the industrial application

of the isolation oxidation of silicon. Analysis for a ﬁnite length elastica or plate is given in

[176], whilst a numerical scheme with subsequent parameter investigation of the more general

system is derived in [177]. Delicate aspects of unusual similarity solutions and asymptotics

can be found in [189]. Some particular features of the 2mth-order PDEs (3.16) were already

considered in [531], where the n-small and waiting-time solutions were noted. Doubly non-

linear equations, such as (3.22), which are relevant to capillary driven ﬂows of thin ﬁlm of

power-law ﬂuids, were derived in [345, 513]. Extra absorption terms in TFEs (see (3.25)) or

the source terms are to model effects of evaporation (certain permeability of the surface may

also be taken into account) or condensation, [451]. Actually, evaporation phenomena of thin

ﬁlms are well known for binary solutions (see references in [261]), which probably hardly

apply to thin ﬁlms on ﬂat surfaces. The Florin problem for the fourth-order TFE with the con-

stant non-zero angle u

= C at the interfaces, which is a lubrication model related to Darcy

ﬂow in a Hele–Shaw cell, was studied in [454].

Discussing other related higher-order models, note that similarity solutions of the equation

with a monotone operator in H

−2

=−(|u|

m−1

xxxx

were studied in [47], where, for m > 1, solutions were proved to be compactly supported and

oscillatory (changing sign) near the interfaces. Other lubrication-type PDEs

=−(u

)

and u

=−|u|

xxxx

(n < 0)

were introduced in [52]. The fourth-order parabolic equation

=−(u(ln u)

)

arises in the context of interface ﬂuctuations in spin systems and semiconductor theory; see

mathematics and references in [64, 307]. The sixth-order PDEs

= (u

xxxxx

m−1

xxxxx

)

and u

={u[

(u(ln u)

)

((lnu)

)

]

}

are obtained, respectively, for power-law ﬂuids spreading on a horizontal substrate [345] and

as a generalized quantum drift-diffusion model for semiconductors, [146, 307]. Critical expo-

nents and asymptotic and singularity phenomena for the eighth-order TFE

=−(u

xxxxxxx

)

are discussed in [189]. Various aspects concerning oscillatory solutions of the Cauchy problem

for TFEs can be found in [174, 175], where further references are given.

Higher-order parabolic PDEs also occur in curve shortening ﬂows for curves in two di-

mensions, whose normal velocity V

is given by the Laplacian of its curvature; see [91]. The

corresponding equation V

≡ N

=−κ

(s is the arclength) for curves y = u(x, t) on the

{x, y}-plane, can be written as (see [82])

=−



√

1+(u

)



√

1+(u

)





The hierarchy of such models of arbitrary order (the third-order PDE belongs to the KdV fam-

ily) goes back to Mullins [428], who proposed classical theory of thermal grooving. Arbitrary

order integrable models V

= D

κ were analyzed in [82]. Other fourth-order parabolic PDEs

162 Exact Solutions and Invariant Subspaces

appear [116] in blow-up analysis of the curve evolution of an immersed plane curve according

to the H

−1

gradient ﬂow.

Exact solutions on invariant subspaces exist for some non-local parabolic equations related

to the b–l model of the propagation of turbulent bursts from a horizontally uniform layer (see

model statement in [26] and [25, p. 291]),

= l(t)b

− k

(t)

in IR

× IR

where k > 0isaﬁxed constant, and l(t)>0 is the measure of the support of a nonnegative

bounded solution, l(t ) = meas supp b(·, t ) ≡ meas {x ∈ IR

: b(x, t )>0}. Aclassof

such equations with invariant subspaces was studied in [224]. Another invariant subspace is

available in the b–ε model of turbulence motion based on the ideas of A.N. Kolmogorov and

L. Prandtl:



= α(

)

− ε,

= β(

)

− γ

where α, β,andγ are positive constants. Here b(x, t ) ≥ 0 denotes the turbulent energy

density, and ε(x, t)≥ 0 is the dissipation rate of turbulent energy. See [27, 26] and [56] for

physical justiﬁcation and further references. For α = β, the ﬁnite-time extinction behavior

can happen on a polynomial subspace invariant under quadratic operators [224].

§3.2.Some details concerning spectra, compact resolvent, and sectorial properties of linear

operators like (3.52) can be found in [163]. Note that, as a rule, rescaled higher-order parabolic

PDEs contain non-symmetric and non-potential operators. The spectral theory in [163, 87] of

similar (but not precisely the same) operators appears in the study of asymptotic blow-up and

global behavior for semilinear 2mth-order parabolic equations

=−(−)

u ±|u|

p−1

[424]. On oscillatory and other features of the CP for the fourth-order TFEs, see [174].

§ 3.4, 3.5. Proposition 3.7 is taken from [343] (transformation (3.60) was ﬁrstusedin[531]

for formal perturbation analysis for small n), where Proposition 3.29 was used for m = 2.

Various results on the FBP and the CP for the sixth-order TFEs are available in [175].

§ 3.6, 3.7. Basic properties of oscillatory solutions of the CP for TFEs of various orders with

more detailed description are given in [174, 175]; see also [227].

§3.8.The KS-type equations are used as a description of the ﬂuctuations of the position of

ﬂame front [202, 203], the motion of ﬂuid on a vertical wall, and chemical reactions with

spatially uniform oscillations on a homogeneous medium. Similar models also occur in solid-

iﬁcation, [201]; see survey [411]. The mKS equation (3.172) is a model for the dynamics of a

hyper-cooled melt [514]. A more general class of such models was introduced and discussed

in [289]. Blow-up in the mKS equations was studied in [50].

§3.9.In the magma equation (3.199), u(x, t) ≥ 0 measures the volume fraction of liquid

phase [519]. The exponents n and l describe permeability and effective viscosity characteriz-

ing the rate of matrix compaction and distension on u. Similar pseudo-parabolic PDEs occur

in modeling of thin ﬁlm ﬂows for poroviscous droplets over a planar substrate [347].

Open problems

• These are formulated throughout the chapter in Section 3.2, Examples 3.8–3.10,

3.14, 3.17, Remark 3.28, Section 3.7, and Examples 3.36, 3.37, 3.39, and 3.41.

§3.3.The original von Mises transformation (see Example 3.10) was introduced in 1927,