# Qualitative Analysis of Coating Flows on a Rotating Horizontal Cylinder

Authors: Marina Chugunova; Roman M. Taranets
Journal: International Journal of Differential Equations
 (2012)

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## Abstract

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchia's lemma for a system of functional equations.


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## Body

## 1. Introduction

The time evolution of thickness of a viscous liquid film spreading over a solid surface under the action of the surface tension and gravity can be described by lubrication models [1–5]. These models approximate the full Navier-Stokes system that describes the motion of the liquid flow. Thin films play an increasingly important role in a wide range of applications, for example, packaging, barriers, membranes, sensors, semiconductor devices, and medical implants [6–8].In this paper we consider the dynamics of a viscous incompressible thin fluid film on the outer surface of a horizontal circular cylinder that is rotating around its axis in the presence of a gravitational field. The motion of the liquid film is governed by four physical effects: viscosity, gravity, surface tension, and centrifugal forces. These are reflected in the parameters:R: the radius of the cylinder, ω: its rate of rotation (assumed constant), g: the acceleration due to gravity, ν: the kinematic viscosity, ρ: the fluid's density, and σ: the surface tension. These parameters yield three independent dimensionless numbers: the Reynolds number Re=(R2ω)/ν, γ=g/(Rω2), and the Weber number We=(ρR3ω2)/σ. The understanding of coating flow dynamics is important for industrial printing process where rotating cylinder transports the coating material in the form of liquid paint. The rotating thin fluid film can exhibit variety of different behaviour including: interesting pattern formations (“shark teeth” and “duck bill” patterns), fluid curtains, hydroplaning drops, and frontal avalanches [8–10]. As a result, the coating flow has been the subject of continuous study since the pioneering model was derived in 1977 by Moffatt (see [11]):
(1.1)∂h∂t+∂∂θ[ωh-13gνRh3cosθ]=0.
The surface tension and inertial effects were neglected in (1.1). Here h(x,t) is the thickness of the fluid film, θ is a rotation angle, and t is a time variable. The linear stability of rigidly rotating films on a rotating circular cylinder under three-dimensional disturbances was examined in [12, 13]. It was shown that the most unstable mode for thin film flows on the surface of a cylinder is the purely axial one that leads to so-called “ring instabilities”. During the past decade, coating and rimming problems attracted many researchers who analyzed different types of flow regime asymptotically [14–17] and numerically [18–20]. For a detailed review of a growing literature on different types of thin film flows please see [21] and references there in.The coating flow is generated by viscous forces due to cylinder's surface motion relative to the fluid. There is no temperature gradient, hence the interface does not experience a shear stress. If the cylinder is fully coated there is only one free boundary where the liquid meets the surrounding air. Otherwise, there is also a free boundary (or contact line) where the air and liquid meet the cylinder's surface.The asymptotic evolution equation for the thickness of the fluid film with the surface tension effect:(1.2)∂h∂t+∂∂θ[ωh-13gνRh3cosθ+13σρR4νh3(∂h∂θ+∂3h∂θ3)]=0,
was derived by Pukhnachev [22] in 1977. It is valid under the assumptions that the fluid film is thin h≪R and its slope is small (1/R)(∂h/∂θ)≪1. Later in 2009, taking into account inertial effects, Kelmanson [23] presented a more general model:
(1.3)∂h∂t+∂∂θ[ωh-13gνRh3cosθ+13σρR4νh3(∂h∂θ+∂3h∂θ3)+13ω2ρνRh3∂h∂θ]=0.
He analyzed, asymptotically and numerically, diverse effects of inertia in both small- and large-surface-tension limits.We should mention that all three lubrication approximation models described above were based on the assumption of the no-slip boundary condition. It is well known [24] that the combination of constant viscosity and no-slip boundary conditions at the liquid-solid interface yields a logarithmic divergence in the rate of dissipation at moving contact line, that is, an infinite energy is needed to make the droplet expand. The most common way to overcome this difficulty is to introduce effective slip conditions (see (2.1)) that indeed removes the force singularity at advancing contact lines (see [25]).The main goal of our paper is to study waiting time phenomenon for the coating flows under an assumption of effective slip conditions, that is, we analyze (2.1) that is a modified version of (1.3). Our approach is based on now well-established nonlinear PDE analysis for degenerate higher order parabolic equations.The sufficient conditions:h0(x)≤A|x|4/n for 0<n<2, |h0x(x)|≤B|x|4/n-1 for 2≤n<3, (where A and B are some positive constants) on nonnegative initial data, h0 for the occurrence of waiting time phenomena were derived by Dal Passo et al. [26] for the classic thin film equation:
(1.4)ht+(|h|nhxxx)x=0.
These results were based on an energy method developed in [27] for quasilinear parabolic equations. To the best of our knowledge, there is only one publication [28], where the waiting time phenomenon in the classic thin film equation (1.4) was discovered for h0(x)~|x|α for 2<α<4/n. The result was obtained by means of matching asymptotic methods and was supported by numerous numerical simulations. For more general nonlinear degenerate parabolic equations with nonlinear lower order terms the waiting time phenomenon was analyzed in [29–31].It is well known [32] that the similarity solutions of the second order nonlinear parabolic equation:
(1.5)ct=(cmcx)x,m>0,
subject to prescribing appropriate initial data, demonstrate the existence of a waiting-time phenomena before the free boundary moves. The comparison theorem, that is not applicable in our case, then enabled a number of results to be obtained about the existence and length of waiting times for general initial data. Our approach is completely different and based on local entropy/energy functional estimates.We also analyze speed of support propagation and obtain an upper bound on it for the modified version of (1.3) (see (2.1)). The first finite speed results for nonnegative generalized solutions of the classic thin film equation (1.4) were obtained in [33, 34] for the case 0<n<2 and 2≤n<3, respectively. For more general types of thin film equations the finite speed of support propagation phenomenon was studied in [35–39] (see also references there in).The outline of our paper is as follows. We first prove forn>0 the long-time existence of a generalized weak solution and then prove that it can have an additional regularity in Section 2. In Sections 3 and 4 we show finite speed support propagation in the “slow” convection case (n>1): for 1<n<3 and waiting time phenomena for 1<n<2, accordingly. The general strategy is to use an extension of Stampacchia's lemma for a system of functional equations (see Lemma  3.1 [26], where this extension is proved for a single equation and Lemma  A.2 in [37], where this extension is proved for systems in the homogeneous case). This result to our knowledge is new and might be of independent interest. We leave as an open problem the “fast” convection case (0<n<1): finite speed of support propagation and sufficient conditions for waiting time phenomenon.


## 2. Existence and Regularity of Solutions

We are interested in the existence of nonnegative generalized weak solutions to the following initial-boundary value problem:(2.1)(P){ht+(f(h)(a0hxxx+a1hx+wx))x=0inQT,∂ih∂xi(-a,t)=∂ih∂xi(a,t)fort>0,i=0,3-,h(x,0)=h0(x)⩾0,
where f(h)=|h|n,  h=h(x,t),  Ω=(-a,a),  QT=(0,T)×Ω,  n>0,  a0>0,  a1≥0, and w(x,t) such that
(2.2)w(x,·)∈W∞1(0,T)fora.e.x∈Ω-,w(·,t)∈W∞2(Ω)fora.e.t∈[0,T].
ote that (1.3) is a particular case of (2.1) that corresponds to n=3 and w(x,t)=cos(x-ωt).We consider a generalized weak solution in the following sense [40, 41].Definition 2.1.
A generalized weak solution of problem(P) is a nonnegative function h satisfying
(2.3)h∈Cx,t1/2,1/8(Q¯T)∩L∞(0,T;H1(Ω)),ht∈L2(0,T;(H1(Ω))′),h∈Cx,t4,1(𝒫),f(h)(a0hxxx+a1hx+wx)∈L2(𝒫),
where 𝒫:={h>0}. The solution h satisfies (2.1) in the following sense:
(2.4)∫0T〈ht(·,t),ϕ〉dt-∬𝒫f(h)(a0hxxx+a1hx+wx)ϕxdxdt=0,
for all ϕ∈C1(QT)∩C(Q-T) with ϕ(-a,·)=ϕ(a,·);
(2.5)h(·,t)→h(·,0)=h0pointwise&stronglyinL2(Ω)ast→0,(2.6)h(-a,t)=h(a,t)∀t∈[0,T],∂ih∂xi(-a,t)=∂ih∂xi(a,t),
for i=1,2,3 at all points of the lateral boundary where  h≠0.Because the second term of (2.4) has an integral over {h>0} rather than over QT, the generalized weak solution is “weaker” than a standard weak solution. Here, {h>0} is short hand for {(x,t)∈Q-T:h(x,t)>0}. This short hand is used throughout: the time interval included in {h>0} is to be inferred from the context it appears in.A key object for proving additional properties of a generalized weak solution is an integral quantity introduced by Bernis and Friedman [42]: the “entropy” ∫G0(h(x,t))dx. The function G0(z) is defined by
(2.7)G0(z)∶={z2-n(2-n)(1-n)+dz+cifn≠1,2,zlnz-z+eifn=1,-lnz+ze+1ifn=2,
where
(2.8)d={1if1<n<2,0otherwise,c={(n-1)1/(1-n)2-nif1<n<2,0otherwise.
By construction, G0 is a nonnegative convex function on [0,∞). For 1≤n≤2, the linear part of G0 is chosen to ensure that G0 has a positive lower bound on [0,∞). Also in the statement of Theorem 2.2 we use an “α-entropy”, ∫‍G0(α)(h(x,t))dx, where
(2.9)G0(α)(z)∶={zlnz-z+eifα=n-1,-lnz+ze+1ifα=n-2,z2-n+α(2-n+α)(1-n+α)+dz+cotherwise,(2.10)d={1ifα∈(n-2,n-1),0otherwise,c={(n-1-α)1/(1+α-n)2+α-nifα∈(n-2,n-1),0otherwise.G0(α) is a nonnegative convex function on [0,∞). The linear part of G0(α) is chosen to ensure that G0(α) has a positive lower bound on [0,∞) if n-2≤α≤n-1. If α=0, the α-entropy is the same as the entropy (2.7).Theorem 2.2.
(a) (Existence). Let n>0 and the nonnegative initial data h0∈H1(Ω), h0(-a)=h0(a) satisfy
(2.11)∫ΩG0(h0)dx<∞.
Then for any time 0<T<∞ there exists a nonnegative generalized weak solution, h, on QT in the sense of the Definition 2.1. Furthermore,
(2.12)h∈L2(0,T;H2(Ω)).
Let
(2.13)ℰ0(T):=12∫Ω(a0hx2(x,T)-a1h2(x,T)-2w(x,T)h(x,T))dx
then the weak solution satisfies
(2.14)ℰ0(T)+∬{h>0}hn(a0hxxx+a1hx+wx)2dxdt≤ℰ0(0)-∬QThwtdxdt.
(b) (Regularity). If the initial data also satisfies
(2.15)∫ΩG0(α)(h0)dx<∞,
for some -1/2<α<1,  α≠0 then the nonnegative generalized weak solution has the extra regularity h(α+2)/2∈L2(0,T;H2(Ω)) and h(α+2)/4∈L4(0,T;W41(Ω)).The theorem above was proved earlier in [41] for the case n=3 only. We note that the analogue of Theorem 4.2 in [42] also holds: there exists a nonnegative weak solution with the integral formulation
(2.16)∫0T〈ht(·,t),ϕ〉dt+a0∬QT(nhn-1hxhxxϕx+hnhxxϕxx)dxdt-∬QThn(a1hx+wx)ϕxdxdt=0.
If initial data satisfy finite α-entropy condition, that is, ∫‍G0(α)(h0)dx<∞ then one can prove existence of nonnegative solutions with some additional regularity properties and use an integral formulation [43] to define them that is similar to that of (2.16) except that the second integral is replaced by the results of one more integration by parts (there are no hxxx terms). It is worth to mention that for the case 0<n<2 the finite entropy assumption in Theorem 2.2 can be omitted because it does not impose any restriction on nonnegative initial data. One needs to impose finite entropy and finite α-entropy conditions on initial data if n≥2 only.


### 2.1. Regularized Problem

Givenδ,  ε>0, a regularized parabolic problem, similar to one that was studied by Bernis and Friedman [42] can be written as:(
P
δ
,
ϵ
)
(2.17)
h
t
+
(
f
δ
ε
(
h
)
(
a
0
h
x
x
x
+
a
1
h
x
+
w
x
)
)
x
=
0
,
(2.18)
∂
i
h
∂
x
i
(
-
a
,
t
)
=
∂
i
h
∂
x
i
(
a
,
t
)
for
t
>
0
,
i
=
0,3
-
,
(2.19)
h
(
x
,
0
)
=
h
0
,
ε
(
x
)
,
where
(2.20)fδε(z)∶=fε(z)+δ=|z|4|z|4-n+ε+δ∀z∈ℝ1,δ>0,ε>0.
The δ>0 in (2.20) makes the problem (2.17) regular (i.e., uniformly parabolic). The parameter ε is an approximating parameter which has the effect of increasing the degeneracy from f(h)~|h|n to fε(h)~h4. The nonnegative initial data, h0, is approximated via
(2.21)h0+εθ⩽h0,ε∈C4+γ(Ω)forsome0<θ<25,∂ih0,ε∂xi(-a)=∂ih0,ε∂xi(a)fori=0,3¯,h0,ε→h0stronglyinH1(Ω)asε→0.
The ε term in (2.21) “lifts” the initial data so that they are smoothing from H1(Ω) to C4+γ(Ω). By Eĭdel’man [44, Theorem 6.3, p.302], the regularized problem has a unique classical solution hδε∈Cx,t4+γ,1+γ/4(Ω×[0,τδε]) for some time τδε>0. For any fixed value of δ and ε, by Eĭdel’man [44, Theorem 9.3, p.316] if one can prove a uniform in time a priori bound |hδε(x,t)|≤Aδε<∞ for some longer time interval [0,Tloc,δε]  (Tloc,δε>τδε) and for all x∈Ω then Schauder-type interior estimates [44, Corollary 2, p.213] imply that the solution hδε can be continued in time to be in Cx,t4+γ,1+γ/4(Ω×[0,Tloc,δε]).Although the solutionhδε is initially positive, there is no guarantee that it will remain nonnegative. The goal is to take δ→0,  ε→0 in such a way that (1)  Tloc,δε→Tloc>0, (2) the solutions hδε converge to a (nonnegative) limit, h, which is a generalized weak solution, and (3)  h inherits certain a priori bounds. This is done by proving various a priori estimates for hδε that are uniform in δ and ε and hold on a time interval [0,Tloc] that is independent of δ and ε. As a result, {hδε} will be a uniformly bounded and equicontinuous (in the Cx,t1/2,1/8 norm) family of functions in Ω-×[0,Tloc]. Taking δ→0 will result in a family of functions {hε} that are classical, positive, unique solutions to the regularized problem with δ=0. Taking ε→0 will then result in the desired generalized weak solution h. This last step is where the possibility of nonunique weak solutions arise; see [40] for simple examples of how such constructions applied to ht=-(|h|nhxxx)x can result in two different solutions arising from the same initial data.


### 2.2. A Priori Estimates

Our first task is to derive a priori estimates for classical solutions of (2.17)–(2.21). The lemmas given in this section are proved in the Section 4.We use an integral quantity based on a functionGδε chosen such that
(2.22)Gδε′′(z)=1fδε(z),Gδε(z)⩾0.
This is analogous to the “entropy” function first introduced by Bernis and Friedman [42].Lemma 2.3.
Leth0ε satisfy (2.21) and be built from a nonnegative function h0 that satisfies the hypotheses of Theorem 2.2. Then there exist δ0>0, ε0>0 and time Tloc>0 such that if δ∈[0,δ0), ε∈[0,ε0), and hδε is a solution of the problem (2.17)–(2.21) with initial data h0ε, then for any T∈[0,Tloc] the following inequalities:
(2.23)∫Ω{hδε,x2(x,T)+2c1a0Gδε(hδε(x,T))}dx+a0∬QTfδε(hδε)hδε,xxx2dxdt≤K1<∞,(2.24)∫ΩGδε(hδε(x,T))dx+a0∬QThδε,xx2dxdt≤K2<∞
hold. The energy ℰδε(t) (see (2.13)) satisfies
(2.25)ℰδε(T)+∬QTfδε(hδε)(a0hδε,xxx+a1hδε,x+wx)2dxdt=ℰδε(0)-∬QThδεwtdxdt.
The time Tloc and the constants Ki are independent of δ and ε.The proof of existence ofδ0,  ε0,  Tloc,  K1, and K2 is constructive; how to find them and what quantities determine them are shown with details in Section 4.Lemma2.3 yields uniform-in-δ-and-ε bounds for ∫‍hδε,x2, ∫‍Gδε(hδε), ∬‍hδε,xx2, and ∬‍fδε(hδε)hδε,xxx2. However, these bounds are found in a different manner than in earlier work for the equation ht=-(|h|nhxxx)x, for example. Although the inequality (2.24) is unchanged, the inequality (2.23) has an extra term involving Gδε. In the proof, this term was introduced to control additional, lower-order terms. This idea of a “blended” ∥hx∥2-entropy bound was first introduced by Shishkov and Taranets for long-wave stable thin film equations with convection [30].The final a priori bounds for positive, classical solutions use the following functions, parameterized byα for α∉{2,3},
(2.26)Gε(α)(z)=G0(α)(z)+εzα-2(α-3)(α-2)⇒(Gε(α)(z))′′=zαfε(z),
where G0(α) is given by (2.9). In the following lemma, we restrict ourselves to the case α∈[-1/2,1]; note that Gε(α)(z)≥0 for such α.Lemma 2.4.
Assumeε0 and Tloc are from Lemma 2.3, δ=0, and ε∈[0,ε0). Assume α∈[-1/2,1] and that hε is a positive, classical solution of the problem (2.17)–(2.21) with initial data h0,ε satisfying Lemma 2.3. If the initial data h0,ε is built from h0 which also satisfies
(2.27)∫ΩG0(α)(h0(x))dx<∞
then there exists K4 such that
(2.28)∫Ω{hε,x2(x,T)+Gε(α)(hε(x,T))}dx+∬QT[βhεαhε,xx2+γhεα-2hε,x4]dxdt≤K4<∞
holds for all T∈[0,Tloc] and K4 is independent of ε and is determined by α,  ε0,  a0,  a1,  wx,  Ω and h0. Here
(2.29)β={a0if0≤α≤1,a01+2α4(1-α)if-12≤α<0,γ={a0α(1-α)6if0≤α≤1,a0(1+2α)(1-α)36f-12≤α<0.
Furthermore, if α∈(-1/2,1)∖{0} then
(2.30){hε(α+2)/2}ε∈(0,ε0)⊂L2(0,Tloc;H2(Ω)),{hε(α+2)/4}ε∈(0,ε0)⊂L4(0,Tloc;W1,4(Ω))
are uniformly bounded.Theα-entropy, ∫‍G0(α)(h)dx, was first introduced for α=-1/2 in [45] and an a priori bound like that of Lemma 2.4 and regularity results like those of Theorem 2.2 were found simultaneously and independently in [40, 43].The proof of existence starts from a construction of a classical solutionhδε on [0,Tloc] that satisfies the hypotheses of Lemma 2.3 if δ∈(0,δ0) and ε∈(0,ε0). Taking the regularizing parameter, δ, to zero, one proves that there is a limit hε and that hε is a generalized weak solution. After that additional nonlinear estimates are required to analyze properties of the limit hε; specifically to show that it is strictly positive, classical, and unique. Hence, the a priori bounds given by Lemmas 2.3 and 2.4 are applicable to hε. This allows us to take the approximating parameter, ε, to zero and to construct the desired nonnegative generalized weak solution of Theorems 2.2 (see, e.g., [41]).


### 2.3. Long-Time Existence of Solutions

Lemma 2.5.
Leth be a generalized solution of Theorem 2.2. Then
(2.31)a04∥h(·,Tloc)∥H1(Ω)2≤ℰ0(0)+K5+K6Tloc,
where ℰ0(0) is defined in (2.13), M=∫‍h0, and
(2.32)K5=∥w∥∞M+263(a0+a1)3/2a0M2+a0+a12M2|Ω|,K6=∥wt∥∞M.Proof of Lemma2.5.
By (2.13),
(2.33)a02∫Ωhx2(x,T)dx≤ℰ0(T)+a12∫Ωh2(x,T)dx+∫Ωh(x,T)w(x,T)dx-∬QThwtdxdt.
The linear-in-time bound (2.14) on ℰ0(Tloc) then implies
(2.34)a02∥h(·,Tloc)∥H12≤ℰ0(0)+a0+a12∫Ωh2dx+(∥w∥∞+∥wt∥∞T)M.
Using the estimate (see [41, Lemma 4.1, page 1837])
(2.35)∥h∥L2(Ω)2≤62/3M4/3(∫Ωhx2dx)1/3+M2|Ω|,
and Young's inequality:
(2.36)a0+a12∫Ωh2dx≤a0+a12(62/3M4/3(∫Ωhx2dx)1/3+M2|Ω|)≤a04∫Ωhx2(x,Tloc)dx+263(a0+a1)3/2a0M2+a0+a12M2|Ω|.
Using this in (2.34), the desired bound (2.31) follows immediately.ThisH1-estimate will be used to extend the short-time existence of a solution to the long-time existence result of Theorem 2.2 (see [41, Proof of Theorem 3, page 1838]).


## 2.1. Regularized Problem

Givenδ,  ε>0, a regularized parabolic problem, similar to one that was studied by Bernis and Friedman [42] can be written as:(
P
δ
,
ϵ
)
(2.17)
h
t
+
(
f
δ
ε
(
h
)
(
a
0
h
x
x
x
+
a
1
h
x
+
w
x
)
)
x
=
0
,
(2.18)
∂
i
h
∂
x
i
(
-
a
,
t
)
=
∂
i
h
∂
x
i
(
a
,
t
)
for
t
>
0
,
i
=
0,3
-
,
(2.19)
h
(
x
,
0
)
=
h
0
,
ε
(
x
)
,
where
(2.20)fδε(z)∶=fε(z)+δ=|z|4|z|4-n+ε+δ∀z∈ℝ1,δ>0,ε>0.
The δ>0 in (2.20) makes the problem (2.17) regular (i.e., uniformly parabolic). The parameter ε is an approximating parameter which has the effect of increasing the degeneracy from f(h)~|h|n to fε(h)~h4. The nonnegative initial data, h0, is approximated via
(2.21)h0+εθ⩽h0,ε∈C4+γ(Ω)forsome0<θ<25,∂ih0,ε∂xi(-a)=∂ih0,ε∂xi(a)fori=0,3¯,h0,ε→h0stronglyinH1(Ω)asε→0.
The ε term in (2.21) “lifts” the initial data so that they are smoothing from H1(Ω) to C4+γ(Ω). By Eĭdel’man [44, Theorem 6.3, p.302], the regularized problem has a unique classical solution hδε∈Cx,t4+γ,1+γ/4(Ω×[0,τδε]) for some time τδε>0. For any fixed value of δ and ε, by Eĭdel’man [44, Theorem 9.3, p.316] if one can prove a uniform in time a priori bound |hδε(x,t)|≤Aδε<∞ for some longer time interval [0,Tloc,δε]  (Tloc,δε>τδε) and for all x∈Ω then Schauder-type interior estimates [44, Corollary 2, p.213] imply that the solution hδε can be continued in time to be in Cx,t4+γ,1+γ/4(Ω×[0,Tloc,δε]).Although the solutionhδε is initially positive, there is no guarantee that it will remain nonnegative. The goal is to take δ→0,  ε→0 in such a way that (1)  Tloc,δε→Tloc>0, (2) the solutions hδε converge to a (nonnegative) limit, h, which is a generalized weak solution, and (3)  h inherits certain a priori bounds. This is done by proving various a priori estimates for hδε that are uniform in δ and ε and hold on a time interval [0,Tloc] that is independent of δ and ε. As a result, {hδε} will be a uniformly bounded and equicontinuous (in the Cx,t1/2,1/8 norm) family of functions in Ω-×[0,Tloc]. Taking δ→0 will result in a family of functions {hε} that are classical, positive, unique solutions to the regularized problem with δ=0. Taking ε→0 will then result in the desired generalized weak solution h. This last step is where the possibility of nonunique weak solutions arise; see [40] for simple examples of how such constructions applied to ht=-(|h|nhxxx)x can result in two different solutions arising from the same initial data.


## 2.2. A Priori Estimates

Our first task is to derive a priori estimates for classical solutions of (2.17)–(2.21). The lemmas given in this section are proved in the Section 4.We use an integral quantity based on a functionGδε chosen such that
(2.22)Gδε′′(z)=1fδε(z),Gδε(z)⩾0.
This is analogous to the “entropy” function first introduced by Bernis and Friedman [42].Lemma 2.3.
Leth0ε satisfy (2.21) and be built from a nonnegative function h0 that satisfies the hypotheses of Theorem 2.2. Then there exist δ0>0, ε0>0 and time Tloc>0 such that if δ∈[0,δ0), ε∈[0,ε0), and hδε is a solution of the problem (2.17)–(2.21) with initial data h0ε, then for any T∈[0,Tloc] the following inequalities:
(2.23)∫Ω{hδε,x2(x,T)+2c1a0Gδε(hδε(x,T))}dx+a0∬QTfδε(hδε)hδε,xxx2dxdt≤K1<∞,(2.24)∫ΩGδε(hδε(x,T))dx+a0∬QThδε,xx2dxdt≤K2<∞
hold. The energy ℰδε(t) (see (2.13)) satisfies
(2.25)ℰδε(T)+∬QTfδε(hδε)(a0hδε,xxx+a1hδε,x+wx)2dxdt=ℰδε(0)-∬QThδεwtdxdt.
The time Tloc and the constants Ki are independent of δ and ε.The proof of existence ofδ0,  ε0,  Tloc,  K1, and K2 is constructive; how to find them and what quantities determine them are shown with details in Section 4.Lemma2.3 yields uniform-in-δ-and-ε bounds for ∫‍hδε,x2, ∫‍Gδε(hδε), ∬‍hδε,xx2, and ∬‍fδε(hδε)hδε,xxx2. However, these bounds are found in a different manner than in earlier work for the equation ht=-(|h|nhxxx)x, for example. Although the inequality (2.24) is unchanged, the inequality (2.23) has an extra term involving Gδε. In the proof, this term was introduced to control additional, lower-order terms. This idea of a “blended” ∥hx∥2-entropy bound was first introduced by Shishkov and Taranets for long-wave stable thin film equations with convection [30].The final a priori bounds for positive, classical solutions use the following functions, parameterized byα for α∉{2,3},
(2.26)Gε(α)(z)=G0(α)(z)+εzα-2(α-3)(α-2)⇒(Gε(α)(z))′′=zαfε(z),
where G0(α) is given by (2.9). In the following lemma, we restrict ourselves to the case α∈[-1/2,1]; note that Gε(α)(z)≥0 for such α.Lemma 2.4.
Assumeε0 and Tloc are from Lemma 2.3, δ=0, and ε∈[0,ε0). Assume α∈[-1/2,1] and that hε is a positive, classical solution of the problem (2.17)–(2.21) with initial data h0,ε satisfying Lemma 2.3. If the initial data h0,ε is built from h0 which also satisfies
(2.27)∫ΩG0(α)(h0(x))dx<∞
then there exists K4 such that
(2.28)∫Ω{hε,x2(x,T)+Gε(α)(hε(x,T))}dx+∬QT[βhεαhε,xx2+γhεα-2hε,x4]dxdt≤K4<∞
holds for all T∈[0,Tloc] and K4 is independent of ε and is determined by α,  ε0,  a0,  a1,  wx,  Ω and h0. Here
(2.29)β={a0if0≤α≤1,a01+2α4(1-α)if-12≤α<0,γ={a0α(1-α)6if0≤α≤1,a0(1+2α)(1-α)36f-12≤α<0.
Furthermore, if α∈(-1/2,1)∖{0} then
(2.30){hε(α+2)/2}ε∈(0,ε0)⊂L2(0,Tloc;H2(Ω)),{hε(α+2)/4}ε∈(0,ε0)⊂L4(0,Tloc;W1,4(Ω))
are uniformly bounded.Theα-entropy, ∫‍G0(α)(h)dx, was first introduced for α=-1/2 in [45] and an a priori bound like that of Lemma 2.4 and regularity results like those of Theorem 2.2 were found simultaneously and independently in [40, 43].The proof of existence starts from a construction of a classical solutionhδε on [0,Tloc] that satisfies the hypotheses of Lemma 2.3 if δ∈(0,δ0) and ε∈(0,ε0). Taking the regularizing parameter, δ, to zero, one proves that there is a limit hε and that hε is a generalized weak solution. After that additional nonlinear estimates are required to analyze properties of the limit hε; specifically to show that it is strictly positive, classical, and unique. Hence, the a priori bounds given by Lemmas 2.3 and 2.4 are applicable to hε. This allows us to take the approximating parameter, ε, to zero and to construct the desired nonnegative generalized weak solution of Theorems 2.2 (see, e.g., [41]).


## 2.3. Long-Time Existence of Solutions

Lemma 2.5.
Leth be a generalized solution of Theorem 2.2. Then
(2.31)a04∥h(·,Tloc)∥H1(Ω)2≤ℰ0(0)+K5+K6Tloc,
where ℰ0(0) is defined in (2.13), M=∫‍h0, and
(2.32)K5=∥w∥∞M+263(a0+a1)3/2a0M2+a0+a12M2|Ω|,K6=∥wt∥∞M.Proof of Lemma2.5.
By (2.13),
(2.33)a02∫Ωhx2(x,T)dx≤ℰ0(T)+a12∫Ωh2(x,T)dx+∫Ωh(x,T)w(x,T)dx-∬QThwtdxdt.
The linear-in-time bound (2.14) on ℰ0(Tloc) then implies
(2.34)a02∥h(·,Tloc)∥H12≤ℰ0(0)+a0+a12∫Ωh2dx+(∥w∥∞+∥wt∥∞T)M.
Using the estimate (see [41, Lemma 4.1, page 1837])
(2.35)∥h∥L2(Ω)2≤62/3M4/3(∫Ωhx2dx)1/3+M2|Ω|,
and Young's inequality:
(2.36)a0+a12∫Ωh2dx≤a0+a12(62/3M4/3(∫Ωhx2dx)1/3+M2|Ω|)≤a04∫Ωhx2(x,Tloc)dx+263(a0+a1)3/2a0M2+a0+a12M2|Ω|.
Using this in (2.34), the desired bound (2.31) follows immediately.ThisH1-estimate will be used to extend the short-time existence of a solution to the long-time existence result of Theorem 2.2 (see [41, Proof of Theorem 3, page 1838]).


## 3. Finite Speed of Support Propagation

Theorem 3.1.
Let1<n<3. Assume h0 is nonnegative, h0∈H1(Ω) and supph0⊂(-r0,r0)⋐Ω. Then the solution h of Theorem 2.2 has finite speed of support propagation, that is, there exists a continuous nondecreasing function Γ(T),  Γ(0)=0 such that supph(T,·)⊂(-r0-Γ(T),r0+Γ(T))⋐Ω for all T≤T0∶=Γ-1(a-r0).In the following theorem, we find the explicit upper bounds of theΓ(T) for a solution of the corresponding Cauchy problem with a compactly supported nonnegative initial data h0∈H1(ℝ1). Note that the definition of generalized weak solution of the Cauchy problem is as Definition 2.1 except that Ω is replaced by ℝ1 and the relation (2.6) is dropped. Using Lemma 2.5, we can show that the upper estimate of Γ(T) from Theorem 3.1 is independent on Ω therefore the solution from Theorem 2.2 can be extended to be identically zero for |x|>r0-Γ(T) and thus is a solution on the line for all T≤T0. Performing a similar procedure in [T0,2T0],…,[mT0,(m+1)T0],…, we obtain a compactly supported nonnegative solution of the Cauchy problem for all T≥0 and Theorem 2.2 holds with Ω=ℝ1.Theorem 3.2.
Let1<n<3. Assume h0 is nonnegative, h0∈H1(ℝ1), supph0⊂(-r0,r0) and h is a solution of the Cauchy problem. Then the following estimates:Γ(T)≤D1(T1/(n+4)+T5/(n+4)) for all T>0 if 1<n<2,Γ(T)≤D2T1/(n+4) for small enough time if 2≤n<3, are valid. Here the constants Di depend on the parameters problem and initial data only.


### 3.1. Proof of Theorem3.1 for the Case 1<n<2

The following lemma contains the local entropy estimate. The proof of Lemma3.3 is similar to (A.16), (A.29), therefore it is omitted.Lemma 3.3.
Letζ∈Ct,x1,2(Q-T) such that suppζ⊂Ω, (ζ4)′=0 on ∂Ω, and ζ4(-a,t)=ζ4(a,t). Assume that -1/2<α<1, and α≠0. Then there exist constants Ci(i=1,2,3) dependent on n,  m,  α,  a0, and a1, independent of Ω, such that for all 0<T<∞(3.1)∫Ωζ4(x,T)G0(α)(h(x,T))dx-∬QT(ζ4)tG0(α)(h)dxdt+C1∬QT(h(α+2)/2)xx2ζ4dxdt≤∫Ωζ4(x,0)G0(α)(h0)dx+C2∬QThα+2(ζ4+ζx4+ζ2ζxx2+ζ2ζx2+ζ3|ζxx|)dxdt+C3∬QThα+1(|ζ3||ζx|+ζ4)dxdt.Let0<n<2, and let supph0⊆(-r0,r0)⋐Ω. For an arbitrary s∈(0,a-r0) and δ>0 we consider the families of sets
(3.2)Ω(s)=Ω∖(-r0-s,r0+s),QT(s)=(0,T)×Ω(s).
We introduce a nonnegative cutoff function η(τ) from the space C2(ℝ1) with the following properties:
(3.3)η(τ)={0ifτ≤0,τ2(3-2τ)if0<τ<1,1ifτ≤1.
Next we introduce our main cut-off functions ηs,δ(x)∈C2(Ω-) such that 0≤ηs,δ(x)≤1 for all x∈Ω- and possess the following properties:
(3.4)ηs,δ(x)=η(|x|-(r0+s)δ)={1,x∈Ω(s+δ),0,x∈Ω∖Ω(s),|(ηs,δ)x|≤3δ,|(ηs,δ)xx|≤6δ2,
for all s>0,  δ>0:r0+s+δ<a. Choosing ζ4(x,t)=ηs,δ(x)e-t/T, from (3.1) we arrive at
(3.5)∫Ω(s+δ)hα-n+2(T)dx+1T∬QT(s+δ)hα-n+2dxdt+C∬QT(s+δ)(h(α+2)/2)xx2dxdt≤Cδ4∬QT(s)hα+2dxdt+Cδ∬QT(s)hα+1dxdt=:C∑i=12‍δ-αi∬QT(s)hξi,
for all s∈(0,a-r0), where (n-1)+<α<1 and 0<δ<1 is enough small. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region Ω(s+δ) to a function v∶=h(α+2)/2 with a=(2ξi)/(α+2),  b=(2(α-n+2))/(α+2),  d=2,  i=0,  j=2, and θi=((α+2)(ξi-α+n-2))/(ξi(4α-3n+8)) under the conditions:
(3.6)α-n+2<ξifori=1,2.
Integrating the resulted inequalities with respect to time and taking into account (3.5), we arrive at the following relations:
(3.7)∬QT(s+δ)hξi≤CT1-(θiξi)/(α+2)(∑i=12‍δ-αi∬QT(s)hξi)1+νi+CT(∑i=12‍δ-αi∬QT(s)hξi)ξi/(α-n+2),
where νi=(4(ξi-α+n-2))/(4α-3n+8). These inequalities are true provided that
(3.8)θiξiα+2<1⟺ξi<5α-4n+10fori=1,2.
Simple calculations show that inequalities (3.6) and (3.8) hold with some (n-1)+<α<1 if and only if 1<n<2. The finite speed of propagations follows from (3.7) by applying Lemma B.3 with s1=0. Hence,
(3.9)supph(T,·)⊂(-r0-Γ(T),r0+Γ(T))⋐ΩforallT:T∈[0,T0],
where T0:=Γ-1(a-r0).


### 3.2. Proof of Theorem3.2 for the Case 1<n<2

We can repeat the previous procedure from Section3.1 for Ω(s)=ℝ1∖(-r0-s,r0+s) and we obtain
(3.10)Gi(s+δ)∶=∬QT(s+δ)hξi≤CT1-(θiξi)/(α+2)(∑i=12‍δ-αi∬QT(s)hξi)1+νi,
instead of (3.7), and
(3.11)Γ(T)=C(T(1-(θ1ξ1)/(α+2))(1+ν2)T(1-(θ2ξ2)/(α+2))ν1(1+ν1)(G(0))ν1)1/(4(1+ν1)(1+ν2))+C(T(1-(θ2ξ2)/(α+2))(1+ν1)T(1-(θ1ξ1)/(α+2))ν2(1+ν2)(G(0))ν2)1/((1+ν1)(1+ν2)),
where
(3.12)G(0)=C(T(1-(θ1ξ1)/(α+2))(1+ν2)(G2(0))1+ν1+T(1-(θ2ξ2)/(α+2))(1+ν1)(G1(0))1+ν2).
Now we need to estimate G(0). With that end in view, we obtain the following estimates:
(3.13)Gi(0)≤C1(C2+C3T)(ξi-1)/(α+5)T1-(ξi-1)/(α+5),i=1,2,
where 1<ξi<α+6, and Ci depends on initial data only. Really, applying the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in Ω=ℝ1 to a function v:=h(α+2)/2 with a=(2ξi)/(α+2),  b=2/(α+2),  d=2,  i=0,  j=2, and θ~i=((α+2)(ξi-1))/(ξi(α+5)) under the condition ξi>1, we deduce that
(3.14)∫ℝ1hξi≤c∥h0∥1(2(3ξi+α+2))/((α+2)(α+5))(∫ℝ1(h(α+2)/2)xx2dx)(ξi-1)/(α+5).
Integrating (3.14) with respect to time and taking into account the Hölder inequality ((ξi-1)/(α+5)<1⇒ξi<α+6), we arrive at the following relations:
(3.15)∬QThξi≤c∥h0∥1(2(3ξi+α+2))/((α+2)(α+5))T1-(ξi-1)/(α+5)(∬QT(h(α+2)/2)xx2dx)(ξi-1)/(α+5).
From (3.15), due to (A.16) (as ε→0) and (2.31), we find (3.13).Inserting (3.13) into (3.12), we obtain after straightforward computations that
(3.16)Γ(T)≤C(T1/(n+4)+T5/(n+4))forallT≥0.


### 3.3. Proof of Theorem3.1 for the Case 4/3<n<3

The following lemma contains the local energy estimate. The proof of Lemma3.4 is Appendix A.Lemma 3.4.
Letn∈(1/2,3) and β>(1-n)/3. Let ζ∈C2(Ω-) such that suppζ in Ω and (ζ6)′=0 on ∂Ω, and ζ(-a)=ζ(a). Then there exist constants Ci(i=1,3¯) dependent on n,  m,  a0, and a1, independent of Ω and ε, such that for any 0<T<∞(3.17)∫Ωζ6hx2(x,T)dx+∫Ωζ4hβ+1(T)dx+C1∬QTζ6(h(n+2)/2)xxx2dxdt≤∫Ωζ6h02(x)dx+∫Ωζ4h0β+1dx+C2∬QThn+2(ζ6+ζx6+ζ3|ζxx|3)dxdt+C3∬QT{χ{ζ>0}hn+3β-1+hnζ6}dxdt,(3.18)∫Ωζ6hx2(x,T)dx+C1∬QTζ6(h(n+2)/2)xxx2dxdt⩽∫Ωζ6h02(x)dx+C2∬QThn+2(ζ6+ζx6+ζ3|ζxx|3)dxdt+C3∬QThnζ6dxdt.Letηs,δ(x) be denoted by (3.4). Setting ζ6(x)=ηs,δ(x) into (3.17), after simple transformations, we obtain
(3.19)∫Ω(s+δ)hx2(x,T)dx+∫Ω(s+δ)hβ+1(T)dx+C∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤Cδ6∬QT(s)hn+2dxdt+C∬QT(s){hn+3β-1+hn}dxdt=:C∑i=13‍δ-αi∬QT(s)hξi,
for all for all s∈(0,a-r0),δ>0:r0+s+δ<a. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region Ω(s+δ) to a function v:=h(n+2)/2 with a=(2ξi)/(n+2),  b=(2(β+1))/(n+2),  d=2,  i=0,  j=3, and θi=((n+2)(ξi-β-1))/(ξi(n+5β+7)) under the conditions:
(3.20)β<ξi-1fori=1,3¯.
Integrating the resulted inequalities with respect to time and taking into account (3.19), we arrive at the following relations:
(3.21)∬QT(s+δ)hξi≤CT1-(θiξi)/(n+2)(∑i=13‍δ-αi∬QT(s)hξi)1+νi+CT(∑i=13‍δ-αi∬QT(s)hξi)ξi/(β+1),
where νi=(6(ξi-β-1))/(n+5β+7). These inequalities are true provided that
(3.22)θiξin+2<1⟺β>ξi-n-86fori=1,3¯.
Simple calculations show that inequalities (3.20) and (3.22) hold with some β∈((2-n)/2,n-1) if and only if 4/3<n<3. Since all integrals on the right-hand sides of (3.21) vanish as T→0, the finite speed of propagations follows from (3.21) by applying Lemma B.3 with s1=0 and sufficiently small T. Hence,
(3.23)supph(T,·)⊂(-r0-Γ(T),r0+Γ(T))⋐ΩforallT:0≤T≤T0.


### 3.4. Proof of Theorem3.2 for the Case 4/3<n<3

Suppose thatΩ(s)=ℝ1∖{x:|x|<s},  QT(s)=(0,T)×Ω(s) for all s>r0, supph0⊆(-r0,r0), and Γ(T)=r(T)-r0. Since the time interval is small, we can assume that r(T)<2r0. Hence, for all s∈(r0,2r0), we can take (up to regularization) ζ=(|x|-s)+ in (3.18). As a result, we obtain
(3.24)12∫Ω(s)(|x|-s)+6hx2dx+δ6C1∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤C4∬QT(s){hn+2+(r(T)-s)+6hn}dxdt,
for all T≤T0,  s∈(r0,2r0). Using the Hardy type inequality
(3.25)∫Ω(s)(|x|-s)+αf2dx≤C0∫Ω(s)(|x|-s)+α+2fx2dx,
where C0=4/(α+1)2 and α>-1, we deduce that
(3.26)∫Ω(s+δ)hdx≤(∫Ω(s+δ)(|x|-s)+4h2dx)1/2(∫Ω(s+δ)(|x|-s)+-4dx)1/2≤(C03δ3)1/2(∫Ω(s)(|x|-s)+6hx2dx)1/2,
whence
(3.27)(∫Ω(s+δ)hdx)2≤C03δ-3∫Ω(s)(|x|-s)+6hx2dx,
for all δ>0, s∈(r0,2r0). Substituting (3.27) in (3.24), we get
(3.28)32C0δ-3supt(∫Ω(s+δ)hdx)2+C1∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤C4δ6∬QT(s){hn+2+Γ6(T)hn}dxdt,
for all T≤T0,  s∈(r0,2r0). By the Nirenberg-Gagliardo, Hölder and Young inequalities, after simple transformations, for ϵi>0, we have
(3.29)C4δ6∬QT(s)hn+2dxdt≤ϵ1∬QT(s)(h(n+2)/2)xxx2dxdt+C(ϵ1)δn+7∫0T(∫Ω(s)hdx)n+2dt,C4Γ6(T)δ6∬QT(s)hndxdt≤ϵ2∬QT(s)(h(n+2)/2)xxx2dxdt+C(ϵ2)(Γ(T)δ)3(n+7)/4∫0T(∫Ω(s)hdx)(3n+1)/4dt.
Substituting the estimates (3.29) in (3.28) and making the standard iterative procedure for small enough 0<ϵi<1, we arrive at the inequality
(3.30)32C0supt(∫Ω(s+δ)hdx)2+C5δ3∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤C6∑i=12‍GT(i)(s)δαi,
where    α1=n+4,  α2=(3(n+3))/4,    GT(1)(s)∶=∫0T(∫Ω(s)hdx)n+2dt,    GT(2)(s):=Γ(3(n+7))/4(T)∫0T(∫Ω(s)hdx)(3n+1)/4dt. Thus, (3.30) yields
(3.31)GT(i)(s+δ)≤C7TΓμi(T)(∑i=12‍GT(i)(s)δαi)βi,
for all s∈(r0,2r0) and 0<δ<s, where μ1=0,  μ2=(3(n+7))/4,  β1=(n+2)/2,  β2=(3n+1)/8. By Lemma B.3, from (3.31) we find that GT(i)(s0)=0, where Γ(T)≤s0(T)=C8(T1/α1+T1/α2Γμ2/α2(T)). As μ2/α2=(n+7)/(n+3)>1 for any T≤T0, we have Γ(T)≤C9T1/(n+4).


## 3.1. Proof of Theorem3.1 for the Case 1<n<2

The following lemma contains the local entropy estimate. The proof of Lemma3.3 is similar to (A.16), (A.29), therefore it is omitted.Lemma 3.3.
Letζ∈Ct,x1,2(Q-T) such that suppζ⊂Ω, (ζ4)′=0 on ∂Ω, and ζ4(-a,t)=ζ4(a,t). Assume that -1/2<α<1, and α≠0. Then there exist constants Ci(i=1,2,3) dependent on n,  m,  α,  a0, and a1, independent of Ω, such that for all 0<T<∞(3.1)∫Ωζ4(x,T)G0(α)(h(x,T))dx-∬QT(ζ4)tG0(α)(h)dxdt+C1∬QT(h(α+2)/2)xx2ζ4dxdt≤∫Ωζ4(x,0)G0(α)(h0)dx+C2∬QThα+2(ζ4+ζx4+ζ2ζxx2+ζ2ζx2+ζ3|ζxx|)dxdt+C3∬QThα+1(|ζ3||ζx|+ζ4)dxdt.Let0<n<2, and let supph0⊆(-r0,r0)⋐Ω. For an arbitrary s∈(0,a-r0) and δ>0 we consider the families of sets
(3.2)Ω(s)=Ω∖(-r0-s,r0+s),QT(s)=(0,T)×Ω(s).
We introduce a nonnegative cutoff function η(τ) from the space C2(ℝ1) with the following properties:
(3.3)η(τ)={0ifτ≤0,τ2(3-2τ)if0<τ<1,1ifτ≤1.
Next we introduce our main cut-off functions ηs,δ(x)∈C2(Ω-) such that 0≤ηs,δ(x)≤1 for all x∈Ω- and possess the following properties:
(3.4)ηs,δ(x)=η(|x|-(r0+s)δ)={1,x∈Ω(s+δ),0,x∈Ω∖Ω(s),|(ηs,δ)x|≤3δ,|(ηs,δ)xx|≤6δ2,
for all s>0,  δ>0:r0+s+δ<a. Choosing ζ4(x,t)=ηs,δ(x)e-t/T, from (3.1) we arrive at
(3.5)∫Ω(s+δ)hα-n+2(T)dx+1T∬QT(s+δ)hα-n+2dxdt+C∬QT(s+δ)(h(α+2)/2)xx2dxdt≤Cδ4∬QT(s)hα+2dxdt+Cδ∬QT(s)hα+1dxdt=:C∑i=12‍δ-αi∬QT(s)hξi,
for all s∈(0,a-r0), where (n-1)+<α<1 and 0<δ<1 is enough small. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region Ω(s+δ) to a function v∶=h(α+2)/2 with a=(2ξi)/(α+2),  b=(2(α-n+2))/(α+2),  d=2,  i=0,  j=2, and θi=((α+2)(ξi-α+n-2))/(ξi(4α-3n+8)) under the conditions:
(3.6)α-n+2<ξifori=1,2.
Integrating the resulted inequalities with respect to time and taking into account (3.5), we arrive at the following relations:
(3.7)∬QT(s+δ)hξi≤CT1-(θiξi)/(α+2)(∑i=12‍δ-αi∬QT(s)hξi)1+νi+CT(∑i=12‍δ-αi∬QT(s)hξi)ξi/(α-n+2),
where νi=(4(ξi-α+n-2))/(4α-3n+8). These inequalities are true provided that
(3.8)θiξiα+2<1⟺ξi<5α-4n+10fori=1,2.
Simple calculations show that inequalities (3.6) and (3.8) hold with some (n-1)+<α<1 if and only if 1<n<2. The finite speed of propagations follows from (3.7) by applying Lemma B.3 with s1=0. Hence,
(3.9)supph(T,·)⊂(-r0-Γ(T),r0+Γ(T))⋐ΩforallT:T∈[0,T0],
where T0:=Γ-1(a-r0).


## 3.2. Proof of Theorem3.2 for the Case 1<n<2

We can repeat the previous procedure from Section3.1 for Ω(s)=ℝ1∖(-r0-s,r0+s) and we obtain
(3.10)Gi(s+δ)∶=∬QT(s+δ)hξi≤CT1-(θiξi)/(α+2)(∑i=12‍δ-αi∬QT(s)hξi)1+νi,
instead of (3.7), and
(3.11)Γ(T)=C(T(1-(θ1ξ1)/(α+2))(1+ν2)T(1-(θ2ξ2)/(α+2))ν1(1+ν1)(G(0))ν1)1/(4(1+ν1)(1+ν2))+C(T(1-(θ2ξ2)/(α+2))(1+ν1)T(1-(θ1ξ1)/(α+2))ν2(1+ν2)(G(0))ν2)1/((1+ν1)(1+ν2)),
where
(3.12)G(0)=C(T(1-(θ1ξ1)/(α+2))(1+ν2)(G2(0))1+ν1+T(1-(θ2ξ2)/(α+2))(1+ν1)(G1(0))1+ν2).
Now we need to estimate G(0). With that end in view, we obtain the following estimates:
(3.13)Gi(0)≤C1(C2+C3T)(ξi-1)/(α+5)T1-(ξi-1)/(α+5),i=1,2,
where 1<ξi<α+6, and Ci depends on initial data only. Really, applying the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in Ω=ℝ1 to a function v:=h(α+2)/2 with a=(2ξi)/(α+2),  b=2/(α+2),  d=2,  i=0,  j=2, and θ~i=((α+2)(ξi-1))/(ξi(α+5)) under the condition ξi>1, we deduce that
(3.14)∫ℝ1hξi≤c∥h0∥1(2(3ξi+α+2))/((α+2)(α+5))(∫ℝ1(h(α+2)/2)xx2dx)(ξi-1)/(α+5).
Integrating (3.14) with respect to time and taking into account the Hölder inequality ((ξi-1)/(α+5)<1⇒ξi<α+6), we arrive at the following relations:
(3.15)∬QThξi≤c∥h0∥1(2(3ξi+α+2))/((α+2)(α+5))T1-(ξi-1)/(α+5)(∬QT(h(α+2)/2)xx2dx)(ξi-1)/(α+5).
From (3.15), due to (A.16) (as ε→0) and (2.31), we find (3.13).Inserting (3.13) into (3.12), we obtain after straightforward computations that
(3.16)Γ(T)≤C(T1/(n+4)+T5/(n+4))forallT≥0.


## 3.3. Proof of Theorem3.1 for the Case 4/3<n<3

The following lemma contains the local energy estimate. The proof of Lemma3.4 is Appendix A.Lemma 3.4.
Letn∈(1/2,3) and β>(1-n)/3. Let ζ∈C2(Ω-) such that suppζ in Ω and (ζ6)′=0 on ∂Ω, and ζ(-a)=ζ(a). Then there exist constants Ci(i=1,3¯) dependent on n,  m,  a0, and a1, independent of Ω and ε, such that for any 0<T<∞(3.17)∫Ωζ6hx2(x,T)dx+∫Ωζ4hβ+1(T)dx+C1∬QTζ6(h(n+2)/2)xxx2dxdt≤∫Ωζ6h02(x)dx+∫Ωζ4h0β+1dx+C2∬QThn+2(ζ6+ζx6+ζ3|ζxx|3)dxdt+C3∬QT{χ{ζ>0}hn+3β-1+hnζ6}dxdt,(3.18)∫Ωζ6hx2(x,T)dx+C1∬QTζ6(h(n+2)/2)xxx2dxdt⩽∫Ωζ6h02(x)dx+C2∬QThn+2(ζ6+ζx6+ζ3|ζxx|3)dxdt+C3∬QThnζ6dxdt.Letηs,δ(x) be denoted by (3.4). Setting ζ6(x)=ηs,δ(x) into (3.17), after simple transformations, we obtain
(3.19)∫Ω(s+δ)hx2(x,T)dx+∫Ω(s+δ)hβ+1(T)dx+C∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤Cδ6∬QT(s)hn+2dxdt+C∬QT(s){hn+3β-1+hn}dxdt=:C∑i=13‍δ-αi∬QT(s)hξi,
for all for all s∈(0,a-r0),δ>0:r0+s+δ<a. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region Ω(s+δ) to a function v:=h(n+2)/2 with a=(2ξi)/(n+2),  b=(2(β+1))/(n+2),  d=2,  i=0,  j=3, and θi=((n+2)(ξi-β-1))/(ξi(n+5β+7)) under the conditions:
(3.20)β<ξi-1fori=1,3¯.
Integrating the resulted inequalities with respect to time and taking into account (3.19), we arrive at the following relations:
(3.21)∬QT(s+δ)hξi≤CT1-(θiξi)/(n+2)(∑i=13‍δ-αi∬QT(s)hξi)1+νi+CT(∑i=13‍δ-αi∬QT(s)hξi)ξi/(β+1),
where νi=(6(ξi-β-1))/(n+5β+7). These inequalities are true provided that
(3.22)θiξin+2<1⟺β>ξi-n-86fori=1,3¯.
Simple calculations show that inequalities (3.20) and (3.22) hold with some β∈((2-n)/2,n-1) if and only if 4/3<n<3. Since all integrals on the right-hand sides of (3.21) vanish as T→0, the finite speed of propagations follows from (3.21) by applying Lemma B.3 with s1=0 and sufficiently small T. Hence,
(3.23)supph(T,·)⊂(-r0-Γ(T),r0+Γ(T))⋐ΩforallT:0≤T≤T0.


## 3.4. Proof of Theorem3.2 for the Case 4/3<n<3

Suppose thatΩ(s)=ℝ1∖{x:|x|<s},  QT(s)=(0,T)×Ω(s) for all s>r0, supph0⊆(-r0,r0), and Γ(T)=r(T)-r0. Since the time interval is small, we can assume that r(T)<2r0. Hence, for all s∈(r0,2r0), we can take (up to regularization) ζ=(|x|-s)+ in (3.18). As a result, we obtain
(3.24)12∫Ω(s)(|x|-s)+6hx2dx+δ6C1∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤C4∬QT(s){hn+2+(r(T)-s)+6hn}dxdt,
for all T≤T0,  s∈(r0,2r0). Using the Hardy type inequality
(3.25)∫Ω(s)(|x|-s)+αf2dx≤C0∫Ω(s)(|x|-s)+α+2fx2dx,
where C0=4/(α+1)2 and α>-1, we deduce that
(3.26)∫Ω(s+δ)hdx≤(∫Ω(s+δ)(|x|-s)+4h2dx)1/2(∫Ω(s+δ)(|x|-s)+-4dx)1/2≤(C03δ3)1/2(∫Ω(s)(|x|-s)+6hx2dx)1/2,
whence
(3.27)(∫Ω(s+δ)hdx)2≤C03δ-3∫Ω(s)(|x|-s)+6hx2dx,
for all δ>0, s∈(r0,2r0). Substituting (3.27) in (3.24), we get
(3.28)32C0δ-3supt(∫Ω(s+δ)hdx)2+C1∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤C4δ6∬QT(s){hn+2+Γ6(T)hn}dxdt,
for all T≤T0,  s∈(r0,2r0). By the Nirenberg-Gagliardo, Hölder and Young inequalities, after simple transformations, for ϵi>0, we have
(3.29)C4δ6∬QT(s)hn+2dxdt≤ϵ1∬QT(s)(h(n+2)/2)xxx2dxdt+C(ϵ1)δn+7∫0T(∫Ω(s)hdx)n+2dt,C4Γ6(T)δ6∬QT(s)hndxdt≤ϵ2∬QT(s)(h(n+2)/2)xxx2dxdt+C(ϵ2)(Γ(T)δ)3(n+7)/4∫0T(∫Ω(s)hdx)(3n+1)/4dt.
Substituting the estimates (3.29) in (3.28) and making the standard iterative procedure for small enough 0<ϵi<1, we arrive at the inequality
(3.30)32C0supt(∫Ω(s+δ)hdx)2+C5δ3∬QT(s+δ)(h(n+2)/2)xxx2dxdt≤C6∑i=12‍GT(i)(s)δαi,
where    α1=n+4,  α2=(3(n+3))/4,    GT(1)(s)∶=∫0T(∫Ω(s)hdx)n+2dt,    GT(2)(s):=Γ(3(n+7))/4(T)∫0T(∫Ω(s)hdx)(3n+1)/4dt. Thus, (3.30) yields
(3.31)GT(i)(s+δ)≤C7TΓμi(T)(∑i=12‍GT(i)(s)δαi)βi,
for all s∈(r0,2r0) and 0<δ<s, where μ1=0,  μ2=(3(n+7))/4,  β1=(n+2)/2,  β2=(3n+1)/8. By Lemma B.3, from (3.31) we find that GT(i)(s0)=0, where Γ(T)≤s0(T)=C8(T1/α1+T1/α2Γμ2/α2(T)). As μ2/α2=(n+7)/(n+3)>1 for any T≤T0, we have Γ(T)≤C9T1/(n+4).


## 4. Waiting Time Phenomenon

LetΩ(s)={x:x≥s} for all s∈ℝ1, and
(4.1)h0(s)∶=∫Ω(s)h0α-n+2(x)dx=0∀s≥0,
where (n-1)+<α<1. Let us assume that the function h0(s) satisfies the flatness conditions. Namely, for every s:s0<s<0 the following estimate:
(4.2)h0(s)≤χmax{(-s)1+(4(α-n+2))/n,(-s)1+(4-3n+4(α-n+2))/(4(n-1))}={χ(-s)1+(4(α-n+2))/nfor43≤n<2,χ(-s)1+(4-3n+4(α-n+2))/(4(n-1))for1<n<43,
is valid.Theorem 4.1.
Let1<n<2. Assume h0 is nonnegative, h0∈H1(ℝ1) and meas{Ω(s)∩supph0}=∅ for all s≥0, that is, the condition (4.1) is valid, and the flatness condition (4.2) also holds.
Then for the solutionh of Theorem 2.2 (with Ω=ℝ1) there exists the time T*=T*(χ)>0 depending on the known parameters only such that
(4.3)supph(t,·)∩Ω(0)=∅∀0<t≤T*,
where χ is the constant from the flatness condition. Note, that T*→+∞ as χ→0.Remark 4.2.
Let the initial datah0∈C(ℝ1) satisfy the following properties:(1)
if1<n<4/3 then we suppose that
(4.4)supx∈Ω(s)h0(x)≤χ(-s)(4-3n+4(α-n+2))/(4(n-1)(α-n+2))forsomeα∈((n-1)+,1);(2)
if4/3≤n<2 then we suppose that supx∈Ω(s)h0(x)≤χ(-s)4/n.
These assumptions on the initial data are sufficient for the validity of flatness condition (4.2) and guarantee the appearance of the WTP, that is, the validation of property (4.3).Remark 4.3.
Note that due to Lemma2.5 we have the estimate hα+2(x,t)≤C(1+t)hα+1(x,t). Therefore, using this inequality in (3.1) with Ω=ℝ1, we could also obtain the waiting time phenomenon by the application of Theorem 2.1 from [46] with w=h(α+2)/2,  l=k=p=2,  q=(2(α-n+2))/(α+2), and s=(2(α+1))/(α+2).Proof of Theorem4.1.
Similar to (3.10) for Ω(s)={x:x≥s} and we obtain
(4.5)Gi(s+δ)∶=∬QT(s+δ)hξi≤KT1-(θiξi)/(α+2)(∑k=12‍δ-αkGk(s)+h0(s))1+νi.
Let us check that all conditions of Lemma B.4 are satisfied. We denote by
(4.6)Gmax(s)∶=maxi=1,2{c02β+1(∑k=12‍(Gk(s))β¯k)βi-1(s)}1/(αiβ),gmax(s)∶=maxi=1,2{2(β+1)/(αiβ)(2β-1∑k=12‍(KT1-(θkξk)/(α+2))β¯k)β¯i/αi(h0(s))(βi-1)/αi},c0=2β-1∑k=12‍(KT1-(θkξk)/(α+2))β¯k,βi=1+νi,β=β1β2.
Taking s=-2δ in (4.5) and passing to the limit δ→∞, due to the boundedness of functions Gk(s) and h0(s), we deduce
(4.7)Gk(-∞)≤KT1-(θkξk)/(α+2)h0βk(-∞).
This implies that the condition (i) of Lemma B.4 is fulfilled. Because of the assumption (4.2) on the function h0(s), we can find T* such that the condition (ii) of Lemma B.4 is valid for all T∈[0,T*]. Here T*=T*(χ) goes to infinity as χ→0. Hence, the application of Lemma B.4 ends the proof.



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*Source: 570283-2012-09-09.xml*