# Research Announcement: Finite–time Blow Up and Long–wave Unstable Thin Film
Equations

Marina Chugunova, M.C. Pugh, Roman M. Taranets

###### Abstract

We study short–time existence, long–time existence, finite speed of
propagation, and finite–time blow–up of nonnegative solutions for long-wave
unstable thin film equations
$h_{t}=-a_{0}(h^{n}h_{xxx})_{x}-a_{1}(h^{m}h_{x})_{x}$ with $n>0$, $a_{0}>0$,
and $a_{1}>0$. The existence and finite speed of propagation results extend
those of [Comm Pure Appl Math 51:625–661, 1998]. For $0<n<2$ we prove the
existence of a nonnegative, compactly–supported, strong solution on the line
that blows up in finite time. The construction requires that the initial data
be nonnegative, compactly supported in $\mathbb{R}^{1}$, be in
$H^{1}(\mathbb{R}^{1})$, and have negative energy. The blow-up is proven for a
large range of $(n,m)$ exponents and extends the results of [Indiana Univ Math
J 49:1323–1366, 2000].

2000 MSC: 35K65, 35K35, 35Q35, 35B05, 35B45, 35G25, 35D05, 35D10, 76A20

keywords: fourth-order degenerate parabolic equations, thin liquid films,
long–wave unstable, finite–time blow–up, finite speed of propagation

##  1 Introduction

Numerous articles have been published since the early sixties concerning the
problem of developing finite-time singularities by solutions of nonlinear
parabolic equations (see the survey papers of Levine [31] and of Bandle and
Brunner [2]). Such problems arise in various applied fields such as combustion
theory, the theory of phase separation in binary alloys, population dynamics
and incompressible fluid flow.

Whether or not there is a finite–time singularity, such as
$\|u(\cdot,t)\|_{\infty}\to\infty$ as $t\to T^{*}<\infty$, is strongly
affected by the nonlinearity in the PDE. For example, consider the semilinear
heat equation on the line:

$u_{t}=u_{xx}+u^{p},$ (1.1)

where $u$ is real–valued. If $p\leqslant 1$ then a solution of an initial
value problem exists for all time. If $1<p\leqslant 3$, then any non-trivial
solution blows up in finite time. If $p>3$ then some initial data yield
solutions that exist for all time and other initial data result in solutions
that have finite–time singularities. The manner in which solutions blow up is
well understood computationally and analytically. The blow up is of a
focussing type: there are isolated points in space around which the graph of
the solution narrows and becomes taller as $t\uparrow T^{*}$, converging to
delta functions centered at the blow–up points. As $t\uparrow T^{*}$, the
behaviour of the solution near the blow–up point(s) becomes more and more
self–similar. Proving this convergence to a self–similar solution uses the
maximum principle, which doesn’t hold for fourth–order equations like the one
we study in this article.

We study the longwave-unstable generalized thin film equation,

$h_{t}=-a_{0}\,(|h|^{n}\,h_{xxx})_{x}-a_{1}\,(|h|^{m}\,h_{x})_{x},$ (1.2)

where $a_{0}>0$, $a_{1}>0$, and $h$ is real valued. Perturbing a constant
steady state slightly, $h_{0}(x)=\bar{h}+\epsilon
h_{1}(x,0)=\bar{h}+\epsilon\cos(\xi x+\phi)$, and linearizing the equation
about $\bar{h}$, the small perturbation $h_{1}(x,t)$ will (approximately)
satisfy $h_{t}=-a_{0}|\bar{h}|^{n}h_{xxxx}-a_{1}|\bar{h}|^{m}h_{xx}$. Hence
the constant steady state is linearly unstable to long wave perturbations:

$\xi^{2}<|\bar{h}|^{m-n}a_{1}/a_{0}\quad\Longrightarrow\quad h_{1}(x,t)\sim
e^{-a_{0}\xi^{2}|\bar{h}|^{n}\left(\xi^{2}-\tfrac{a_{1}}{a_{0}}|\bar{h}|^{m-n}\right)t}\cos(\xi
x+\phi)\quad\mbox{grows}.$ (1.3)

Such equations arise in the modelling of fluids and materials. For example,
equation (1.2) with $n=m=1$ describes a thin jet in a Hele-Shaw cell [16]
where $h$ represents the thickness of the jet and $x$ is the axial direction;
if $n=3$ and $m=-1$ equation (1.2) describes Van der Waals driven rupture of
thin films [44] where $h$ represents the thickness of the film; if $n=m=3$ the
equation models fluid droplets hanging from a ceiling [21] with $h$
representing the thickness of the film, and finally if $n=0$ and $m=1$ the
equation is a modified Kuramoto-Sivashinsky equation and describes the
solidification of a hyper-cooled melt [9] where $h$ desribes the deviation
from flatness of a near planar interface. We note that in the first three
cases the solution $h$ must be nonnegative for the model to make physical
sense.

Hocherman and Rosenau [28] considered whether or not equation (1.2) could have
solutions that blow up in finite time. They conjectured that if $n>m$ then
solutions might blow up in finite time but if $n<m$ they would exist for all
time. Indeed, this conjecture is natural if one considers the linear stability
of a constant steady state $\bar{h}$: if $n>m$ the unstable band (1.3) grows
as $\bar{h}\to\infty$ suggesting that $n=m$ is critical.

Hocherman and Rosenau considered general, real–valued solutions. However, if
$n>0$ equation (1.2) may have solutions that are nonnegative for all time.
Bertozzi and Pugh [12] proposed that111In fact, the article considers (1.2)
with general coefficients: $f(h)$ and $g(h)$ instead of $|h|^{n}$ and
$|h|^{m}$ respectively. In the following, for simplicity, their results are
discussed for the power–law case. if the boundary conditions are such that the
mass, $\int h(x,t)\;dx$, is conserved then mass conservation, combined with
the nonnegativity results in a different balance: $m=n+2$ instead of $m=n$.
For such cases they introduced the regimes

$\begin{cases}m<n+2\Longrightarrow\mbox{subcritical regime}\\\
m=n+2\Longrightarrow\mbox{critical regime}\\\
m>n+2\Longrightarrow\mbox{supercritical regime}\end{cases}$

In [12], Bertozzi and Pugh considered equation (1.2) on a finite interval with
periodic boundary conditions. For a subset of the subcritical ($m<n+2$) regime
they proved some global-in-time results. Specifically, they proved that given
positive initial data, $h_{0}>0$, there is a nonnegative weak solution of
(1.2) that exists for all time if $0<n\leqslant m<n+2$. By restricting $n$
further to $0<n<3$ they can consider nonnegative initial data, $h_{0}\geqslant
0$. They prove that there a nonnegative weak solution that exists for all time
and also prove the local entropy bound needed for the finite speed of
propagation proof for $0<n<2$. For the critical ($m=n+2$) regime, they prove
that the above results will hold if the mass $\int h_{0}\;dx$ is sufficiently
small. Also, they provided numerical simulations suggesting that other initial
data can result in solutions that blow up in finite time and conjectured that
this is also true for the supercritical ($m>n+2$) regime.

In [13], they considered equation (1.2) on the line and found some analytical
results for the critical and supercritical ($m\geqslant n+2$) regimes in the
special case of $n=1$. They introduced a large class of ‘‘negative energy’’
initial data and proved that given initial data $h_{0}$ with compact support
and negative energy there is a nonnegative weak solution $h$ that blows up in
finite time: there is a time $T^{*}<\infty$ such that the weak solution
$h(\cdot,t)$ exists on $[0,T^{*})$ and

$\limsup_{t\to
T^{*}}\|h(\cdot,t)\|_{L^{\infty}}=\infty\qquad\mbox{and}\qquad\limsup_{t\to
T^{*}}\|h(\cdot,t)\|_{H^{1}}=\infty.$

The blow–up time $T^{*}$ depends only on $m$ and $H^{1}$-norm of the initial
data. We note that uniqueness of nonnegative weak solutions of (1.2) is an
open problem. Indeed, there are simple counterexamples to uniqueness for the
simplest equation $h_{t}=-(h^{n}h_{xxx})_{x}$ (see, e.g. [4]) although it is
hypothesized that solutions are unique if one considers the question within a
sufficiently restrictive class of weak solutions. For this reason, one cannot
exclude the possibility that the initial data $h_{0}$ might also be achieved
by a different weak solution, one that exists for all time.

Their proof relied on a second moment argument, found formally by Bernoff [8]:
if $h$ is a smooth compactly-supported solution of (1.2) on $[0,T^{*})$ then
the second moment of $h$ satisfies

$\int\limits_{-\infty}^{\infty}x^{2}h(x,t)\;dx\leqslant\int\limits_{-\infty}^{\infty}x^{2}h_{0}(x)\;dx+6\mathcal{E}(0)\,t$
(1.4)

holds for all $t\in[0,T^{*})$. Here, $\mathcal{E}(0)$ is the energy of the
initial data:

$\mathcal{E}(0):=\int\limits_{-\infty}^{\infty}\left\\{\tfrac{a_{0}}{2}{h_{0}}_{x}^{2}(x)-\tfrac{a_{1}}{m(m+1)}h_{0}^{m+1}(x)\right\\}\;dx.$
(1.5)

As a result, there could never be a global-in-time nonnegative smooth solution
with negative-energy initial data: for such a solution the left–hand side of
(1.4) would always be nonnegative but the right–hand side would become
negative in finite time. (This argument is strictly formal because, to date,
no-one has constructed nonnegative, compactly-supported, smooth solutions on
the line.) The blow-up result is found by first proving the short-time
existence of a nonnegative, compactly-supported, weak solution: it exists on
$[0,T_{0}]$ where the larger $\|h_{0}\|_{H^{1}}$ is, the smaller $T_{0}$ will
be. Also, the constructed solution satisfies the second moment inequality
(1.4) at time $t=T_{0}$. By ‘‘time–stepping’’ the short-time existence result,
they construct a solution on $[0,T^{*})$ such that the second moment
inequality (1.4) holds at a sequence of times $T_{i}$ with $T_{i}\to T^{*}$.
It then follows immediately that $T^{*}$ must be finite and therefore the
$\limsup_{t\to T^{*}}\|h(\cdot,t)\|_{H^{1}}$ must be infinite.

### Outline of results

The main results of this paper are: short-time existence of nonnegative strong
solutions on $[-a,a]$ given nonnegative initial data, finite speed of
propagation for these solutions if their initial data had compact support
within $(-a,a)$, and finite-time blow-up for solutions of the Cauchy problem
that have initial data with negative energy.

First, we consider equation (1.2) on a bounded interval $[-a,a]$ with periodic
boundary conditions. Given nonnegative initial data that has finite
‘‘entropy’’, in Theorem 1 we prove the short-time existence of a nonnegative
weak solution if $n>0$ and $m\geqslant n/2$. The solution is a ‘‘generalized
weak solution’’ as described in Section 2 and the entropy is introduced in
Section 3. Additional regularity is proven in Theorem 2: there is a second
type of entropy such that if this ‘‘$\alpha$-entropy’’ is also finite for the
initial data then there is a ‘‘strong solution’’ which satisfies Theorem 1 and
also has some additional regularity. We note that the work [12] described
above was primarily concerned with long-time222 Throughout this article, we
use phrases like “long-time”, “global-in-time” and “exist for all time” as
shorthand for the types of large-time results that have been proven in the
thin film literature to date: given a time $T<\infty$ there is a solution
$h(\cdot,t)$ for $t\in[0,T]$. Specifically, $T$ can be taken arbitrarily
large.  existence results: for this reason the authors only addressed the
existence theory for the subcritical ($m<n+2$) case. However, given finite-
entropy initial data their methods easily imply a short-time result for
general $n>0$ as long as $m\geqslant n$. Our advance is prove the results for
the larger range of $m\geqslant n/2$. The left plot in Figure 1 presents the
$(n,m)$ parameter range for which our short-time existence results hold. The
darker region represents the parameters for which the methods of [12] would
have yielded the results. The lighter region represents the extended area
where our methods also yield results.

Figure 1:  Left: the darker region represents the exponents $(n,m)$ for which
the methods of [12] yield short-time existence results like Theorems 1 and 2.
The larger, lighter region represents the additional exponents for which we’ve
proved our theorems. Middle: For $(n,m)$ in the shaded region, we prove that
the strong solutions of Theorem 2 can have finite speed of propagation
(Theorem 3). Right: For $(n,m)$ in the shaded region, we construct solutions
on the line that blow up in finite time (Theorem 4). The heavy solid line
denotes the $(n,m)$ for which this had already been done by [13].

In Theorem 3 we prove that if the initial data $h_{0}$ has compact support
then the strong solution of Theorem 2 will have finite speed of propagation.
Specifically, if the support of the initial data satisfies
$\mbox{supp}(h_{0})\subseteq[-r_{0},r_{0}]\subset(-a,a)$ then there is a
nondecreasing function $\Gamma(t)$ and a time $T_{speed}$ such that
$\mbox{supp}(h(\cdot,t))\subseteq[-r_{0}-\Gamma(t),r_{0}+\Gamma(t)]\subset(-a,a)$
for every time $t\in[0,T_{speed}]$. For $0<n<2$, we further prove that there
is a constant $C$ such that $\Gamma(t)\leqslant Ct^{1/(n+4)}$. This power law
behaviour is the same as has been found for $h_{t}=-(|h|^{n}h_{xxx})_{x}$ for
$0<n<2$ [5] and $2\leqslant n<3$ [29]; it corresponds to the rate of expansion
of the self–similar source–type solution.

The middle plot in Figure 1 presents the $(n,m)$ parameter range for which we
were able to prove finite speed of propagation. While we were successful in
proving finite speed of propagation for the entire expected range of
$m\geqslant n/2$ if $1/2<n<3$, if $0<n\leqslant 1/2$ we could prove it for
only a subset of $m\geqslant n/2$. This technical obstruction is discussed
further in Section 7. The $n$ values are restricted to $n<3$ because for
$n\geq 3$ if initial data has compact support in $(-a,a)$ then it will have
infinite entropy and will not be admissible initial data for Theorems 1 and 2.

To prove Theorem 3 we start by proving a local entropy estimate similar to
that of [5] for $0<n<2$ and a local energy estimate similar to that of [6, 29]
for $1/2<n<3$. Using these and well-chosen ‘‘localization’’ (or ‘‘test’’)
functions, we find systems of functional inequalities. In [24], Giacomelli and
Shishkov proved an extension of Stampacchia’s lemma to systems of
inequalities, we use this to then finish the proof.

A Stampacchia-like lemma for a single inequality was used by [29] to prove
finite speed of propagation for $h_{t}=-(|h|^{n}h_{xxx})_{x}$ and [18] proved
an extension of Stampacchia’s lemma (also for a single inequality) to study
waiting time phenomena. Similar approaches were subsequently used to study
finite speed of propagation and waiting time phenomena for related equations,
see [1, 19, 25, 27, 42, 23, 39, 40, 41]. Further, there are finite speed of
propagation and waiting time results [42, 35] that use the extension of
Stampacchia’s lemma to systems of [24] as well as other types of extensions to
systems [41].

Having proven finite speed of propagation in Theorem 3, we use this to prove a
short-time existence result for the Cauchy problem. Specifically, for the
range of exponents $(n,m)$ of Theorem 3 given nonnegative initial data with
bounded support in $\mathbb{R}^{1}$ we construct a nonnegative, compactly
supported strong solution on $\mathbb{R}^{1}\times[0,T_{0}]$ that satisfies
the bounds and regularity of Theorems 1 and 2 (with those bounds taken over
$\mathbb{R}^{1}$ rather than $\Omega$). The larger the $H^{1}$ norm of the
initial data, the shorter the time $T_{0}$. In Lemma 8.1 we prove that for a
subset of these exponents (see Figure 1), the entropy of the solution
satisfies a second-moment inequality at time $T_{0}$

$e^{-\widetilde{B}(T_{0})}\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h(x,T_{0}))\,dx}\leqslant\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\int\limits_{0}^{T_{0}}{e^{-\widetilde{B}(t)}\left(k_{1}\mathcal{E}_{0}(0)+k_{2}\int\limits_{-\infty}^{\infty}{x^{2}h^{2}_{xx}\,dx}\right)\,dt}$

where $k_{1}=2(4-n)$, $k_{2}=3a_{0}(n-1)/2$, $\mathcal{E}_{0}(0)$ is the
energy of the initial data (1.5),

$\widetilde{G}_{0}(z)=\tfrac{1}{2-n}z^{2-n},\qquad\mbox{and}\qquad\widetilde{B}(t)=\tfrac{a_{1}^{2}|(1-n)(2-n)|}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{t}{\|h(.,\tau)\|_{L^{\infty}(\Omega)}^{2m-n}\,d\tau}.$

Note that if $n=1$ then the second-moment inequality for the entropy reduces
to the second moment inequality (1.4) used by [13].

‘‘Time-stepping’’ this existence result, we construct a nonnegative, compactly
supported, strong solution on $\mathbb{R}^{1}\times[0,T^{*})$ such that our
second-moment inequality for the entropy holds for a sequence of times
$T_{i}\to T^{*}$. The left-hand side of the inequality is nonnegative. If the
initial data has negative energy, then the second term on the right-hand side
has an integrand which has the possibility of becoming negative. In Theorem 4
we prove that for a range of $(n,m)$ values (see Figure 1) that if
$T^{*}=\infty$ then the constructed solution would yield a right-hand side
that becomes negative in finite time, an impossibility. Hence $T^{*}$ must be
finite and therefore the $\limsup_{t\to T^{*}}\|h(\cdot,t)\|_{H^{1}}$ must be
infinite.

Ideally, we would have proven finite speed of propagation for all $(n,m)$ with
$0<n<3$ and all $n/2\leqslant m$ and would have proven the finite time blow up
result for all $(n,m)$ with $0<n<3$ and all $m\geqslant n+2$. We believe the
obstructions are technical ones.

We close by noting that our blow–up result does not give qualitative
information such as proving that there’s a focussing singularity, as suggested
by numerical simulations [12]. However, there has been some detailed study of
the critical regime $m=n+2$. There, a critical mass $M_{c}$ has been
identified [43], there are self-similar, compactly-supported, source-type
solutions with masses in $[0,M_{c})$ [3], and there are self-similar,
compactly supported, solutions with masses in $(M_{c},M_{1})$ that blow up in
finite time [37]. Further, these self-similar blow-up solutions have been
shown to be linearly stable [38].

##  2 Generalized weak solution

We study the existence of a nonnegative weak solution, $h(x,t)$, of the
initial–boundary value problem

$\displaystyle(\textup{P})$ $\displaystyle
h_{t}+\left(f(h)\left(a_{0}h_{xxx}+a_{1}D^{\prime\prime}(h)h_{x}\right)\right)_{x}=0\text{
in }Q_{T},\hfill$ (2.1) $\displaystyle(\textup{P})$
$\displaystyle\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)\text{ for
}t>0,\,i=\overline{0,3},\hfill$ (2.2) $\displaystyle(\textup{P})$
$\displaystyle\qquad\qquad h(0,x)=h_{0}(x)\geqslant 0,\hfill$ (2.3)

where $a_{0}>0$, $a_{1}\geqslant 0$, $f(h)=|h|^{n}$,
$D^{\prime\prime}(h)=|h|^{m-n}$, $n>0$, $m>0$, $\Omega=(-a,a)$, and
$Q_{T}=\Omega\times(0,T)$. We consider a weak solution like that considered in
[4, 5]:

######  Definition 2.1 (generalized weak solution).

Let $n>0$, $m>0$, $a_{0}>0$, and $a_{1}\geqslant 0$. A function $h$ is a
generalized weak solution of the problem $(\textup{P})$ if

$\displaystyle h\in C^{1/2,1/8}_{x,t}(\overline{Q}_{T})\cap
L^{\infty}(0,T;H^{1}(\Omega)),$ (2.4) $\displaystyle h_{t}\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime}),$ (2.5) $\displaystyle h\in
C^{4,1}_{x,t}(\mathcal{P}),\,\,\,\sqrt{f(h)}\,\left(a_{0}h_{xxx}+a_{1}D^{\prime\prime}(h)h_{x}\right)\in
L^{2}(\mathcal{P}),\,\,$ (2.6)

where $\mathcal{P}=\overline{Q}_{T}\setminus(\\{h=0\\}\cup\\{t=0\\})$ and $h$
satisfies (2.1) in the following sense:

$\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt-\iint\limits_{\mathcal{P}}{f(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}(h)h_{x})\phi_{x}\,dxdt}\
=0$ (2.7)

for all $\phi\in C^{1}(Q_{T})$ with $\phi(-a,\cdot)=\phi(a,\cdot)$;

$\displaystyle h(\cdot,t)\to h(\cdot,0)=h_{0}\mbox{ pointwise \& strongly in
$L^{2}(\Omega)$ as $t\to 0$},$ (2.8) $\displaystyle h(-a,t)=h(a,t)\;\forall
t\in[0,T]\;\mbox{and}\;\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)$ (2.9)
$\displaystyle\mbox{for}\;i=\overline{1,3}\;\mbox{at all points of the lateral
boundary where $\\{h\neq 0\\}$.}$

Because the second term of (2.7) has an integral over $\mathcal{P}$ rather
than over $Q_{T}$, the generalized weak solution is ‘‘weaker’’ than a standard
weak solution. Also note that the first term of (2.7) uses $h_{t}\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime})$; this is different from the definition of
weak solution first introduced by Bernis and Friedman [7]; there, the first
term was the integral of $h\phi_{t}$ integrated over $Q_{T}$. We do not
require a test function $\phi$ to be zero at both ends $t=0$ and $t=T$ that is
crucial for our construction of a continuation of the weak solution.

##  3 Main results

Our main results are: the short-time existence of a nonnegative generalized
weak solution (Theorem 1), the short-time existence of a nonnegative strong
solution (Theorem 2), finite-speed of propagation and finite-time blow-up for
some of these strong solutions (Theorems 3 and 4 respectively).

The short-time existence of generalized weak solutions relies on an integral
quantity: $\int G_{0}(h(x,t))\;dx$. This ‘‘entropy’’ was introduced by Bernis
and Friedman [7], where

$G_{0}(z):=\begin{cases}\tfrac{z^{2-n}}{(2-n)(1-n)}&\text{if
}n\in(0,1)\cup(2,\infty)\\\ \tfrac{z^{2-n}}{(2-n)(1-n)}+z+b&\text{if
}n\in(1,2)\\\ z\ln z-z+1&\text{ if }n=1\\\ -\ln z+z/e&\text{ if
}n=2.\end{cases}\Longrightarrow\quad G_{0}^{\prime\prime}(z)=\tfrac{1}{z^{n}}$
(3.1)

with $b$ chosen to ensure that $G_{0}(z)\geqslant 0$ for all $z\geqslant 0$.

######  Theorem 1 (Existence).

Let $n>0,\ m\geqslant\tfrac{n}{2}$, $a_{0}>0$, and $a_{1}\geqslant 0$ in
equation (1.2). Assume that the nonnegative initial data $h_{0}\in
H^{1}(\Omega)$ has finite entropy

$\int\limits_{\Omega}{G_{0}(h_{0}(x))\,dx}<\infty,$ (3.2)

where $G_{0}$ is given by (3.1) and either 1) $h_{0}(-a)=h_{0}(a)=0$ or 2)
$h_{0}(-a)=h_{0}(a)\neq 0$ and $\tfrac{\partial^{i}h_{0}}{\partial
x^{i}}(-a)=\tfrac{\partial^{i}h_{0}}{\partial x^{i}}(a)\text{ holds for
}i=1,2,3$. Then for some time $T_{loc}>0$ there exists a nonnegative
generalized weak solution $h$ on $Q_{T_{loc}}$ in the sense of the definition
2.1. Furthermore,

$h\in L^{2}(0,T_{loc};H^{2}(\Omega)).$ (3.3)

If

$D_{0}(z)=\begin{cases}\tfrac{z^{m-n+2}}{(m-n+1)(m-n+2)}&\mbox{if}\;m-n\notin\\{-2,-1\\}\\\
-\log(z)&\mbox{if}\;m-n=-2\\\
z\log(z)&\mbox{if}\;m-n=-1\end{cases},\quad\Longrightarrow\quad
D_{0}^{\prime\prime}(z)=z^{m-n},$ (3.4)
$\mathcal{E}_{0}(T):=\int\limits_{\Omega}{\\{\tfrac{a_{0}}{2}h_{x}^{2}(x,T)-a_{1}D_{0}(h(x,T))\\}\,dx},\qquad
B_{1}(T):=\tfrac{a_{1}^{2}}{a_{0}}\int\limits_{0}^{T}{\|h(.,t)\|^{2m-n}_{L^{\infty}(\Omega)}\,dt},$
(3.5)

and

$B_{2}(T):=\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{4m-n}\;dt+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{2m-n}\;dt,$

then the weak solution satisfies

$\mathcal{E}_{0}(T_{loc})+\iint\limits_{\\{h>0\\}}{h^{n}(a_{0}h_{xxx}+a_{1}h^{m-n}h_{x})^{2}\,dxdt}\leqslant\mathcal{E}_{0}(0),$
(3.6) $\int\limits_{\Omega}{h_{x}^{2}(x,T_{loc})\,dx}\leqslant
e^{B_{1}(T_{loc})}\int\limits_{\Omega}{h_{0x}^{2}(x)\,dx},$ (3.7)

and

$\int\limits_{\Omega}{\\{h_{x}^{2}(x,T_{loc})+G_{0}(h(x,T_{loc})\\}\,dx}\leqslant
e^{B_{2}(T_{loc})}\int\limits_{\Omega}{\\{h_{0x}^{2}(x)+G_{0}(h_{0}(x))\,dx}.$
(3.8)

The time of existence, $T_{loc}$, is determined by $a_{0}$, $a_{1}$,
$|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, and $\int G_{0}(h_{0})$.

There is nothing special about the time $T_{loc}$ in the bounds (3.6), (3.7),
and (3.8); given a countable collection of times in $[0,T_{loc}]$, one can
construct a weak solution for which these bounds will hold at those times.
Also, we note that the analogue of Theorem 4.2 in [7] also holds: there exists
a nonnegative weak solution with the integral formulation

$\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt+a_{0}\iint\limits_{Q_{T}}(nh^{n-1}h_{x}h_{xx}\phi_{x}+h^{n}h_{xx}\phi_{xx})\;dxdt-
a_{1}\iint\limits_{Q_{T}}{h^{m}h_{x}\phi_{x}\;dxdt}=0.$ (3.9)

We note that the existence theory for the long-wave stable case, $a_{1}<0$,
has already been considered by [10, 20].

Theorem 2 states that if the initial data also has finite ‘‘$\alpha$-entropy’’
then a solution can be constructed which satisfies Theorem 1 and has
additional regularity. This solution is called a ‘‘strong’’ solution; see [11,
4, 10]. The $\alpha$-entropy, defined below, was discovered by Leo Kadanoff
[14]. Let

$G_{0}^{(\alpha)}(z)=\begin{cases}\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}&\mbox{if
}\alpha\in(-\infty,n-2)\cup(n-1,\infty)\\\
\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}+z+c&\mbox{if
}\alpha\in(n-2,n-1)\\\ z\ln z-z+1&\mbox{if }\alpha=n-1\\\ -\ln z+z/e&\mbox{if
}\alpha=n-2,\end{cases}G_{0}^{(\alpha)^{\prime\prime}}(z)=\tfrac{z^{\alpha}}{z^{n}}$
(3.10)

where $c$ is chosen to ensure that $G_{0}^{(\alpha)}(z)\geqslant 0$ for all
$z\geqslant 0$.

######  Theorem 2 (Regularity).

Assume the initial data $h_{0}$ satisfies the hypotheses of Theorem 1 and also
has finite $\alpha$-entropy for some $\alpha\in(-1/2,1)$, $\alpha\neq 0$,

$\int\limits_{\Omega}{G_{0}^{(\alpha)}(h_{0}(x))\,dx}<\infty.$ (3.11)

Then for some time time $T_{loc}^{(\alpha)}\in(0,T_{loc}]$ there exists a
nonnegative generalized weak solution on $Q_{T_{loc}^{(\alpha)}}$ that
satisfies Theorem 1 and has the additional regularity

$h^{\tfrac{\alpha+2}{2}}\in L^{2}(0,T_{loc}^{(\alpha)};H^{2}(\Omega))\text{
and }h^{\tfrac{\alpha+2}{4}}\in
L^{2}(0,T_{loc}^{(\alpha)};W^{1}_{4}(\Omega)).$ (3.12)

######  Theorem 3 (Finite Speed of Propagation).

Consider the range of exponents $0<n\leqslant\tfrac{1}{2}$ and $n/2<m<6-n$ or
$\tfrac{1}{2}<n<3$ and $m\geqslant n/2$ (see Figure 1). Assume the initial
data satisfies the hypotheses of Theorem 2 and also has compact support in
$(-a,a)$: $\mbox{supp}(h_{0})\subseteq[-r_{0},r_{0}]\subset(-a,a)$. Then there
exists a time $0<T_{speed}\leqslant T_{loc}^{(\alpha)}$ and a nondecreasing
function $\Gamma(t)\in C([0,T_{speed}])$ such that the strong solution from
Theorem 2, has finite speed propagation, i.e.

$\mbox{supp}(h(\cdot,t))\subseteq[-r_{0}-\Gamma(t),r_{0}+\Gamma(t)]\subset(-a,a)$

for all $t\in[0,T_{speed}]$. Furthermore, if $0<n<2$ there exists a constant
$C$ which depends on $n$, $m$, $\alpha$, and $\int h_{0}$ such that
$\Gamma(t)\leqslant Ct^{1/(n+4)}$.

######  Theorem 4 (Finite Time Blow Up).

Consider the range of exponents $0<n\leqslant\tfrac{1}{2}$ and $4-n\leqslant
m<6-n$ or $\tfrac{1}{2}<n\leqslant 1$ and $m\geqslant 4-n$ or $1<n<2$ and
$m\geqslant n+2$ (see Figure 1). Assume the nonnegative initial data $h_{0}$
has compact support, negative energy (3.5)

$\mathcal{E}_{0}(0)=\int\limits_{-\infty}^{\infty}{\\{\tfrac{a_{0}}{2}h_{0x}^{2}-a_{1}D_{0}(h_{0})\\}\,dx}<0,$

and satisfies the hypotheses of Theorem 2 with $\Omega=\mathbb{R}^{1}$. Then
there exists a finite time $T^{*}<\infty$ and a nonnegative, compactly
supported, strong solution $h$ on $\mathbb{R}^{1}\times[0,T^{*})$ such that

$\mathop{\limsup}\limits_{t\to
T^{*}}\|h(.,t)\|_{H^{1}(\mathbb{R}^{1})}=\mathop{\limsup}\limits_{t\to
T^{*}}\|h(.,t)\|_{L^{\infty}(\mathbb{R}^{1})}=+\infty.$ (3.13)

The solution satisfies the bounds of Theorems 1 and 2 with
$\Omega=\mathbb{R}^{1}$.

##  4 Proof of Existence of Generalized Solutions

Our proof of existence of generalized weak solutions defined in the section 2
follows the main concept of the proof from [7].

###  4.1 Regularized Problem

Given $\delta,\varepsilon>0$, a regularized parabolic problem, similar to that
of Bernis and Friedman [7], is considered:

$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle
h_{t}+\left({f_{\delta\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})}\right)_{x}=0,\qquad\hfill$
(4.1) $\displaystyle(\textup{P}_{\delta,\varepsilon})$
$\displaystyle\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)\text{ for
}t>0,\,i=\overline{0,3},\hfill$ (4.2)
$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle\qquad\qquad
h(x,0)=h_{0,\varepsilon}(x)\hfill$ (4.3)

where

$f_{\delta\varepsilon}(z):=f_{\varepsilon}(z)+\delta=\tfrac{|z|^{4+n}}{|z|^{4}+\varepsilon|z|^{n}}+\delta,\
D^{\prime\prime}_{\varepsilon}(z):=\tfrac{|z|^{m-n}}{1+\varepsilon|z|^{m-n}},\varepsilon>0,\delta>0.$
(4.4)

We note that $f_{\delta\varepsilon}\in C^{1+\gamma}(\mathbb{R}^{1})$ and
$f_{\delta\varepsilon}D^{\prime\prime}_{\varepsilon}\in
C^{1+\gamma}(\mathbb{R}^{1})$ for some $\gamma\in(0,1)$. The $\delta>0$ in
(4.4) makes the problem (4.1) regular (i.e. uniformly parabolic). The
parameter $\varepsilon$ is an approximating parameter which has the effect of
increasing the degeneracy from $f(h)\sim|h|^{n}$ to $f_{\varepsilon}(h)\sim
h^{4}$. For $\varepsilon>0$, the nonnegative initial data, $h_{0}$, is
approximated via

$\begin{gathered}h_{0}+\varepsilon^{\theta}\leq h_{0,\varepsilon}\in
C^{4+\gamma}(\overline{\Omega})\text{ for some }0<\theta<\tfrac{2}{5},\\\
\tfrac{\partial^{i}h_{0,\varepsilon}}{\partial
x^{i}}(-a)=\tfrac{\partial^{i}h_{0,\varepsilon}}{\partial x^{i}}(a)\text{ for
}i=\overline{0,3},\\\ h_{0,\varepsilon}\to h_{0}\text{ strongly in
}H^{1}(\Omega)\text{ as }\varepsilon\to 0.\end{gathered}$ (4.5)

The effect of $\varepsilon>0$ in (4.5) is to both ‘‘lift’’ the initial data,
making it positive, and to smooth the initial data from $H^{1}(\Omega)$ to
$C^{4+\gamma}(\Omega)$.

By Eĭdelman [22, Theorem 6.3, p.302], the regularized problem has a unique
classical solution $h_{\delta\varepsilon}\in
C_{x,t}^{4+\gamma,1+\gamma/4}(\Omega\times[0,\tau_{\delta\varepsilon}])$ for
some time $\tau_{\delta\varepsilon}>0$. For any fixed value of $\delta$ and
$\varepsilon$, by Eĭdelman [22, Theorem 9.3, p.316] if one can prove a uniform
in time an a priori bound $|h_{\delta\varepsilon}(x,t)|\leqslant
A_{\delta\varepsilon}<\infty$ for some longer time interval
$[0,T_{loc,\delta\varepsilon}]\quad(T_{loc,\delta\varepsilon}>\tau_{\delta\varepsilon}$)
and for all $x\in\Omega$ then Schauder-type interior estimates [22, Corollary
2, p.213] imply that the solution $h_{\delta\varepsilon}$ can be continued in
time to be in
$C_{x,t}^{4+\gamma,1+\gamma/4}(\Omega\times[0,T_{loc,\delta\varepsilon}])$.

Although the solution $h_{\delta\varepsilon}$ is initially positive, there is
no guarantee that it will remain nonnegative. The goal is to take $\delta\to
0$, $\varepsilon\to 0$ in such a way that a) $T_{loc,\delta\varepsilon}\to
T_{loc}>0$, b) the solutions $h_{\delta\varepsilon}$ converge to a
(nonnegative) limit, $h$, which is a generalized weak solution, and c) $h$
inherits certain a priori bounds. This is done by proving various a priori
estimates for $h_{\delta\varepsilon}$ that are uniform in $\delta$ and
$\varepsilon$ and hold on a time interval $[0,T_{loc}]$ that is independent of
$\delta$ and $\varepsilon$. As a result, $\\{h_{\delta\varepsilon}\\}$ will be
a uniformly bounded and equicontinuous (in the $C_{x,t}^{1/2,1/8}$ norm)
family of functions in $\bar{\Omega}\times[0,T_{loc}]$. Taking $\delta\to 0$
will result in a family of functions $\\{h_{\varepsilon}\\}$ that are
classical, positive, unique solutions to the regularized problem with
$\delta=0$. Taking a subsequence of $\varepsilon$ going to zero will then
result in the desired generalized weak solution $h$. This last step is where
the possibility of nonunique weak solutions arise; see [4] for simple examples
of how such a construction applied to $h_{t}=-(|h|^{n}h_{xxx})_{x}$ can result
in there being two different solutions arising from the same initial data.

###  4.2 A priori estimates

We start by proving a priori estimates for classical solutions of the
regularized problem (4.1)–(4.5). Appendix A contains the proofs of the lemmas
in this section.

We introduce function $G_{\delta\varepsilon}$ chosen such that

$G^{\prime\prime}_{\delta\varepsilon}(z)=\tfrac{1}{f_{\delta\varepsilon}(z)}\quad\mbox{and}\quad
G_{\delta\varepsilon}(z)\geqslant 0\quad\forall z\in\mathbb{R}^{1}.$ (4.6)

This is analogous to the ‘‘entropy’’ function (3.1) introduced by Bernis and
Friedman [7].

######  Lemma 4.1.

Let $n>0$ and $m\geqslant n/2$. There exist $\delta_{0}>0$,
$\varepsilon_{0}>0$, and time $T_{loc}>0$ such that if
$\delta\in[0,\delta_{0})$, $\varepsilon\in(0,\varepsilon_{0})$ and
$h_{\delta\varepsilon}$ is a classical solution of the problem (4.1)–(4.5)
with initial data $h_{0,\delta\varepsilon}$ that is built from a nonnegative
function $h_{0}$ satisfying the hypotheses of Theorem 1 then for any
$T\in[0,T_{loc}]$ the following inequalities hold true:

$\displaystyle\int\limits_{\Omega}{\\{h_{\delta\varepsilon,x}^{2}(x,T)+\tfrac{a_{1}^{2}}{2a_{0}^{2}}(1+\delta)G_{\delta\varepsilon}(h_{\delta\varepsilon}(x,T))\\}\,dx}$
(4.7) $\displaystyle\hskip
72.26999pt+a_{0}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h_{\delta\varepsilon})h^{2}_{\delta\varepsilon,xxx}\,dxdt}\leqslant
K_{1}<\infty,$
$\int\limits_{\Omega}G_{\delta\varepsilon}(h_{\delta\varepsilon}(x,T))\;dx+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}h_{\delta\varepsilon,xx}^{2}\;dxdt\leqslant
K_{2}<\infty.$ (4.8)

The energy $\mathcal{E}_{\delta\varepsilon}(t)$ (see (3.5)) satisfies:

$\mathcal{E}_{\delta\varepsilon}(T)+\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})^{2}}\;dxdt=\mathcal{E}_{\delta\varepsilon}(0).$
(4.9)

The time $T_{loc}$ and the constants $K_{1}$, and $K_{2}$ are independent of
$\delta$ and $\varepsilon$.

The existence of $\delta_{0}$, $\varepsilon_{0}$, $T_{loc}$, $K_{1}$, $K_{2}$,
and $C_{0}$ is constructive; how to find them and what quantities determine
them is shown in Section A.

Lemma 4.1 yields uniform-in-$\delta$-and-$\varepsilon$ bounds for $\int
h_{\delta\varepsilon,x}^{2}$, $\int
G_{\delta\varepsilon}(h_{\delta\varepsilon})$, $\iint
f_{\delta\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,xxx}^{2}$,
and $\iint h_{\delta\varepsilon,xx}^{2}$ which will be key in constructing a
nonnegative generalized weak solution. However, these bounds are found in a
different manner than in earlier work for the equation
$h_{t}=-(|h|^{n}h_{xxx})_{x}$, for example. Although the inequality (4.8) is
unchanged, the inequality (4.7) has an extra term involving
$G_{\delta\varepsilon}$. In the proof, this term was introduced to control
additional, lower–order terms. This idea of a ‘‘blended’’
$\|h_{x}\|_{2}$–entropy bound was first introduced by Shishkov and Taranets
especially for long-wave stable thin film equations with convection [34].

######  Lemma 4.2.

Assume $\varepsilon_{0}$ and $T_{loc}$ are from Lemma 4.1, $\delta=0$, and
$\varepsilon\in(0,\varepsilon_{0})$. If $h_{\varepsilon}$ is a positive,
classical solution of the problem (4.1)–(4.5) with initial data
$h_{0,\varepsilon}$ satisfying Lemma 4.1 then

$\int\limits_{\Omega}{h_{\varepsilon,x}^{2}(x,T)\,dx}\leqslant
e^{B_{1,\varepsilon}(T)}\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx},$
(4.10)
$\int\limits_{\Omega}{\\{h_{\varepsilon,x}^{2}(x,T)+G_{\varepsilon}(h_{\varepsilon}(x,T))\\}\,dx}\leqslant
e^{B_{2,\varepsilon}(T)}\int\limits_{\Omega}{\\{h_{0\varepsilon,x}^{2}+G_{\varepsilon}(h_{0,\varepsilon})\\}\,dx}$
(4.11)

hold true for all $T\in[0,T_{loc}]$. Here

$B_{1,\varepsilon}(T):=\tfrac{a_{1}^{2}}{a_{0}}\int\limits_{0}^{T}\|h_{\varepsilon}(\cdot,t)\|_{\infty}^{2m-n}\;dt,$
$B_{2,\varepsilon}(T):=\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\int\limits_{0}^{T}\|h_{\varepsilon}(\cdot,t)\|_{\infty}^{4m-n}\;dt+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}\|h_{\varepsilon}(\cdot,t)\|_{\infty}^{2m-n}\;dt.$

The final a priori bound uses the following functions, parametrized by
$\alpha$, chosen such that $G_{\varepsilon}^{(\alpha)}\geqslant 0$ and
$G_{\varepsilon}^{(\alpha)\prime\prime}(z)=z^{\alpha}/f_{\varepsilon}(z)$:

$G_{\varepsilon}^{(\alpha)}(z)=\begin{cases}\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}&\mbox{if
}\alpha\in(-\infty,n-2)\cup(n-1,\infty)\\\
\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}+z+c&\mbox{if
}\alpha\in(n-2,n-1)\\\ z\ln
z-z+1+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}&\mbox{if
}\alpha=n-1\\\ -\ln
z+z/e+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}&\mbox{if
}\alpha=n-2,\end{cases}$ (4.12)

######  Lemma 4.3.

Assume $\varepsilon_{0}$ and $T_{loc}$ are from Lemma 4.1, $\delta=0$, and
$\varepsilon\in(0,\varepsilon_{0})$. Assume $h_{\varepsilon}$ is a positive,
classical solution of the problem (4.1)–(4.5) with initial data
$h_{0,\varepsilon}$ that is built from a nonnegative function $h_{0}$
satisfying the hypotheses of Theorem 2 then there exists $K_{3}$,
$\varepsilon_{0}^{(\alpha)}$ and $T_{loc}^{(\alpha)}$ with
$0<\varepsilon_{0}^{(\alpha)}\leqslant\varepsilon_{0}$ and
$0<T_{loc}^{(\alpha)}\leqslant T_{loc}$ such that

$\int\limits_{\Omega}{\\{h_{\varepsilon,x}^{2}(x,T)+G_{\varepsilon}^{(\alpha)}(h_{\varepsilon}(x,T))\\}\,dx}+\iint\limits_{Q_{T}}\left[\beta
h_{\varepsilon}^{\alpha}h_{\varepsilon,xx}^{2}+\gamma
h_{\varepsilon}^{\alpha-2}h_{\varepsilon,x}^{4}\right]\;dx\,dt\leqslant
K_{3}<\infty$ (4.13)

holds for all $T\in[0,T_{loc}^{(\alpha)}]$. $K_{3}$ is determined by $\alpha$,
$\varepsilon_{0}$, $a_{0}$, $a_{1}$, $\Omega$ and $h_{0}$. Here,

$\beta=\begin{cases}a_{0}&\mbox{if }0<\alpha<1,\\\
a_{0}\tfrac{1+2\alpha}{4(1-\alpha)}&\mbox{if }-1/2<\alpha<0\end{cases}$

and

$\gamma=\begin{cases}a_{0}\tfrac{\alpha(1-\alpha)}{6}&\mbox{if }0<\alpha<1,\\\
a_{0}\tfrac{(1+2\alpha)(1-\alpha)}{36}&\mbox{if }-1/2<\alpha<0.\end{cases}$

Furthermore,

$h_{\varepsilon}^{\tfrac{\alpha+2}{2}}\in B_{R_{1}}(0)\subset
L^{2}(0,T_{loc};H^{2}(\Omega))\quad\mbox{and}\quad
h_{\varepsilon}^{\tfrac{\alpha+2}{4}}\in B_{R_{2}}(0)\subset\in
L^{2}(0,T_{loc};W^{1}_{4}(\Omega))$ (4.14)

where the radii $R_{1}$ and $R_{2}$ are independent of $\varepsilon$.

The $\alpha$–entropy, $\int G_{0}^{(\alpha)}(h)\;dx$, was first introduced for
$\alpha=-1/2$ in [14] and an a priori bound like that of Lemma 4.3 and
regularity results like those of Theorem 2 were found simultaneously and
independently in [4] and [11].

###  4.3 Proof of existence and regularity of solutions

Bound (4.7) yields uniform $L^{\infty}$ control for classical solutions
$h_{\delta\varepsilon}$, allowing the time of existence
$T_{loc,\delta\varepsilon}$ to be taken as $T_{loc}$ for all
$\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0})$. The
existence theory starts by constructing a classical solution
$h_{\delta\varepsilon}$ on $[0,T_{loc}]$ that satisfy the hypotheses of Lemma
4.1 if $\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0})$. The
a priori bounds of Lemma 4.1 then allow one to take the regularizing
parameter, $\delta$, to zero and prove that there is a limit $h_{\varepsilon}$
and that $h_{\varepsilon}$ is a generalized weak solution. One then proves
additional regularity for $h_{\varepsilon}$; specifically that it is strictly
positive, classical, and unique. It then follows that the a priori bounds
given by Lemmas 4.1, 4.2, and 4.3 apply to $h_{\varepsilon}$. This allows one
to take the approximating parameter, $\varepsilon$, to zero and construct the
desired generalized weak solution of Theorems 1 and 2:

######  Lemma 4.4.

Assume that the initial data $h_{0,\varepsilon}$ satisfies (4.5) and is built
from a nonnegative function $h_{0}$ that satisfies the hypotheses of Theorem
1. Fix $\delta=0$ and $\varepsilon\in(0,\varepsilon_{0})$ where
$\varepsilon_{0}$ is from Lemma 4.1. Then there exists a unique, positive,
classical solution $h_{\varepsilon}$ on $[0,T_{loc}]$ of problem
($\mbox{P}_{0,\varepsilon}$), see (4.1)–(4.5), with initial data
$h_{0,\varepsilon}$ where $T_{loc}$ is the time from Lemma 4.1.

The proof uses a number of arguments like those presented by Bernis & Friedman
[7] and we refer to that article as much as possible.

###### Proof.

Fix $\varepsilon\in(0,\varepsilon_{0})$ and assume $\delta\in(0,\delta_{0})$.
Because $G_{\delta\varepsilon}(z)\geqslant 0$, the bound (4.7) yields a
uniform-in-$\delta$-and-$\varepsilon$ upper bound on
$|h_{\delta\varepsilon}(x,T)|$ for
$(x,T)\in\overline{\Omega}\times[0,T_{loc}]$. As discussed in Subsection 4.1,
this allows the classical solution $h_{\delta\varepsilon}$ to be extended from
$[0,\tau_{\delta\varepsilon}]$ to $[0,T_{loc}]$.

By Section 2 of [7], the a priori bound (4.7) on
$\|h_{x}(\cdot,T_{loc})\|_{2}$ implies that $h_{\delta\varepsilon}\in
C^{1/2,1/8}_{x,t}(\overline{Q_{T_{loc}}})$ and that
$\\{h_{\delta\varepsilon}\\}$ is a uniformly bounded, equicontinuous family in
$\overline{Q_{T_{loc}}}$. By the Arzela-Ascoli theorem, there is a subsequence
$\\{\delta_{k}\\}$, so that $h_{\delta_{k}\varepsilon}$ converges uniformly to
a limit $h_{\varepsilon}\in C^{1/2,1/8}_{x,t}(\overline{Q_{T_{loc}}})$.

We now argue that $h_{\varepsilon}$ is a generalized weak solution, using
methods similar to those of [7, Theorem 3.1].

By construction, $h_{\varepsilon}$ is in
$C^{1/2,1/8}_{x,t}(\overline{Q_{T_{loc}}})$, satisfying the first part of
(2.4). The strong convergence $h_{\varepsilon}(\cdot,t)\to
h_{\varepsilon}(\cdot,0)$ in $L^{2}(\Omega)$ follows immediately. The uniform
convergence of $h_{\delta_{k}\varepsilon}$ to $h_{\varepsilon}$ implies the
pointwise convergence $h(\cdot,t)\to h(\cdot,0)=h_{0}$, and so
$h_{\varepsilon}$ satisfies (2.8).

Because $h_{\delta\varepsilon}$ is a classical solution,

$\iint\limits_{Q_{T}}{h_{\delta\varepsilon,t}\phi\;dxdt}-\iint\limits_{Q_{T_{loc}}}{f_{\delta\varepsilon}(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})\phi_{x}\;dxdt}=0.$
(4.15)

The bound (4.7) yields a uniform bound on

$\delta\iint\limits_{Q_{T_{loc}}}h_{\delta\varepsilon,xxx}^{2}\;dxdt$

for $\delta\in(0,\delta_{0})$. It follows that

$\delta_{k}\iint\limits_{Q_{T_{loc}}}(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})\phi_{x}\,dxdt\to
0\qquad\mbox{as $\delta_{k}\to 0$.}$

Introducing the notation

$H_{\delta\varepsilon}:=f_{\delta\varepsilon}(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})$
(4.16)

the integral formulation (4.15) can be written as

$\iint\limits_{Q_{T}}{h_{\delta\varepsilon,t}\phi
dxdt}=\iint\limits_{Q_{T_{loc}}}H_{\delta\varepsilon}(x,t)\phi_{x}(x,t)\;dxdt.$
(4.17)

By the $L^{\infty}$ control of $h_{\delta\varepsilon}$ and the energy bound
(4.9), $H_{\delta\varepsilon}$ is uniformly bounded in $L^{2}(Q_{T_{loc}})$.
Taking a further subsequence of $\\{\delta_{k}\\}$ yields
$H_{\delta_{k}\varepsilon}$ converging weakly to a function $H_{\varepsilon}$
in $L^{2}(Q_{T_{loc}})$. The regularity theory for uniformly parabolic
equations implies that $h_{\delta\varepsilon,t}$, $h_{\delta\varepsilon,x}$,
$h_{\delta\varepsilon,xx}$, $h_{\delta\varepsilon,xxx}$, and
$h_{\delta\varepsilon,xxxx}$ converge uniformly to $h_{\varepsilon,t}$, …,
$h_{\varepsilon,xxxx}$ on any compact subset of $\\{h_{\varepsilon}>0\\}$,
implying (2.9) and the first part of (2.6). Note that because the initial data
$h_{0,\delta\varepsilon}$ is in $C^{4}$ the regularity extends all the way to
$t=0$ which is excluded in the definition of $\mathcal{P}$ in (2.6).

The energy $\mathcal{E}_{\delta\varepsilon}(T_{loc})$ is not necessarily
positive. However, the a priori bound (4.7), combined with the $L^{\infty}$
control on $h_{\delta\varepsilon}$, ensures that
$\mathcal{E}_{\delta\varepsilon}(T_{loc})$ has a uniform lower bound. As a
result, the bound (4.9) yields a uniform bound on

$\iint\limits_{Q_{T_{loc}}}f_{\delta\varepsilon}\left(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x}\right)^{2}\;dxdt.$

Using this, one can argue that for any $\sigma>0$

$\iint\limits_{\\{h_{\varepsilon}<\sigma\\}}\left|H_{\delta_{k}\varepsilon}\phi_{x}\right|\;dxdt\leqslant
C\sigma^{n/2}$

for some $C$ independent of $\delta$, $\varepsilon$, and $\sigma$. Taking
$\delta_{k}\to 0$ and using that $\sigma$ is arbitrary, we conclude

$H_{\delta_{k}\varepsilon}\to
H_{\varepsilon}=f_{\varepsilon}\left(h_{\varepsilon})(a_{0}h_{\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\varepsilon,x}\right)\,\chi_{\\{h_{\varepsilon}>0\\}}.$

As a result, taking $\delta_{k}\to 0$ in (4.17) implies $h_{\varepsilon}$
satisfies (2.7) and the second part of (2.6).

The bound (4.7) yields a uniform bound on $\int
h_{\delta\varepsilon,x}^{2}(x,T)\;dx$ for every $T\in[0,T_{loc}]$. As a
result, $\\{h_{\delta_{k}\varepsilon}\\}$ is uniformly bounded in

$L^{\infty}(0,T_{loc};H^{1}(\Omega)).$

Therefore, another refinement of the sequence $\\{\delta_{k}\\}$ yields
$\\{h_{\delta_{k}\varepsilon}\\}$ weakly convergent in this space. As a
result, $h_{\varepsilon}\in L^{\infty}(0,T_{loc};H^{1}(\Omega))$ and the
second part of (2.4) holds.

Having proven then $h_{\varepsilon}$ is a generalized weak solution, we now
prove that $h_{\varepsilon}$ is a strictly positive, classical, unique
solution. This uses the entropy $\int
G_{\delta\varepsilon}(h_{\delta\varepsilon})$ and the a priori bound (4.8).
This bound is, up to the coefficient $a_{0}$, identical to the a priori bound
(4.17) in [7]. By construction, the initial data $h_{0,\varepsilon}$ is
positive (see (4.5)), hence $\int
G_{\varepsilon}(h_{0,\varepsilon})\;dx<\infty$. Also, by construction
$f_{\varepsilon}(z)\sim z^{4}$ for $z\ll 1$. This implies that the generalized
weak solution $h_{\varepsilon}$ is strictly positive [7, Theorem 4.1]. Because
the initial data $h_{0,\varepsilon}$ is in $C^{4}(\bar{\Omega})$, it follows
that $h_{\varepsilon}$ is a classical solution in
$C^{4,1}_{x,t}(\overline{Q_{T_{loc}}})$. This implies that
$h_{\varepsilon}(\cdot,t)\to h_{\varepsilon}(\cdot,0)$ strongly333 Unlike the
definition of weak solution given in [7], Definition 2.1 does not include that
the solution converges to the initial data strongly in $H^{1}(\Omega)$.  in
$C^{1}(\bar{\Omega})$. The proof of Theorem 4.1 in [7] then implies that
$h_{\varepsilon}$ is unique. ∎

###### Proof of Theorem 1.

As in the proof of Lemma 4.4, following [7], there is a subsequence
$\\{\varepsilon_{k}\\}$ such that $h_{\varepsilon_{k}}$ converges uniformly to
a function $h\in C^{1/2,1/8}_{x,t}$ which is a generalized weak solution in
the sense of Definition 2.1 with $f(h)=|h|^{n}$.

The initial data is assumed to have finite entropy: $\int G_{0}(h_{0})<\infty$
where $G_{0}$ is given by (3.1). This, combined with $f(h)=|h|^{n}$, implies
that the generalized weak solution $h$ is nonnegative and, if $n\in[2,4)$ the
set of points $\\{h=0\\}$ in $Q_{T_{loc}}$ has zero measure and $h$ is
positive, smooth, and unique if $n\geqslant 4$ [7, Theorem 4.1].

To prove (3.6), start by taking $T=T_{loc}$ in the a priori bound (4.9). As
$\varepsilon_{k}\to 0$, the right-hand side of (4.9) is unchanged. Now,
consider the $\varepsilon_{k}\to 0$ limit of

$\mathcal{E}_{\varepsilon_{k}}(T_{loc})=\int\limits_{\Omega}{(\tfrac{a_{0}}{2}h_{\varepsilon_{k},x}^{2}(x,T_{loc})-a_{1}D_{\varepsilon_{k}}(h_{\varepsilon_{k}}(x,T_{loc})))\,dx},$

where $D_{\varepsilon}$ is defined in (4.4). By the uniform convergence of
$h_{\varepsilon_{k}}$ to $h$, the second term in the energy converges strongly
as $\varepsilon_{k}\to 0$. Hence the bound (4.9) yields a uniform bound on
$\\{\int_{\Omega}h_{\varepsilon_{k},x}^{2}(x,T_{loc})\;dx\\}$. Taking a
further refinement of $\\{\varepsilon_{k}\\}$, yields
$h_{\varepsilon_{k},x}(\cdot,T_{loc})$ converging weakly in $L^{2}(\Omega)$.
In a Hilbert space, the norm of the weak limit is less than or equal to the
$\liminf$ of the norms of the functions in the sequence, hence
$\mathcal{E}_{0}(T_{loc})\leqslant\liminf_{\varepsilon_{k}\to
0}\mathcal{E}_{\varepsilon_{k}}(T_{loc}).$ A uniform bound on $\iint
f_{\varepsilon}(h_{\varepsilon})\left(a_{0}h_{\varepsilon,xxx}+\dots\right)^{2}\;dx$
also follows from (4.9). Hence
$\sqrt{f_{\varepsilon_{k}}(h_{\varepsilon_{k}})}\left(a_{0}h_{\varepsilon_{k},xxx}+\dots\right)$
converges weakly in $L^{2}(Q_{T_{loc}})$, after taking a further subsequence.
We write the weak limit as two integrals: one over $\\{h=0\\}$ and one over
$\\{h>0\\}$. We can determine the weak limit on $\\{h>0\\}$: as in the proof
of Lemma 4.4, the regularity theory for uniformly parabolic equations allows
one to argue that the weak limit is $h^{n/2}\left(a_{0}h_{xxx}+\dots\right)$
on $\\{h>0\\}$. Using that 1) the norm of the weak limit is less than or equal
to the $\liminf$ of the norms of the functions in the sequence and that 2) the
$\liminf$ of a sum is greater than or equal to the sum of the $\liminf$s and
dropping the nonnegative term arising from the integral over $\\{h=0\\}$
results in the desired bound (3.6).

It follows from (4.8) that $h_{\varepsilon_{k},xx}$ converges weakly to some
$v$ in $L^{2}(Q_{T_{loc}})$, combining with strong convergence in
$L^{2}(0,T;H^{1}(\Omega))$ of $h_{\varepsilon_{k}}$ to $h$ by Lemma D.1 and
with the definition of weak derivative, we obtain that $v=h_{xx}$ and $h\in
L^{2}(0,T_{loc};H^{2}(\Omega))$ that implies (3.3). Hence
$h_{\varepsilon,t}\to h_{t}\text{ weakly in
}L^{2}(0,T;(H^{1}(\Omega))^{\prime})$ that implies (2.5). By Lemma D.2 we also
have $h\in C([0,T_{loc}],L^{2}(\Omega))$. ∎

###### Proof of Theorem 2.

Fix $\alpha\in(-1/2,1)$. The initial data $h_{0}$ is assumed to have finite
entropy $\int G_{0}^{(\alpha)}(h_{0}(x))\;dx<\infty$, hence Lemma 4.3 holds
for the approximate solutions $\\{h_{\varepsilon_{k}}\\}$ where this sequence
of approximate solutions is assumed to be the one at the end of the proof of
Theorem 1. By (4.14),

$\left\\{h_{\varepsilon_{k}}^{\tfrac{\alpha+2}{2}}\right\\}\quad\mbox{is
uniformly bounded in $\varepsilon_{k}$ in $L^{2}(0,T_{loc};H^{2}(\Omega))$}$

and

$\left\\{h_{\varepsilon_{k}}^{\tfrac{\alpha+2}{4}}\right\\}\quad\mbox{is
uniformly bounded in $\varepsilon_{k}$ in
$L^{2}(0,T_{loc};W^{1}_{4}(\Omega))$}.$

Taking a further subsequence in $\\{\varepsilon_{k}\\}$, it follows from the
proof of [17, Lemma 2.5, p.330] that these sequences converge weakly in
$L^{2}(0,T_{loc};H^{2}(\Omega))$ and $L^{2}(0,T_{loc};W^{1}_{4}(\Omega))$, to
$h^{\tfrac{\alpha+2}{2}}$ and $h^{\tfrac{\alpha+2}{4}}$ respectively. ∎

##  5 The subcritical regime: long–time existence of solutions

######  Lemma 5.1.

Let $h\in H^{1}(\Omega)$ be a nonnegative function and let
$M=\int\limits_{\Omega}{h(x)\,dx}$. Then

$\|h\|_{L^{p}(\Omega)}^{p}\leqslant
k_{1}M^{\tfrac{p+2}{3}}\biggl{(}\int\limits_{\Omega}{h^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}+k_{2}M^{p},\
\ p\geqslant 1,$ (5.1)

where $k_{1}=2^{\tfrac{4-p}{3}}3^{\tfrac{2(p-1)}{3}}(1-\epsilon)^{-1}$,
$k_{2}=|\Omega|^{1-p}(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{1-p}$, and
$\epsilon\in(0,1)$.

Note that by taking $h$ to be a constant function, one finds that the constant
$k_{2}M^{p}$ in (5.1) is sharp when $\epsilon\to 1$.

###### Proof.

Let $v=h-M/|\Omega|$. By (A.3),

$\|v\|_{L^{p}(\Omega)}^{p}\leqslant(\tfrac{3}{2})^{\tfrac{2(p-1)}{3}}\biggl{(}\int\limits_{\Omega}{v^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}\biggl{(}\int\limits_{\Omega}{|v|\,dx}\biggr{)}^{\tfrac{p+2}{3}}.$

Hence, due to the inequality

$|a-b|^{p}\geqslant(1-\epsilon)a^{p}-c_{0}(\epsilon,p)b^{p}$

for any $a\geqslant 0,\ b\geqslant 0,\ p>1,\ \epsilon\in(0,1)$,
$c_{0}(\epsilon,p)\geqslant\tfrac{1-\epsilon}{(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{p-1}}$,

$\|h\|_{L^{p}(\Omega)}^{p}\leqslant\tfrac{1}{1-\epsilon}(\tfrac{3}{2})^{\tfrac{2(p-1)}{3}}\biggl{(}\int\limits_{\Omega}{h^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}\biggl{(}\int\limits_{\Omega}{\left|h-\tfrac{M}{|\Omega|}\right|\,dx}\biggr{)}^{\tfrac{p+2}{3}}+\\\
\tfrac{1}{(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{p-1}}\tfrac{M^{p}}{|\Omega|^{p-1}}\leqslant\tfrac{1}{1-\epsilon}(\tfrac{3}{2})^{\tfrac{2(p-1)}{3}}\biggl{(}\int\limits_{\Omega}{h^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}(2M)^{\tfrac{p+2}{3}}+\\\
\tfrac{1}{(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{p-1}}\tfrac{M^{p}}{|\Omega|^{p-1}}.$

∎

######  Lemma 5.2.

Let $0<n/2\leqslant m<n+2$. Let $h$ be the generalized solution of Theorem 1.
Then

$\tfrac{a_{0}}{4}\|h(.,T_{loc})\|^{2}_{H^{1}(\Omega)}\leqslant\mathcal{E}_{0}(0)+c_{1}M^{\tfrac{m-n+4}{2-m+n}}+c_{2}M^{m-n+2}+c_{3}M^{2},$
(5.2)

where $\mathcal{E}_{0}(0)$ is defined in (3.5), and $M=\int h_{0}$. Moreover,
if $m=n+2$ and $0<M<M_{c}$ then

$\tfrac{a_{0}}{4}\|h(.,T_{loc})\|^{2}_{H^{1}(\Omega)}\leqslant\tfrac{2}{3(1-c_{4}M^{2})}\mathcal{E}_{0}(0)+\tfrac{c_{5}}{1-c_{4}M^{2}}M^{4}+c_{6}M^{2}.$
(5.3)

###### Proof of Lemma 5.2.

We present the proof for the $m-n\neq-1,-2$ case in (3.4), leaving the
$m-n=-1,-2$ cases to the reader. The first step is to find a priori bound
(5.2) that is the analogue of Proposition 2.2 in [12]. From (3.6) we deduce

$\tfrac{a_{0}}{2}\int\limits_{\Omega}{h_{x}^{2}\,dx}\leqslant\mathcal{E}_{0}(0)+\tfrac{a_{1}}{(m-n+1)(m-n+2)}\int\limits_{\Omega}{h^{m-n+2}\,dx}.$
(5.4)

Due to (5.1), we have

$\int\limits_{\Omega}{h^{m-n+2}\,dx}\leqslant
k_{1}M^{\tfrac{m-n+4}{3}}\Bigl{(}\int\limits_{\Omega}{h_{x}^{2}\,dx}\Bigr{)}^{\tfrac{m-n+1}{3}}+k_{2}M^{m-n+2}.$
(5.5)

Thus, from (5.4), in view of (5.5), we find that

$\tfrac{a_{0}}{2}\int\limits_{\Omega}{h_{x}^{2}\,dx}-\tfrac{a_{1}k_{1}M^{\tfrac{m-n+4}{3}}}{(m-n+1)(m-n+2)}\Bigl{(}\int\limits_{\Omega}{h_{x}^{2}\,dx}\Bigr{)}^{\tfrac{m-n+1}{3}}\leqslant\mathcal{E}_{0}(0)+\tfrac{a_{1}k_{2}M^{m-n+2}}{(m-n+1)(m-n+2)}.$
(5.6)

Moreover, due to (5.1), in view of the Young inequality (A.6), we have

$\int\limits_{\Omega}{h^{2}\,dx}\leqslant
6^{2/3}M^{\tfrac{4}{3}}\Bigl{(}\int\limits_{\Omega}{h_{x}^{2}\,dx}\Bigr{)}^{\tfrac{1}{3}}+\tfrac{M^{2}}{|\Omega|}\leqslant\tfrac{1}{4}\int\limits_{\Omega}{h_{x}^{2}\,dx}+(\tfrac{8\sqrt{3}}{3}+|\Omega|^{-1})M^{2}.$
(5.7)

In the subcritical ($m<n+2$) case, $\tfrac{m-n+1}{3}<1$ and using (5.6) and
(5.7) we deduce (5.2) with

$\begin{gathered}c_{1}=\bigl{(}\tfrac{a_{1}k_{1}}{(m-n+1)(m-n+2)}\bigr{)}^{\tfrac{3}{2-m+n}}\bigl{(}\tfrac{8(m-n+1)}{3a_{0}}\bigr{)}^{\tfrac{m-n+1}{2-m+n}}\tfrac{2-m+n}{3},\\\
c_{2}=\tfrac{a_{1}k_{2}}{(m-n+1)(m-n+2)},\
c_{3}=\tfrac{a_{0}}{2}(\tfrac{8\sqrt{3}}{3}+|\Omega|^{-1}).\end{gathered}$
(5.8)

In the critical ($m=n+2$) case, $\tfrac{m-n+1}{3}=1$. If
$M<M_{c}=\bigl{(}\tfrac{6a_{0}}{a_{1}k_{1}}\bigr{)}^{\tfrac{1}{2}}$ then using
(5.6) we arrive at

$\int\limits_{\Omega}{h_{x}^{2}\,dx}\leqslant\tfrac{12}{6a_{0}-a_{1}k_{1}M^{2}}\bigl{(}\mathcal{E}_{0}(0)+\tfrac{a_{1}k_{2}}{12}M^{4}\bigr{)}$
(5.9)

Using (5.7), from (5.9) we obtain (5.3) with

$c_{4}=\tfrac{a_{1}k_{1}}{6a_{0}},c_{5}=\tfrac{a_{1}k_{2}}{18},\
c_{6}=\tfrac{a_{0}}{3}(\tfrac{8\sqrt{3}}{3}+|\Omega|^{-1}).$ (5.10)

∎

Under certain conditions, a bound closely related to (5.2) implies that if the
solution of Theorem 1 is initially constant then it will remain constant for
all time:

######  Theorem 5.

Assume $m=n$, the coefficient $a_{1}\geqslant 0$ in (2.1), and
$|\Omega|^{2}<a_{0}/|a_{1}|$. If the initial data is constant, $h_{0}\equiv
C>0$, then the solution of Theorem 1 satisfies $h(x,t)=C$ for all
$x\in\bar{\Omega}$ and all $t>0$.

For the long–wave unstable case ($a_{1}>0$) the hypotheses correspond to the
domain not being ‘‘too large’’. We note that it is not yet known whether or
not solutions from Theorem 1 are unique and so Theorem 5 does have content: it
ensures that the approximation method isn’t producing unexpected (nonconstant)
solutions from constant initial data.

###### Proof.

Consider the approximate solution $h_{\varepsilon}$. The definition of
$\mathcal{E}_{\varepsilon}(T)$ combined with the uniform-in-time bound (4.9)
implies

$\tfrac{a_{0}}{2}\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\leqslant\mathcal{E}_{\varepsilon}(0)+\tfrac{|a_{1}|}{2}\int\limits_{\Omega}h_{\varepsilon}^{2}(x,T)\;dx.$
(5.11)

Letting $M_{\varepsilon}=\int h_{0,\varepsilon}\,dx$ and applying Poincaré’s
inequality (A.2) to $v_{\varepsilon}=h_{\varepsilon}-M_{\varepsilon}/|\Omega|$
and using $\int h_{\varepsilon}^{2}\,dx=\int
v_{\varepsilon}^{2}\,dx+M_{\varepsilon}^{2}/|\Omega|$ yields

$\left(\tfrac{a_{0}}{2}-\tfrac{|a_{1}|\,|\Omega|^{2}}{2}\right)\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,t)\;dx\leq\mathcal{E}_{\varepsilon}(0)+\tfrac{|a_{1}|M_{\varepsilon}^{2}}{2|\Omega|}.$

If $h_{0,\varepsilon}\equiv C_{\varepsilon}=C+\varepsilon^{\theta}$ this
becomes

$\left(\tfrac{a_{0}}{2}-\tfrac{|a_{1}||\Omega|^{2}}{2}\right)\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\leqslant(|a_{1}|-a_{1})\tfrac{C_{\varepsilon}^{2}|\Omega|}{2}.$

If $a_{1}\geqslant 0$ and $|\Omega|^{2}<a_{0}/|a_{1}|$ then $\int
h_{\varepsilon,x}^{2}(x,T)\;dx=0$ for all $T\in[0,T_{\varepsilon,loc}]$ and
that this, combined with the continuity in space and time of
$h_{\varepsilon}$, implies that $h_{\varepsilon}\equiv C_{\varepsilon}$ on
$Q_{T_{\varepsilon,loc}}$.

Taking the sequence $\\{\varepsilon_{k}\\}$ that yields convergence to the
solution $h$ of Theorem 1, $h\equiv C$ on $Q_{T_{loc}}$. ∎

This $H^{1}$ control in time of the generalized solution given by Lemma 5.2 is
now used to extend the short–time existence result of Theorem 1 to a long–time
existence result:

######  Theorem 6.

Let $0<n/2\leqslant m<n+2$. Let $T_{g}$ be an arbitrary positive finite
number. The generalized weak solution $h$ of Theorem 1 can be continued in
time from $[0,T_{loc}]$ to $[0,T_{g}]$ in such a way that $h$ is also a
generalized weak solution and satisfies all the bounds of Theorem 1 (with
$T_{loc}$ replaced by $T_{g}$).

Similarly, the short–time existence of strong solutions (see Theorem 2) can be
extended to long–time existence.

###### Proof.

To construct a weak solution up to time $T_{g}$, one applies the local
existence theory iteratively, taking the solution at the final time of the
current time interval as initial data for the next time interval.

Introduce the times

$0=T_{0}<T_{1}<T_{2}<\dots<T_{N}<\dots\quad\mbox{where}\quad
T_{N}:=\sum_{n=0}^{N-1}T_{n,loc}$ (5.12)

and $T_{n,loc}$ is the interval of existence (A.21) for a solution with
initial data $h(\cdot,T_{n})$:

$T_{n,loc}:=\tfrac{9}{20c_{11}(\gamma_{1}-1)}\min\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,T_{n})+\tfrac{2c_{2}}{a_{0}}G_{0}(h(x,T_{n}))\;dx\right)^{-(\gamma_{1}-1)}\right\\}$
(5.13)

where $\gamma_{1}=\max\\{3,2m-n\\}$ and $c_{2}$ and $c_{11}$ are given in the
proof of Lemma 4.1.

The proof proceeds by contradiction. Assume there exists initial data $h_{0}$,
satisfying the hypotheses of Theorem 1, that results in a weak solution that
cannot be extended arbitrarily in time:

$\sum_{k=0}^{\infty}T_{n,loc}=T^{*}<\infty\quad\Longrightarrow\quad\lim_{n\to\infty}T_{n,loc}=0.$

From the definition (5.13) of $T_{n,loc}$, this implies

$\lim_{n\to\infty}\int\limits_{\Omega}(h_{x}^{2}(x,T_{n})+\tfrac{2c_{2}}{a_{0}}G_{0}(h(x,T_{n})))\;dx=\infty.$
(5.14)

By (5.2),

$\tfrac{a_{0}}{4}\int\limits_{\Omega}h_{x}^{2}(x,T_{n})\;dx\leq\mathcal{E}_{0}(T_{n-1})+K,$

where $K=c_{1}M^{\tfrac{m-n+4}{2-m+n}}+c_{2}M^{m-n+2}+c_{3}M^{2}$. By (3.6),

$\mathcal{E}_{0}(T_{n-1})\leqslant\mathcal{E}_{0}(T_{n-2})\leqslant\dots\mathcal{E}_{0}(0).$

Combining these,

$\tfrac{a_{0}}{4}\int\limits_{\Omega}h_{x}^{2}(x,T_{n})\;dx\leq\mathcal{E}_{0}(0)+K.$
(5.15)

By assumption, $T_{n}\to T^{*}<\infty$ as $n\to\infty$ hence $\int
h_{x}^{2}(x,T_{n})$ remains bounded. Assumption (5.14) then implies that $\int
G_{0}(h(x,T_{n}))\to\infty$ as $n\to\infty$.

To continue the argument, we step back to the approximate solutions
$h_{\varepsilon}$. Let $T_{n,\varepsilon}$ be the analogue of $T_{n}$ and
$T_{n,loc,\varepsilon}$, defined by (A.20), be the analogue of $T_{n,loc}$. By
(A.16),

$\displaystyle\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n,\varepsilon}))\;dx\leqslant\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n-1,\varepsilon}))\;dx$
(5.16) $\displaystyle\hskip
144.54pt+c_{10}\int\limits_{T_{n-1,\varepsilon}}^{T_{n,\varepsilon}}\max\left\\{1,\Bigl{(}\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\Bigr{)}^{\gamma_{2}}\right\\}\;dT$

with $\gamma_{2}=3$. Using the bound (4.9), one can prove the analogue of
Lemma 5.2 for the approximate solution $h_{\varepsilon}$. However the bound
(5.2) would be replaced by a bound on $\|h_{\varepsilon}(\cdot,T)\|_{H^{1}}$
which holds for all $T\in[0,T_{\varepsilon,loc}]$. This bound would then be
used to prove a bound like (5.15) to prove boundedness of $\int
h_{\varepsilon,x}^{2}(x,T)$ for all $T\in[0,T_{n,\varepsilon}]$. Using this
bound,

$\displaystyle\int\limits_{T_{n-1,\varepsilon}}^{T_{n,\varepsilon}}\Bigl{(}\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\Bigr{)}^{\gamma_{2}}dT\leq\bigl{(}\tfrac{4}{a_{0}}\bigr{)}^{\gamma_{2}}\int\limits_{T_{n-1,\varepsilon}}^{T_{n,\varepsilon}}\left(\mathcal{E}_{\varepsilon}(0)+K\right)^{\gamma_{2}}\,dT$
$\displaystyle\hskip
72.26999pt=\bigl{(}\tfrac{4}{a_{0}}\bigr{)}^{\gamma_{2}}\left(\mathcal{E}_{\varepsilon}(0)+K\right)^{\gamma_{2}}\,T_{n-1,loc,\varepsilon}.$
(5.17)

If the initial data is such that $4/a_{0}\;(\mathcal{E}_{\varepsilon}(0)+K)<1$
then before using (5.17) in (5.16) we replace $K$ by a larger value so that
$4/a_{0}\;(\mathcal{E}_{\varepsilon}(0)+K)>1$. Using (5.17) in (5.16), it
follows that

$\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n,\varepsilon}))\;dx\leqslant\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n-1,\varepsilon}))\;dx+\beta\,T_{n-1,loc,\varepsilon}$
(5.18)

for
$\beta=c_{10}\bigl{(}\tfrac{4}{a_{0}}\bigr{)}^{\gamma_{2}}\left(\mathcal{E}_{\varepsilon}(0)+K\right)^{\gamma_{2}}$.
Here $\beta$ depends on $|\Omega|$, the coefficients of the PDE, and on the
initial data $h_{0,\varepsilon}$.

One now takes the sequence $\\{\varepsilon_{k}\\}$ that was used to construct
the weak solution of Theorem 1 on the interval $[T_{n-1},T_{n}]$. Taking
$\varepsilon_{k}\to 0$, (5.18) yields

$\int\limits_{\Omega}G_{0}(h(x,T_{n}))\;dx\leq\int\limits_{\Omega}G_{0}(h(x,T_{n-1}))\,dx+\beta\,T_{n-1,loc}.$
(5.19)

Applying (5.19) iteratively,

$\displaystyle\int\limits_{\Omega}G_{0}(h(x,T_{n}))\;dx\leqslant\int\limits_{\Omega}G_{0}(h_{0}(x))\;dx+\beta\sum_{k=0}^{n-1}T_{k,loc}$
$\displaystyle\hskip
72.26999pt=\int\limits_{\Omega}G_{0}(h_{0}(x))\;dx+\beta\,T_{n}.$

This upper bound proves that $\int G_{0}(h(x,T_{n}))$ cannot diverge to
infinity as $n\to\infty$, finishing the proof. ∎

##  6 Strong positivity

######  Proposition 6.1.

Let $m\geqslant n/2$. Assume the initial data $h_{0}$ satisfies $h_{0}(x)>0$
for all $x\in\omega\subseteq\Omega$ where $\omega$ is an open interval. Then

  1. 1.

if $n>3/2$ and $\alpha\in(-1/2,\min\\{1,n-2\\})$ then the strong solution from
Theorem 2 satisfies $h(x,T)>0$ for almost every $x\in\omega$, for all
$T\in[0,T_{loc}^{(\alpha)}]$;

  2. 2.

if $n>2$ and $\alpha\in(-1/2,\min\\{1,3n/4-2\\})$ then the strong solution
from Theorem 2 satisfies $h(x,T)>0$ for all $x\in\omega$, for almost every
$T\in[0,T_{loc}^{(\alpha)}]$;

  3. 3.

if $n\geqslant 7/2$ and $m\geqslant n-1/2$ then the strong solution from
Theorem 2 satisfies $h(x,T)>0$ for all
$(x,T)\in\overline{Q}_{T_{loc}^{(\alpha)}}$.

The proof of Proposition 6.1 depends on a local version of the a priori bound
(4.13) of Lemma 4.3:

######  Lemma 6.1.

Let $\omega\subseteq\Omega$ be an open interval and $\zeta\in
C^{2}(\bar{\Omega})$ such that $\zeta>0$ on $\omega$,
$\text{supp}\,\zeta=\overline{\omega}$, and $(\zeta^{4})^{\prime}=0$ on
$\partial\Omega$. If $\omega=\Omega$, choose $\zeta$ such that
$\zeta(-a)=\zeta(a)>0$. Let $\xi:=\zeta^{4}$.

If the initial data $h_{0}$ and the time $T_{loc}^{(\alpha)}$ are as in
Theorem 2 then for all $T\in[0,T_{loc}^{(\alpha)}]$ the strong solution $h$
from Theorem 2 satisfies

$\int\limits_{\Omega}{\xi(x)\;G_{0}^{(\alpha)}(h(x,T))}\;dx<\infty$ (6.1)

The proof of Lemma 6.1 is given in Appendix A. The proof of Proposition 6.1 is
essentially a combination of the proofs of Theorem 6.1 and Corollary 4.5 in
[7] and is provided here for the reader’s convenience.

###### Proof of Proposition 6.1.

Choose the test function $\zeta(x)$ to satisfy the hypotheses of Lemma 6.1.
Hence, (6.1) holds for every $T\in[0,T_{loc}^{(\alpha)}]$.

Proof of 1): Assume it is not true that $h(x,T)>0$ for almost every
$x\in\omega$, for all $T\in[0,T_{loc}^{(\alpha)}]$. Then there is a time
$T\in[0,T_{loc}^{(\alpha)}]$ such that the set
$\\{x\;:\;h(x,T)=0\\}\cap\omega$ has positive measure. Then because
$\alpha-n+2<0$,

$\infty>\int\limits_{\Omega}\xi(x)h^{\alpha-n+2}(x,T)\;dx\geqslant\int\limits_{\\{h(\cdot,T)=0\\}\cap\omega}\xi(x)h^{\alpha-n+2}(x,T)\;dx=\infty.$

This contradiction implies there can be no time at which $h$ vanishes on a set
of positive measure in $\omega$, as desired.

Proof of 2): We start by noting that by (3.12),
$(h^{\tfrac{\alpha+2}{2}})_{xx}(\cdot,T)\in L^{2}(\Omega)$ for almost all
$T\in[0,T_{loc}^{(\alpha)}]$ hence $h^{\tfrac{\alpha+2}{2}}(\cdot,T)\in
C^{3/2}(\Omega)$ for almost all $T\in[0,T_{loc}^{(\alpha)}]$. To prove 2), it
therefore suffices to show that if $T$ such that
$h^{\tfrac{\alpha+2}{2}}(\cdot,T)\in C^{3/2}(\Omega)$ then $h(x,T)>0$ on
$\omega$. Assume this is not true and there is an $x_{0}\in\omega$ and $T_{0}$
such that $h(x_{0},T_{0})=0$ and $h^{\tfrac{\alpha+2}{2}}(\cdot,T_{0})\in
C^{3/2}(\Omega)$. Then there is a $L$ such that

$h^{\tfrac{\alpha+2}{2}}(x,T_{0})=|h^{\tfrac{\alpha+2}{2}}(x,T_{0})-h^{\tfrac{\alpha+2}{2}}(x_{0},T_{0})|\leqslant
L|x-x_{0}|^{3/2}.$

Hence

$\infty>\int\limits_{\Omega}{\xi(x)h^{\alpha-n+2}(x,T_{0})}\;dx\geqslant
L^{\tfrac{2(\alpha-n+2)}{\alpha+2}}\int\limits_{\Omega}\xi(x)|x-x_{0}|^{-\tfrac{3(n-\alpha-2)}{\alpha+2}}\;dx=\infty.$

This contradiction implies there can be no point $x_{0}$ such that
$h(x_{0},T_{0})=0$, as desired. Note that we used $\xi>0$ on $\omega$ and
$x_{0}\in\omega$ to conclude that the integral diverges.

Proof of 3): Taking $\alpha=-\tfrac{1}{2}$ in (A.43), the approximate solution
$h_{\varepsilon}$ satisfies

$\int\limits_{\Omega}{G^{(-1/2)}_{\varepsilon}(h_{\varepsilon}(x,T))\,dx}\leqslant\int\limits_{\Omega}{G^{(-1/2)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n-\tfrac{1}{2}}h^{2}_{\varepsilon,x}\,dxdt}.$
(6.2)

Now we use the estimate

$\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n-\tfrac{1}{2}}h_{\varepsilon,x}^{2}\,dxdt}=-\tfrac{2}{2m-2n+1}\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n+\tfrac{1}{2}}h_{\varepsilon,xx}\,dxdt}\\\
\leqslant\tfrac{2}{2m-2n+1}\left(\iint\limits_{Q_{T}}{h^{2}_{\varepsilon,xx}\,dxdt}\right)^{\tfrac{1}{2}}\left(\iint\limits_{Q_{T}}{h_{\varepsilon}^{2m-2n+1}\,dxdt}\right)^{\tfrac{1}{2}}.$

Using $m\geqslant n-1/2$ and the $L^{\infty}$ bound on $h_{\varepsilon}$ from
(4.7) as well as (4.8), we obtain

$\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n-\tfrac{1}{2}}h_{\varepsilon,x}^{2}\,dxdt}\leqslant
C\sqrt{T},\text{ i.\,e.
}\iint\limits_{Q_{T}}{\left(h_{\varepsilon}^{\tfrac{2m-2n+3}{4}}\right)_{x}^{2}\,dxdt}\leqslant
C\sqrt{T}.$ (6.3)

Returning to (6.2) and taking $\varepsilon_{k}\to 0$ along the subsequence
that yielded the strong solution $h$, we find that

$\int\limits_{\Omega}{G^{(-1/2)}_{0}(h)\,dx}=\tfrac{1}{(n-3/2)(n-1/2)}\int\limits_{\Omega}{h^{3/2-n}(x,T)\,dx}\leqslant
K<\infty.$ (6.4)

We now prove $h(x,T)>0$ for all $x\in\bar{\Omega}$, and for every
$T\in[0,T_{loc}^{(\alpha)}]$. By (2.4), $h(\cdot,T)\in C^{1/2}(\bar{\Omega})$
for all $T\in[0,T_{loc}^{(\alpha)}]$. Assume $T_{0}$ is such that
$h(x_{0},T_{0})=0$ at some $x_{0}\in\Omega$. Then there is a $L$ such that

$h(x,T_{0})=|h(x,T_{0})-h(x_{0},T_{0})|\leqslant L|x-x_{0}|^{1/2}.$

Hence since $n\geqslant 7/2$

$\infty>\int\limits_{\Omega}{h^{\tfrac{3}{2}-n}(x,T_{0})}\;dx\geq
L^{\tfrac{3-2n}{2}}\int\limits_{\Omega}|x-x_{0}|^{-\tfrac{2n-3}{4}}\;dx=\infty.$

This contradiction implies there can be no point $x_{0}$ such that
$h(x_{0},T_{0})=0$, as desired. ∎

##  7 Finite speed of propagation

###  7.1 Local entropy estimate

######  Lemma 7.1.

Let $\zeta\in C^{1,2}_{t,x}(\bar{Q}_{T})$ such that
$\text{supp}\,\zeta\subset\Omega$, $(\zeta^{4})_{x}=0$ on $\partial\Omega$,
and $\zeta^{4}(-a,t)=\zeta^{4}(a,t)$. Assume that $-\tfrac{1}{2}<\alpha<1$,
and $\alpha\neq 0$. Then there exist a weak solution $h(x,t)$ in the sense of
the theorem Theorem 2, constants $C_{i}\ (i=1,2,3)$ dependent on
$n,\,m,\,\alpha,\,a_{0}$, and $a_{1}$, independent of $\Omega$, such that for
all $T\leqslant T_{loc}^{(\alpha)}$

$\int\limits_{\Omega}{\zeta^{4}(x,T)G^{(\alpha)}_{0}(h(x,T))\,dx}-\iint\limits_{Q_{T}}{(\zeta^{4})_{t}G^{(\alpha)}_{0}(h)\,dxdt}+C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant\\\
\int\limits_{\Omega}{\zeta^{4}(x,0)G^{(\alpha)}_{0}(h_{0})\,dx}+C_{2}\iint\limits_{Q_{T}}{h^{\alpha+2}(\zeta_{x}^{4}+\zeta^{2}\zeta_{xx}^{2})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{2(m-n+1)+\alpha}\zeta^{4}\,dxdt}.$
(7.1)

###### Sketch of Proof of Lemma 7.1.

In the following, we denote the positive, classical solution $h_{\varepsilon}$
constructed in Lemma 4.3 by $h$ (whenever there is no chance of confusion).

Let $\phi(x,t)=\zeta^{4}(x,t)$. Recall the entropy function
$G^{(\alpha)}_{\varepsilon}(z)$ defined by (4.12). Multiplying (4.1) by
$\phi(x,t)G^{\prime(\alpha)}_{\varepsilon}(h_{\varepsilon})$, and integrating
over $Q_{T}$ yields

$\int\limits_{\Omega}{\phi(x,T)G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}-\int\limits_{\Omega}{\phi(x,0)G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}-\\\
\iint\limits_{Q_{T}}{\phi_{t}(x,t)G^{(\alpha)}_{\varepsilon}(h)\,dxdt}=\iint\limits_{Q_{T}}{\phi_{x}f_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}+\\\
\iint\limits_{Q_{T}}{\phi
h^{\alpha}(a_{0}h_{x}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2})\,dxdt}=:I_{1}+I_{2}.$
(7.2)

We now bound the terms $I_{1}$ and $I_{2}$. First,

$I_{1}=-a_{0}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)h_{xx}\,dxdt}-a_{0}\iint\limits_{Q_{T}}{\phi_{x}(h^{\alpha}+f^{\prime}_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h))h_{x}h_{xx}\,dxdt}-\\\
a_{1}\iint\limits_{Q_{T}}{\phi_{xx}F^{(\alpha)}_{\varepsilon}(h)\,dxdt}=-a_{0}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)h_{xx}\,dxdt}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{\phi_{xx}h^{\alpha}h_{x}^{2}\,dxdt}+\\\
\tfrac{a_{0}\alpha}{2}\iint\limits_{Q_{T}}{\phi_{x}h^{\alpha-1}h_{x}^{3}\,dxdt}-a_{0}\iint\limits_{Q_{T}}{\phi_{x}f^{\prime}_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)h_{x}h_{xx}\,dxdt}-a_{1}\iint\limits_{Q_{T}}{\phi_{xx}F^{(\alpha)}_{\varepsilon}(h)\,dxdt},$
(7.3)

where
$F^{(\alpha)}_{\varepsilon}(z):=\int\limits_{0}^{z}{f_{\varepsilon}(s)G^{\prime(\alpha)}_{\varepsilon}(s)g^{\prime}_{\varepsilon}(s)\,ds}$,

$I_{2}=-a_{0}\iint\limits_{Q_{T}}{(\phi_{x}h^{\alpha}h_{x}h_{xx}+\alpha\phi
h^{\alpha-1}h^{2}_{x}h_{xx}+\phi h^{\alpha}h^{2}_{xx})\,dxdt}+\\\
a_{1}\iint\limits_{Q_{T}}{\phi
h^{\alpha}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{\phi_{xx}h^{\alpha}h_{x}^{2}\,dxdt}+\tfrac{5a_{0}\alpha}{6}\iint\limits_{Q_{T}}{\phi_{x}h^{\alpha-1}h_{x}^{3}\,dxdt}+\\\
\tfrac{a_{0}\alpha(\alpha-1)}{3}\iint\limits_{Q_{T}}{\phi
h^{\alpha-2}h_{x}^{4}\,dxdt}-a_{0}\iint\limits_{Q_{T}}{\phi
h^{\alpha}h^{2}_{xx}\,dxdt}+a_{1}\iint\limits_{Q_{T}}{\phi
h^{\alpha}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}.$ (7.4)

One easily finds that for all $\varepsilon>0$ and all $z\geqslant 0$

$|f_{\varepsilon}(z)G^{\prime(\alpha)}_{\varepsilon}(z)|\leqslant\tfrac{z^{\alpha+1}}{|\alpha-n+1|},\
|f_{\varepsilon}^{\prime}(z)G^{\prime(\alpha)}_{\varepsilon}(z)|\leqslant\tfrac{sz^{\alpha}}{|\alpha-n+1|},\
|F_{\varepsilon}(z)|\leqslant\tfrac{z^{m-n+\alpha+2}}{|\alpha-n+1|(m-n+\alpha+2)}+o(\varepsilon).$

Using these bounds, and the Cauchy inequality, we bound $I_{1}+I_{2}$:

$I_{1}+I_{2}\leqslant\tfrac{a_{0}\alpha(\alpha-1)}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h_{x}^{4}\phi\,dxdt}-a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\phi\,dxdt}+\\\
a_{1}\iint\limits_{Q_{T}}{h^{m-n+\alpha}h_{x}^{2}\phi\,dxdt}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h_{x}^{2}|\phi_{xx}|\,dxdt}+\\\
\tfrac{a_{0}}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha+1}|h_{xx}||\phi_{xx}|\,dxdt}+\tfrac{a_{0}s}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha}|h_{x}||h_{xx}||\phi_{x}|\,dxdt}+\\\
+\tfrac{4a_{0}\alpha}{3}\iint\limits_{Q_{T}}{h^{\alpha-1}h_{x}^{3}\phi_{x}\,dxdt}+\tfrac{a_{1}}{|\alpha-n+1|(m-n+\alpha+2)}\iint\limits_{Q_{T}}{h^{m-n+\alpha+2}|\phi_{xx}|\,dxdt}.$
(7.5)

Due to (7.5), we deduce from (7.2) that

$\int\limits_{\Omega}{\phi(x,T)G^{(\alpha)}_{\varepsilon}(h(T))\,dx}-\iint\limits_{Q_{T}}{\phi_{t}(x,t)G^{(\alpha)}_{\varepsilon}(h)\,dxdt}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\phi\,dxdt}+\\\
\tfrac{a_{0}\alpha(1-\alpha)}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h_{x}^{4}\phi\,dxdt}\leqslant\int\limits_{\Omega}{\phi(x,0)G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\\\
a_{1}\iint\limits_{Q_{T}}{h^{m-n+\alpha}h_{x}^{2}\phi\,dxdt}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h_{x}^{2}|\phi_{xx}|\,dxdt}+\\\
\tfrac{a_{0}}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha+1}|h_{xx}||\phi_{xx}|\,dxdt}+\tfrac{a_{0}s}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha}|h_{x}||h_{xx}||\phi_{x}|\,dxdt}+\\\
\tfrac{4a_{0}\alpha}{3}\iint\limits_{Q_{T}}{h^{\alpha-1}h_{x}^{3}\phi_{x}\,dxdt}+\tfrac{a_{1}}{|\alpha-n+1|(m-n+\alpha+2)}\iint\limits_{Q_{T}}{h^{m-n+\alpha+2}|\phi_{xx}|\,dxdt},$
(7.6)

where $\alpha\neq n-1$. Recalling $\phi=\zeta^{4}$, and using the Young’s
inequality (A.6) and simple transformations, from (7.6) we find that

$\int\limits_{\Omega}{\zeta^{4}(x,T)G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}-\iint\limits_{Q_{T}}{(\zeta^{4})_{t}G^{(\alpha)}_{\varepsilon}(h)\,dxdt}+\\\
C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant\int\limits_{\Omega}{\zeta^{4}(x,0)G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon})\,dx}+\\\
C_{2}\iint\limits_{Q_{T}}{h^{\alpha+2}(\zeta_{x}^{4}+\zeta^{2}\zeta_{xx}^{2})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{2(m-n+1)+\alpha}\zeta^{4}\,dxdt}.$
(7.7)

We now argue that the $\varepsilon\to 0$ limit of the right-hand side of (7.7)
is finite and bounded by $K$, allowing us to apply Fatou’s lemma to the left-
hand side of (7.7), concluding

$\int\limits_{\Omega}\zeta^{4}(x,T)\;G^{(\alpha)}_{0}(h(x,T))\;dx-\iint\limits_{Q_{T}}{(\zeta^{4})_{t}G^{(\alpha)}_{0}(h)\,dxdt}+C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant
K<\infty$

for every $T\in[0,T_{loc}^{(\alpha)}]$, as desired. (Note that in taking
$\varepsilon\to 0$ we will choose the exact same sequence $\varepsilon_{k}$
that was used to construct the weak solution $h$ of Theorem 2. Also, in
applying Fatou’s lemma we used the fact that $\\{h=0\\}$ having measure zero
in $Q_{T_{loc}^{(\alpha)}}$ implies $\\{h(\cdot,T)\\}$ has measure zero in
$\Omega$.)

It suffices to show that
$\int\zeta^{4}(x,0)G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon})\to\int\zeta^{4}(x,0)G^{(\alpha)}_{0}(h_{0})<\infty$
as $\varepsilon\to 0$ (the rest of items is bonded as $h\in
L^{\infty}(0,T_{loc}^{(\alpha)};H^{1}(\Omega))$). This uses the Lebesgue
Dominated Convergence Theorem. First, note that

$G^{(\alpha)}_{\varepsilon}(z)=\tfrac{z^{\alpha-n+2}}{(\alpha-n+2)(\alpha-n+1)}+\tfrac{\varepsilon
z^{\alpha-2}}{(\alpha-2)(\alpha-3)}=G^{(\alpha)}_{0}(z)+\tfrac{\varepsilon
z^{\alpha-2}}{(\alpha-2)(\alpha-3)},$

hence if $h_{0}(x)>0$ then

$G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))=G^{(\alpha)}_{0}(h_{0}(x)+\varepsilon^{\theta})+\tfrac{\varepsilon(h_{0}(x)+\varepsilon^{\theta})^{\alpha-2}}{(\alpha-2)(\alpha-3)}\leqslant
G^{(\alpha)}_{0}(h_{0}(x)+\varepsilon^{\theta})+\tfrac{\varepsilon^{1-\theta(2-\alpha)}}{|(\alpha-2)(\alpha-3)|}.$

Because $h_{0}$ has finite entropy ($\int G^{(\alpha)}_{0}(h_{0})<\infty$) it
is positive almost everywhere in $\Omega$. Using this and the fact that
$\theta$ was chosen so that $\theta<1/(2-\alpha)<2/5$, we have
$|\phi(x,0)\,G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))|\leqslant\phi(x,0)(G^{(\alpha)}_{0}(h_{0}(x))+c)\leqslant
C(G_{0}(h_{0}(x))+c)$ almost everywhere in $x$ and for all
$\varepsilon<\varepsilon_{0}$. The dominating function is in $L^{1}$, because
$h_{0}$ has finite entropy.

It remains to show pointwise convergence
$\phi(x,0)G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))\to\phi(x,0)G_{0}(h_{0}(x))$
almost everywhere in $x$:

$\displaystyle\left|G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|\leq\left|G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))\right|$
$\displaystyle\hskip
28.90755pt+\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|=\tfrac{\varepsilon
h_{0,\varepsilon}^{\alpha-2}(x)}{(\alpha-2)(\alpha-3)}+\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|$
$\displaystyle\hskip
28.90755pt\leqslant\tfrac{\varepsilon^{1-\theta(2-\alpha)}}{|(\alpha-2)(\alpha-3)|}+\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|$

As before, $\tfrac{\varepsilon^{1-\theta(2-\alpha)}}{|(\alpha-2)(\alpha-3)|}$
goes to zero by the choice of $\theta$. The term
$\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|$
goes to zero for almost every $x\in\Omega$ because $G^{(\alpha)}_{0}(z)$ is
continuous everywhere except at $z=0$.

The proof is similar for the case $\alpha=n-1$ and $\alpha=n-2$. ∎

###  7.2 Proof of Theorem 3 for the case $0<n<2$

Let $0<n<2$, and let $\text{supp}\,h_{0}\subseteq(-r_{0},r_{0})\Subset\Omega$.
For an arbitrary $s\in(0,a-r_{0})$ and $\delta>0$ we consider the families of
sets

$\Omega(s)=\Omega\setminus(-r_{0}-s,r_{0}+s),\ Q_{T}(s)=(0,T)\times\Omega(s).$
(7.8)

We introduce a nonnegative cutoff function $\eta(\tau)$ from the space
$C^{2}(\mathbb{R}^{1})$ with the following properties:

$\eta(\tau)=\left\\{\begin{aligned} \hfill 0\ &\ \tau\leqslant 0,\\\
\hfill\tau^{2}(3-2\tau)\ &\ 0<\tau<1,\\\ \hfill 1\ &\ \tau\geqslant
1.\end{aligned}\right.$ (7.9)

Next we introduce our main cut-off functions $\eta_{s,\delta}(x)\in
C^{2}(\bar{\Omega})$ such that $0\leqslant\eta_{s,\delta}(x)\leqslant 1\
\forall\,x\in\bar{\Omega}$ and possess the following properties:

$\eta_{s,\delta}(x)=\eta\bigl{(}\tfrac{|x|-(r_{0}+s)}{\delta}\bigr{)}=\left\\{\begin{aligned}
\hfill 1\;&,x\in\Omega(s+\delta),\\\ \hfill 0\;&,x\in\Omega(s),\\\
\end{aligned}\right.\ \ \ |(\eta_{s,\delta})_{x}|\leqslant\tfrac{3}{\delta},\
|(\eta_{s,\delta})_{xx}|\leqslant\tfrac{6}{\delta^{2}}$ (7.10)

for all $s>0,\ \delta>0:r_{0}+s+\delta<a$. Choosing
$\zeta^{4}(x,t)=\eta_{s,\delta}(x)e^{-\tfrac{t}{T}}$, from (7.1) we arrive at

$\int\limits_{\Omega(s+\delta)}{h^{\alpha-n+2}(T)\,dx}+\tfrac{1}{T}\iint\limits_{Q_{T}(s+\delta)}{h^{\alpha-n+2}\,dxdt}+C\iint\limits_{Q_{T}(s+\delta)}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dxdt}\leqslant\\\
\tfrac{C}{\delta^{4}}\iint\limits_{Q_{T}(s)}{h^{\alpha+2}\,dxdt}+C\iint\limits_{Q_{T}(s)}{h^{2(m-n+1)+\alpha}\,dxdt}=:C\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}$
(7.11)

for all $s>0$, where we consider $(n-1)_{+}<\alpha<1$. We apply the Gagliardo-
Nirenberg interpolation inequality (see Lemma D.4) in the region
$\Omega(s+\delta)$ to a function $v:=h^{\tfrac{\alpha+2}{2}}$ with
$a=\tfrac{2\xi_{i}}{\alpha+2},\ b=\tfrac{2(\alpha-n+2)}{\alpha+2},\ d=2,\
i=0,\ j=2$, and
$\theta_{i}=\tfrac{(\alpha+2)(\xi_{i}-\alpha+n-2)}{\xi_{i}(4\alpha-3n+8)}$
under the conditions:

$\alpha-n+2<\xi_{i}\text{ for }i=1,2.$ (7.12)

Integrating the resulted inequalities with respect to time and taking into
account (7.11), we arrive at the following relations:

$\iint\limits_{Q_{T}(s+\delta)}{h^{\xi_{i}}}\leqslant
C\,T^{1-\tfrac{\theta_{i}\xi_{i}}{\alpha+2}}\Biggl{(}\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{1+\nu_{i}}\\!\\!\\!\\!\\!+C\,T\Biggl{(}\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{\tfrac{\xi_{i}}{\alpha-n+2}}\\!\\!\\!\\!,$
(7.13)

where $\nu_{i}=\tfrac{4(\xi_{i}-\alpha+n-2)}{4\alpha-3n+8}$. These
inequalities are true provided that

$\tfrac{\theta_{i}\xi_{i}}{\alpha+2}<1\Leftrightarrow\xi_{i}<5\alpha-4n+10\text{
for }i=1,2.$ (7.14)

Simple calculations show that inequalities (7.12) and (7.14) hold with some
$(n-1)_{+}<\alpha<1$ if and only if

$0<n<2,\ \ \tfrac{n}{2}<m<6-n.$

Since all integrals on the right-hand sides of (7.13) vanish as $T\to 0$, the
finite speed of propagations follows from (7.13) by applying Lemma D.5 with
$s_{1}=0$ and sufficiently small $T$. Hence,

$\textnormal{supp}\,h(T,.)\subset(-r_{0}-\Gamma_{0}(T),r_{0}+\Gamma_{0}(T))\Subset\Omega\
\text{for all }T:0\leqslant T\leqslant T_{speed}.$ (7.15)

Due to (7.15), we can consider the solution $h(x,t)$ with compact support in
the whole space $\mathbb{R}^{1}$ and for $T\leqslant T_{speed}$. In this case,
we can repeat the previous procedure for
$\Omega(s)=\mathbb{R}^{1}\setminus(-r_{0}-s,r_{0}+s)$ and we obtain

$G_{i}(s+\delta):=\iint\limits_{Q_{T}(s+\delta)}{h^{\xi_{i}}}\leqslant
C\,T^{1-\tfrac{\theta_{i}\xi_{i}}{\alpha+2}}\Biggl{(}\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{1+\nu_{i}},$
(7.16)

instead of (7.13), and

$\Gamma_{0}(T)=C\bigl{(}T^{(1-\tfrac{\theta_{1}\xi_{1}}{\alpha+2})(1+\nu_{2})}T^{(1-\tfrac{\theta_{2}\xi_{2}}{\alpha+2})\nu_{1}(1+\nu_{1})}(G(0))^{\nu_{1}}\bigr{)}^{\tfrac{1}{4(1+\nu_{1})(1+\nu_{2})}},$
(7.17)
$H(s)=CT^{(1-\tfrac{\theta_{2}\xi_{2}}{\alpha+2})(1+\nu_{1})}T^{(1-\tfrac{\theta_{1}\xi_{1}}{\alpha+2})\nu_{2}(1+\nu_{2})}(G_{2}(0))^{\nu_{2}},$

where

$G(0)=C(T^{(1-\tfrac{\theta_{1}\xi_{1}}{\alpha+2})(1+\nu_{2})}(G_{2}(0))^{1+\nu_{1}}+T^{(1-\tfrac{\theta_{2}\xi_{2}}{\alpha+2})(1+\nu_{1})}(G_{1}(0))^{1+\nu_{2}}).$
(7.18)

Now we need to estimate $G(0)$. With that end in view, we obtain the following
estimates:

$G_{i}(0)\leqslant
C_{1}\,(C_{2}+C_{3}T)^{\tfrac{\xi_{i}-1}{\alpha+5}}T^{1-\tfrac{\xi_{i}-1}{\alpha+5}},\
\ i=1,2.$ (7.19)

where $1<\xi_{i}<\alpha+6$, and $C_{i}$ depends on $h_{0}(x)$ only. Really,
applying the Gagliardo-Nirenberg interpolation inequality (see Lemma D.4) in
$\Omega=\mathbb{R}^{1}$ to a function $v:=h^{\tfrac{\alpha+2}{2}}$ with
$a=\tfrac{2\xi_{i}}{\alpha+2},\ b=\tfrac{2}{\alpha+2},\ d=2,\ i=0,\ j=2$, and
$\widetilde{\theta}_{i}=\tfrac{(\alpha+2)(\xi_{i}-1)}{\xi_{i}(\alpha+5)}$
under the condition $\xi_{i}>1$, we deduce that

$\int\limits_{\mathbb{R}^{1}}{h^{\xi_{i}}dx}\leqslant
c\,\|h_{0}\|_{1}^{\tfrac{2(3\xi_{i}+\alpha+2)}{(\alpha+2)(\alpha+5)}}\Bigl{(}\int\limits_{\mathbb{R}^{1}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dx}\Bigr{)}^{\tfrac{\xi_{i}-1}{\alpha+5}}.$
(7.20)

Integrating (7.20) with respect to time and taking into account the Hölder
inequality ($\tfrac{\xi_{i}-1}{\alpha+5}\leqslant
1\Rightarrow\xi_{i}<\alpha+6\Rightarrow m\leqslant n+2$), we arrive at the
following relations:

$G_{i}(0)\leqslant
c\,\|h_{0}\|_{1}^{\tfrac{2(3\xi_{i}+\alpha+2)}{(\alpha+2)(\alpha+5)}}T^{1-\tfrac{\xi_{i}-1}{\alpha+5}}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dx}\Bigr{)}^{\tfrac{\xi_{i}-1}{\alpha+5}},\
m\leqslant n+2.$ (7.21)

From (7.21), due to (A.32) and (5.2), we find (7.19).

Inserting (7.19) into (7.18), we obtain after straightforward computations
that

$\Gamma_{0}(T)\leqslant\Gamma(T)=C\,T^{\tfrac{1}{n+4}}$ (7.22)

for $T\leqslant T_{speed}$, where $\tfrac{n}{2}<m\leqslant n+2$, and
$T_{speed}$ finds from the condition $H(0)<1$.

In the case of $m>n+2$, we have (7.21) for $i=1$ only, i. e.

$G_{1}(0)\leqslant
c\,\|h_{0}\|_{1}^{\tfrac{2(3\xi_{1}+\alpha+2)}{(\alpha+2)(\alpha+5)}}T^{1-\tfrac{\xi_{1}-1}{\alpha+5}}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dx}\Bigr{)}^{\tfrac{\xi_{1}-1}{\alpha+5}}.$
(7.23)

Hence

$G_{1}(0)\leqslant
C(C+C(1-(1-\tfrac{k_{1}(2m-n)\|h_{0}\|_{H^{1}}^{2m-n}}{2}T)^{\tfrac{m+2}{2m-n}})T^{1-\tfrac{\xi_{1}-1}{\alpha+5}}\leqslant
C_{0}T^{1-\tfrac{\xi_{1}-1}{\alpha+5}}.$ (7.24)

Next we need estimate the term $G_{2}(0)$. Really, by using the inequality
(A.3), we obtain

$G_{2}(0)\leqslant
b_{2}\|h_{0}\|_{1}^{\tfrac{\xi_{2}+2}{3}}\int\limits_{0}^{T}{\|h_{x}\|_{2}^{\tfrac{2}{3}(\xi_{2}-1)}dt}=b_{2}\|h_{0}\|_{1}^{\tfrac{\xi_{2}+2}{3}}\|h_{0}\|_{H^{1}}^{\tfrac{2(\xi_{2}-1)}{3}}\times\\\
\int\limits_{0}^{T}{(1-\tfrac{k_{1}(2m-n)\|h_{0}\|_{H^{1}}^{2m-n}}{2}t)^{-\tfrac{2(\xi_{2}-1)}{3(2m-n)}}dt}=b_{2}\|h_{0}\|_{1}^{\tfrac{\xi_{2}+2}{3}}\|h_{0}\|_{H^{1}}^{-\tfrac{2m+n-2(\alpha+1))}{3}}\times\\\
\tfrac{6}{k_{1}(2m+n-2(\alpha+1))}\bigl{(}1-(1-\tfrac{k_{1}(2m-n)\|h_{0}\|_{H^{1}}^{2m-n}}{2}T)^{\tfrac{2m+n-2(\alpha+1)}{3(2m-n)}}\bigr{)}\leqslant
C_{0}$ (7.25)

for all $T\leqslant T_{loc}$, where $C_{0}$ depends on $h_{0}(x)$. Inserting
(7.19) for $i=1$ and (7.25) into (7.18), we obtain (7.22).

###  7.3 Local energy estimate

######  Lemma 7.2.

Let $n\in\bigl{(}\tfrac{1}{2},3\bigr{)}$, and $m>\tfrac{2(n-1)_{+}}{3}$, and
$\beta>\tfrac{1-n}{3}$. Let $\zeta\in C^{2}(\bar{\Omega})$ such that
$\text{supp}\,\zeta$ in $\Omega$, and $(\zeta^{6})^{\prime}=0$ on
$\partial\Omega$, and $\zeta(-a)=\zeta(a)$. Then there exist constants $C_{i}\
(i=1,2,3)$ dependent on $n,\,m,\,a_{0}$, and $a_{1}$, independent of $\Omega$
and $\varepsilon$, such that for all $T\leqslant T_{loc}$

$\int\limits_{\Omega}{\zeta^{6}h^{2}_{x}(x,T)\,dx}+\int\limits_{\Omega}{\zeta^{4}h^{\beta+1}(T)\,dx}+C_{1}\iint\limits_{Q_{T}}{\zeta^{6}(h^{\tfrac{n+2}{2}})_{xxx}^{2}\,dxdt}\leqslant\int\limits_{\Omega}{\zeta^{6}h_{0}^{2}(x)\,dx}+\\\
\int\limits_{\Omega}{\zeta^{4}h_{0}^{\beta+1}\,dx}+C_{2}\iint\limits_{Q_{T}}{h^{n+2}(\zeta_{x}^{6}+\zeta^{3}|\zeta_{xx}|^{3})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt}+\\\
C_{4}\iint\limits_{Q_{T}}{\\{h^{n+2\beta}\zeta_{x}^{2}+\chi_{\\{\zeta>0\\}}h^{n+3\beta-1}+h^{\tfrac{6m-n+6\beta+4}{5}}\zeta^{\tfrac{12}{5}}\zeta_{x}^{\tfrac{6}{5}}+h^{\tfrac{3m-n+3\beta+1}{2}}\zeta^{3}\\}\,dxdt}.$
(7.26)

###### Sketch of Proof of Lemma 7.2.

Let $\phi(x)=\zeta^{6}(x)$. Multiplying (4.1) by $-(\phi(x)h_{x})_{x}$, and
integrating on $Q_{T}$, yields

$\tfrac{1}{2}\int\limits_{\Omega}{\phi(x)h^{2}_{x}(x,T)\,dx}-\tfrac{1}{2}\int\limits_{\Omega}{\phi(x)h_{0\varepsilon,x}^{2}(x)\,dx}=\\\
-\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})(\phi_{xx}h_{x}+2\phi_{x}h_{xx}+\phi
h_{xxx})\,dxdt}=\\\
-\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\phi_{xx}h_{x}\,dxdt}-\\\
2\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\phi_{x}h_{xx}\,dxdt}-\\\
\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\phi
h_{xxx}\,dxdt}=:I_{1}+I_{2}+I_{3}.$ (7.27)

We now bound the terms $I_{1}$, $I_{2}$ and $I_{3}$. First,

$I_{1}=-a_{0}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)h_{xxx}h_{x}\,dxdt}-a_{1}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=\\\
-6a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h_{xxx}h_{x}\zeta^{4}(5\zeta_{x}^{2}+\zeta\zeta_{xx})\,dxdt}-6a_{1}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\zeta^{4}(5\zeta_{x}^{2}+\zeta\zeta_{xx})\,dxdt}\leqslant\\\
\epsilon_{1}\iint\limits_{Q_{T}}{\zeta^{6}\\{f_{\varepsilon}(h)h_{xxx}^{2}+h^{n-4}h_{x}^{6}\\}\,dxdt}+C(\epsilon_{1})\iint\limits_{Q_{T}}{h^{n+2}(\zeta_{x}^{6}+\zeta^{3}|\zeta_{xx}|^{3})\,dxdt}+\\\
C(\epsilon_{1})\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt},$ (7.28)
$I_{2}=-2a_{0}\iint\limits_{Q_{T}}{\phi_{x}f_{\varepsilon}(h)h_{xxx}h_{xx}\,dxdt}-2a_{1}\iint\limits_{Q_{T}}{\phi_{x}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{xx}h_{x}\,dxdt}=\\\
-12a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h_{xxx}h_{xx}\zeta^{5}\zeta_{x}\,dxdt}-12a_{1}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{xx}h_{x}\zeta^{5}\zeta_{x}\,dxdt}\leqslant\\\
\epsilon_{2}\iint\limits_{Q_{T}}{\zeta^{6}\\{f_{\varepsilon}(h)h_{xxx}^{2}+h^{n-2}h_{x}^{2}h^{2}_{xx}+h^{n-1}|h_{xx}|^{3}\\}\,dxdt}+\\\
C(\epsilon_{2})\iint\limits_{Q_{T}}{h^{n+2}\zeta_{x}^{6}\,dxdt}+C(\epsilon_{2})\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt},$
(7.29) $I_{3}=-a_{0}\iint\limits_{Q_{T}}{\phi
f_{\varepsilon}(h)h_{xxx}^{2}\,dxdt}-a_{1}\iint\limits_{Q_{T}}{\phi
f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{xxx}h_{x}\,dxdt}\leqslant\\\
-a_{0}\iint\limits_{Q_{T}}{\zeta^{6}f_{\varepsilon}(h)h_{xxx}^{2}\,dxdt}+\epsilon_{3}\iint\limits_{Q_{T}}{\zeta^{6}(f_{\varepsilon}(h)h^{2}_{xxx}+h^{n-4}h_{x}^{6})\,dxdt}+\\\
C(\epsilon_{3})\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt}.$ (7.30)

Now, multiplying (4.1) by $\zeta^{4}(h+\gamma)^{\beta}$,
$\beta>\tfrac{1-n}{3},\ \gamma>0$ and integrating on $Q_{T}$, using the
Young’s inequality (A.6), letting $\gamma\to 0$, we obtain the following
estimate

$\int\limits_{\Omega}{\zeta^{4}h^{\beta+1}(T)\,dx}\leqslant\int\limits_{\Omega}{\zeta^{4}h_{0\varepsilon}^{\beta+1}\,dx}+\epsilon_{4}\iint\limits_{Q_{T}}{\zeta^{6}\\{f_{\varepsilon}(h)h_{xxx}^{2}+h^{n-4}h_{x}^{6}\\}\,dxdt}+\\\
C(\epsilon_{4})\iint\limits_{Q_{T}}{\\{h^{n+2\beta}\zeta_{x}^{2}+\chi_{\\{\zeta>0\\}}h^{n+3\beta-1}+h^{\tfrac{6m-n+6\beta+4}{5}}\zeta^{\tfrac{12}{5}}\zeta_{x}^{\tfrac{6}{5}}+h^{\tfrac{3m-n+3\beta+1}{2}}\zeta^{3}\\}\,dxdt},$
(7.31)

where $\beta>\max\\{\tfrac{1-n}{3},-m+\tfrac{n-1}{3}\\}=\tfrac{1-n}{3}$ as
$m>\tfrac{2(n-1)_{+}}{3}$.

If we now add inequalities (7.27) and (7.31), in view of (7.28)–(7.30), then,
applying Lemma D.3, choosing $\epsilon_{i}>0$, and letting $\varepsilon\to 0$,
we obtain (7.26). ∎

###  7.4 Proof of Theorem 3 for the case $\tfrac{1}{2}<n<3$

Let $\eta_{s,\delta}(x)$ denote by (7.10). Setting
$\zeta^{6}(x)=\eta_{s,\delta}(x)$ into (7.26), after simple transformations,
we obtain

$\int\limits_{\Omega(s+\delta)}{h^{2}_{x}(x,T)\,dx}+\int\limits_{\Omega(s+\delta)}{h^{\beta+1}(T)\,dx}+C\iint\limits_{Q_{T}(s+\delta)}{(h^{\tfrac{n+2}{2}})_{xxx}^{2}\,dxdt}\leqslant\\\
\tfrac{C}{\delta^{6}}\iint\limits_{Q_{T}(s)}{h^{n+2}\,dxdt}+\tfrac{C}{\delta^{2}}\iint\limits_{Q_{T}(s)}{h^{n+2\beta}\,dxdt}+\tfrac{C}{\delta^{\tfrac{6}{5}}}\iint\limits_{Q_{T}(s)}{h^{\tfrac{6m-n+6\beta+4}{5}}\,dxdt}+\\\
C\iint\limits_{Q_{T}(s)}{\\{h^{3m-2n+2}+h^{n+3\beta-1}+h^{\tfrac{3m-n+3\beta+1}{2}}\\}\,dxdt}=:C\sum\limits_{i=1}^{6}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}$
(7.32)

for all for all $s>0,\ \delta>0:r_{0}+s+\delta<a$. We apply the Gagliardo-
Nirenberg interpolation inequality (Lemma D.4) in the region
$\Omega(s+\delta)$ to a function $v:=h^{\tfrac{n+2}{2}}$ with
$a=\tfrac{2\xi_{i}}{n+2},\ b=\tfrac{2(\beta+1)}{n+2},\ d=2,\ i=0,\ j=3$, and
$\theta_{i}=\tfrac{(n+2)(\xi_{i}-\beta-1)}{\xi_{i}(n+5\beta+7)}$ under the
conditions:

$\beta<\xi_{i}-1\text{ for }i=\overline{1,6}.$ (7.33)

Integrating the resulted inequalities with respect to time and taking into
account (7.32), we arrive at the following relations:

$\iint\limits_{Q_{T}(s+\delta)}{h^{\xi_{i}}}\leqslant
C\,T^{1-\tfrac{\theta_{i}\xi_{i}}{n+2}}\Biggl{(}\sum\limits_{i=1}^{6}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{1+\nu_{i}}\\!\\!\\!\\!\\!+C\,T\Biggl{(}\sum\limits_{i=1}^{5}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{\tfrac{\xi_{i}}{\beta+1}}\\!\\!\\!\\!,$
(7.34)

where $\nu_{i}=\tfrac{6(\xi_{i}-\beta-1)}{n+5\beta+7}$. These inequalities are
true provided that

$\tfrac{\theta_{i}\xi_{i}}{n+2}<1\Leftrightarrow\beta>\tfrac{\xi_{i}-n-8}{6}\text{
for }i=\overline{1,6}.$ (7.35)

Simple calculations show that inequalities (7.33) and (7.35) hold with some
$\beta>\tfrac{1-n}{3}$ if and only if

$\tfrac{1}{2}<n<3,\ \ m>\tfrac{n}{2}.$

Since all integrals on the right-hand sides of (7.34) vanish as $T\to 0$, the
finite speed of propagations follows from (7.34) by applying Lemma D.5 with
$s_{1}=0$ and sufficiently small $T$. Hence,

$\textnormal{supp}\,h(T,.)\subset(-r_{0}-\Gamma(T),r_{0}+\Gamma(T))\Subset\Omega\
\text{for all }T:0\leqslant T\leqslant T_{speed}.$ (7.36)

##  8 Finite time blow up

######  Lemma 8.1.

Let $0<n<2,\,m\geqslant\max\\{n+2,4-n\\}$. Then the weak solution $h$ from
Theorem 1 satisfies the second-moment inequality:

$e^{-\widetilde{B}(T)}\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h(x,T))\,dx}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\\\
\int\limits_{0}^{T}{\biggl{(}k_{1}\mathcal{E}_{0}(0)+W^{\prime}(t)+k_{2}\int\limits_{\Omega}{x^{2}h^{2}_{xx}\,dx}\biggr{)}e^{-\widetilde{B}(t)}dt}$
(8.1)

for all $T\in[0,T_{loc}]$, where $k_{1}=2(4-n),\
k_{2}=\tfrac{3a_{0}(n-1)}{2}$. Here

$\widetilde{G}_{0}(z)=\tfrac{1}{2-n}z^{2-n},\
\widetilde{B}(T):=\tfrac{a_{1}^{2}|1-n|(2-n)}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}{\|h(.,\tau)\|_{L^{\infty}(\Omega)}^{2m-n}\,d\tau},$
$W(T):=\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}mD_{0}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}.$
(8.2)

Moreover, if $h(\cdot,t)$ has a compact support in $\Omega$ then
$W^{\prime}(t)\equiv 0$.

The proof of Lemma 8.1 is given in Appendix B Proof of second moment estimate.

###### Proof of Theorem 4.

Assume $0<n<2$ and $m$ satisfies the hypotheses of Theorem 3. By Theorems 2,
3, there is a compactly supported strong solution $h$ for $t\in[0,T_{0}]$.
Hence, we can assume that $W^{\prime}(t)=0$ in Lemma 8.1.

First, we construct a sequence of times $T_{0}<T_{1}<...$ and extend the
strong solution $h$ from the time interval $[0,T_{i}]$ to the time interval
$[0,T_{i+1}]$. Taking it as an initial datum, we obtain a time interval of
existence

$T_{1}-T_{0}=\tfrac{1}{2d_{7}(\gamma_{3}-1)}\biggl{(}\int\limits_{\Omega}{\\{\tfrac{1}{2}h_{x}^{2}(x,T_{0})+G_{0}^{(\alpha)}(h(x,T_{0}))\\}\,dx}\biggr{)}^{-(\gamma_{3}-1)},$

by (A.39). Applying Theorem 3 to the time interval $[T_{0},T_{1}]$, we have a
strong solution with compact support that satisfies all a priori estimates
with the time interval $[0,T_{0}]$ replaced by $[0,T_{1}]$. In this way, we
construct a nonnegative, compactly supported, strong solution on
$\mathbb{R}^{1}\times[0,T^{*})$ where

$T^{*}=\lim_{i\to\infty}T_{i}.$

If $T^{*}<\infty$ then

$\tfrac{1}{2e_{7}(\gamma_{3}-1)}\biggl{(}\int\limits_{\Omega}{\\{\tfrac{1}{2}h_{x}^{2}(x,T_{i})+G_{0}^{(\alpha)}(h(x,T_{i}))\\}\,dx}\biggr{)}^{1-\gamma_{3}}=T_{i+1}-T_{i}\to
0.$

Hence, the $H^{1}$ norms at times $T_{i}$ must blow up. And, due to (3.8), the
$L^{\infty}$-norm of the solution at times $T_{i}$ must also blow up.
Therefore, to finish the proof it suffices to prove that $T^{*}<\infty$.

Let

$V(0)=\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}\quad\mbox{and}\quad
V(T_{i}):=e^{-\widetilde{B}(T_{i})}\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h(T_{i}))\,dx}.$
(8.3)

Using that $k_{1}>0$ and $\mathcal{E}_{0}(T_{i})\leqslant\mathcal{E}_{0}(0)$,
we apply the inequality (8.1) iteratively to find

$V(T_{i})\leqslant
V(0)+k_{1}\mathcal{E}_{0}(0)\int\limits_{0}^{T_{i}}e^{-\widetilde{B}(t)}\;dt+k_{2}\int\limits_{0}^{T_{i}}\int\limits_{-\infty}^{\infty}x^{2}h_{xx}^{2}(x,t)\;dxe^{-\widetilde{B}(t)}\;dt$
(8.4)

Case $n=1$: [13] In this case, $\widetilde{B}(t)=0$ and $k_{2}=0$. Hence (8.4)
becomes

$V(T_{i})\leqslant V(0)+k_{1}\mathcal{E}_{0}(0)\,T_{i}.$

If $T^{*}=\infty$ then the right-hand side will become negative, contradicting
that the left-hand side is nonnegative. Hence $T^{*}<\infty$.

Case $0<n<1$: In this case, $k_{2}<0$ and (8.4) implies

$V(T_{i})\leqslant
V(0)+k_{1}\mathcal{E}_{0}(0)\int\limits_{0}^{T_{i}}e^{-\widetilde{B}(t)}\;dt$
(8.5)

We wish to argue that if $T^{*}=\infty$ then the right-hand side of (8.5) must
become negative, leading to a contradition.

Let $g(t)=\int\limits_{0}^{t}{e^{-\widetilde{B}(s)}ds}$. Using (A.19), we have
$\|h(x,t)\|_{\infty}\leqslant K_{1}(T_{i}-t)^{-\tfrac{1}{2(2m-n-1)}}$ for all
$t<T_{i}$.

$\widetilde{B}(s)\leqslant\tfrac{a_{1}^{2}(1-n)(2-n)}{2a_{0}(m-n+1)^{2}}K_{1}^{2m-n}\int\limits_{0}^{s}{(T_{i}-s)^{-\tfrac{2m-n}{2(2m-n-1)}}\,d\tau}=K_{2}\bigl{(}T_{i}^{\tfrac{2m-n-2}{2(2m-n-1)}}-(T_{i}-s)^{\tfrac{2m-n-2}{2(2m-n-1)}}\bigr{)},$

whence

$g(T_{i})\geqslant
e^{-K_{2}T_{i}^{\tfrac{2m-n-2}{2(2m-n-1)}}}\int\limits_{0}^{T_{i}}{e^{K_{2}(T_{i}-s)^{\tfrac{2m-n-2}{2(2m-n-1)}}}ds}=\\\
\tfrac{\int\limits_{0}^{T_{i}}{e^{K_{2}s^{\tfrac{2m-n-2}{2(2m-n-1)}}}ds}}{e^{K_{2}T_{i}^{\tfrac{2m-n-2}{2(2m-n-1)}}}}\sim
T_{i}^{\tfrac{2m-n}{2(2m-n-1)}}\text{ as }T_{i}\to+\infty.$ (8.6)

Since $\mathcal{E}_{0}(0)<0$, if $T_{i}\to+\infty$ then for large times the
right-hand side of (8.5) would be negative: an impossibility. Therefore
$\mathop{\lim}\limits_{i\to+\infty}T_{i}=T^{*}<\infty$.

Case $1<n<2$: In this case, we write inequality (8.4) in the form:

$V(T_{i})\leqslant V(0)+k_{1}\mathcal{E}_{0}(0)\,g(T_{i})+k_{2}\,f(T_{i})$
(8.7)

where

$f(T_{i})=\int\limits_{0}^{T_{i}}e^{-\widetilde{B}(t)}\int\limits_{-\infty}^{\infty}x^{2}h_{xx}^{2}(x,t)\;dx\;dt.$

Assume $T^{*}=\infty$ and that $f(T_{i})$ grows slower than $g(T_{i})$ as
$T_{i}\to\infty$. Then the right–hand side of (8.7) will become negative in
finite time, which is an impossibility. It therefore suffices to show that
$f(T_{i})$ grows more slowly than $g(T_{i})$.

Due to (A.16) and (A.19) as $\varepsilon\to 0$, we have

$\tfrac{a_{0}}{2}\int\limits_{0}^{T_{i}}\int\limits_{-\infty}^{\infty}{h^{2}_{xx}\,dxdt}\leqslant\int\limits_{-\infty}^{\infty}{G_{0}(h_{0})\,dx}+C\int\limits_{0}^{T_{i}}{\|h_{x}\|_{2}^{2(m-n+1)}dt}\leqslant\\\
\int\limits_{-\infty}^{\infty}{G_{0}(h_{0})\,dx}+C\int\limits_{0}^{T_{i}}{(T_{i}-t)^{-\tfrac{m-n+1}{2m-n-1}}dt}=\int\limits_{-\infty}^{\infty}{G_{0}(h_{0})\,dx}+\tfrac{C(2m-n-1)}{m-2}T_{i}^{\tfrac{m-2}{2m-n-1}}.$
(8.8)

In view of (7.22), we know that $|x|\leqslant C(h_{0})$ for all $t\leqslant
T_{i}$. Hence, due to (8.8), we deduce

$f(T_{i})\leqslant\int\limits_{-\infty}^{\infty}{x^{2}h^{2}_{xx}\,dxdt}\leqslant
C\int\limits_{0}^{T_{i}}\int\limits_{-\infty}^{\infty}{h^{2}_{xx}\,dxdt}\leqslant
C(1+T_{i}^{\tfrac{m-2}{2m-n-1}}).$

Comparing with the exponent in (8.6), we see that $f(T_{i})$ grows more slowly
than $g(T_{i})$, as desired.

∎

##  Appendix A Proofs of A Priori Estimates

The first observation is that the periodic boundary conditions imply that
classical solutions of equation (4.1) conserve mass:

$\int\limits_{\Omega}{h_{\delta\varepsilon}(x,t)\,dx}=\int\limits_{\Omega}{h_{0,\delta\varepsilon}(x)\,dx}=M_{\delta\varepsilon}<\infty\text{
for all }t>0.$ (A.1)

Further, (4.5) implies $M_{\delta\varepsilon}\to M=\int h_{0}$ as
$\varepsilon,\delta\to 0$. The initial data in this article have $M>0$, hence
$M_{\delta\varepsilon}>0$ for $\delta$ and $\varepsilon$ sufficiently small.

Also, we will relate the $L^{p}$ norm of $h$ to the $L^{p}$ norm of its zero-
mean part as follows:

$|h(x)|\leqslant\left|h(x)-\tfrac{M}{|\Omega|}\right|+\tfrac{M}{|\Omega|}\Longrightarrow\|h\|_{p}^{p}\leqslant
2^{p-1}\;\|v\|_{p}^{p}+\left(\tfrac{2}{|\Omega|}\right)^{p-1}\;M^{p}$

where $v:=h-M/|\Omega|$ and we have assumed that $M\geqslant 0$. We will use
the Poincaré inequality which holds for any zero-mean function in
$H^{1}(\Omega)$

$\|v\|_{p}^{p}\leqslant b_{1}\|v_{x}\|_{p}^{p}\qquad 1\leqslant p<\infty$
(A.2)

with $b_{1}=|\Omega|^{p}$.

Also used will be an interpolation inequality [30, Theorem 2.2, p. 62] for
functions of zero mean in $H^{1}(\Omega)$:

$\|v\|_{p}^{p}\leqslant b_{2}\,\|v_{x}\|_{2}^{ap}\;\|v\|_{r}^{(1-a)p}$ (A.3)

where $r\geqslant 1$, $p\geqslant r$,

$a=\tfrac{1/r-1/p}{1/r+1/2},\qquad b_{2}=\left(1+r/2\right)^{ap}.$

It follows that for any zero-mean function $v$ in $H^{1}(\Omega)$

$\|v\|_{p}^{p}\leqslant
b_{3}\|v_{x}\|_{2}^{p},\quad\Longrightarrow\quad\|h\|_{p}^{p}\leqslant
b_{4}\|h_{x}\|_{2}^{p}+b_{5}M_{\delta\varepsilon}^{p}$ (A.4)

where

$b_{3}=\begin{cases}b_{1}\;|\Omega|^{(2-p)/p}&\mbox{if}\quad 1\leqslant
p\leqslant 2\\\ b_{1}^{(p+2)/2}\;b_{2}&\mbox{if}\quad
2<p<\infty\end{cases},\quad b_{4}=2^{p-1}\,b_{3},\quad
b_{5}=\left(\tfrac{2}{|\Omega|}\right)^{p-1}$

To see that (A.4) holds, consider two cases. If $1\leqslant p<2$, then by
(A.2), $\|v\|_{p}$ is controlled by $\|v_{x}\|_{p}$. By the Hölder inequality,
$\|v_{x}\|_{p}$ is then controlled by $\|v_{x}\|_{2}$. If $p>2$ then by (A.3),
$\|v\|_{p}$ is controlled by $\|v_{x}\|_{2}^{a}\|v\|_{2}^{1-a}$ where
$a=1/2-1/p$. By the Poincaré inequality, $\|v\|_{2}^{1-a}$ is controlled by
$\|v_{x}\|_{2}^{1-a}$.

If $0<p<1$ then, instead of (A.4), we obtain

$\|h\|_{p}^{p}\leqslant\widetilde{b}_{4}\|h_{x}\|_{2}^{p}+\widetilde{b}_{5}M_{\delta\varepsilon}^{p}$
(A.5)

where

$\widetilde{b}_{4}=|\Omega|^{1-\tfrac{p}{2}}\,b_{4}^{\tfrac{p}{2}},\quad\widetilde{b}_{5}=|\Omega|^{1-\tfrac{p}{2}}\,b_{5}^{\tfrac{p}{2}}.$

The Cauchy inequality $ab\leqslant\epsilon a^{2}+b^{2}/(4\epsilon)$ with
$\epsilon>0$ will be used often as will Young’s inequality

$ab\leqslant\epsilon a^{p}+\tfrac{b^{q}}{q\,(\epsilon
p)^{q/p}},\qquad\tfrac{1}{p}+\tfrac{1}{q}=1,\;\epsilon>0.$ (A.6)

###### Sketch of Proof of Lemma 4.1.

In the following, we denote the classical solution $h_{\delta\varepsilon}$ by
$h$ whenever there is no chance of confusion.

To prove the bound (4.7) one starts by multiplying (4.1) by $-h_{xx}$,
integrating over $Q_{T}$, and using the periodic boundary conditions (4.2)
yields

$\displaystyle\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
(A.7) $\displaystyle\hskip
14.45377pt=\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}(x)\,dx}-a_{1}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}h_{xxx}\,dxdt}.$

The Cauchy inequality is used to bound some terms on the right-hand of (A.7):

$a_{1}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}h_{xxx}\,dxdt}\leqslant\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}+\\\
\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)(D^{\prime\prime}_{\varepsilon}(h))^{2}h_{x}^{2}\,dxdt}.$
(A.8)

Using (A.8) in (A.7) yields

$\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+\\\
\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)(D^{\prime\prime}_{\varepsilon}(h))^{2}h_{x}^{2}\,dxdt}\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+\\\
\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{|h|^{2m-n}h_{x}^{2}\,dxdt}+\tfrac{\delta}{\varepsilon^{2}}\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}.$
(A.9)

Above, we used the bounds $f_{\delta\varepsilon}(z)\leqslant|z|^{n}+\delta$,
$D^{\prime\prime}_{\varepsilon}(z)\leqslant|z|^{m-n}$, and
$D^{\prime\prime}_{\varepsilon}(z)\leqslant\varepsilon^{-1}$. By the Cauchy
inequality, bound (A.4), (A.3) and bound (A.5),

$\displaystyle\iint\limits_{Q_{T}}|h|^{2m-n}h_{x}^{2}\;dxdt\leqslant\tfrac{1}{2}\iint\limits_{Q_{T}}h^{2(2m-n)}\;dxdt+\tfrac{1}{2}\iint\limits_{Q_{T}}h_{x}^{4}\;dxdt$
$\displaystyle\leqslant\tfrac{b_{4}}{2}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n}\;dt+\tfrac{b_{5}}{2}\;M_{\delta\varepsilon}^{2(2m-n)}\;T+\tfrac{b_{2}}{2}\int\limits_{0}^{T}\|h_{xx}(\cdot,t)\|_{2}\;\|h_{x}(\cdot,t)\|_{2}^{3}\;dt$
$\displaystyle\hskip
28.90755pt\leqslant\tfrac{1}{2}\iint\limits_{Q_{T}}h_{xx}^{2}\;dxdt+\tfrac{b_{2}^{2}}{8}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{3}\;dt+\tfrac{b_{4}}{2}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n}+c_{1}\;T,$
(A.10)

where $c_{1}=M_{\delta\varepsilon}^{2(2m-n)}\;b_{5}/2$,
$m\geqslant\tfrac{n}{2}$. From (A.9), due to (A.10), we arrive at

$\displaystyle\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
$\displaystyle\hskip
14.45377pt\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+c_{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+c_{3}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{3}\;dt$
$\displaystyle\hskip
43.36243pt+c_{4}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n}\;dt+c_{5}\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}+c_{6}\;T$
$\displaystyle\hskip
14.45377pt\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+c_{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{1}}\right\\}\;dt$
(A.11)

where $\gamma_{1}=\max\\{3,2m-n\\}$, $m\geqslant\tfrac{n}{2}$,

$\displaystyle c_{2}=\tfrac{a_{1}^{2}}{4a_{0}},\quad
c_{3}=\tfrac{a_{1}^{2}b_{2}^{2}}{16a_{0}},\quad
c_{4}=\tfrac{a_{1}^{2}b_{4}}{4a_{0}},\
c_{5}=\tfrac{a_{1}^{2}}{2a_{0}}\tfrac{\delta}{\varepsilon^{2}}$ $\displaystyle
c_{6}=\tfrac{a_{1}^{2}}{2a_{0}}c_{1},\quad c_{7}=c_{3}+c_{4}+c_{5}+c_{6}.$

Now, multiplying (4.1) by $G^{\prime}_{\delta\varepsilon}(h)$, integrating
over $Q_{T}$, and using the periodic boundary conditions (4.2), we obtain

$\int\limits_{\Omega}{G_{\delta\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}=\int\limits_{\Omega}{G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{D^{\prime\prime}_{\varepsilon}(h)h^{2}_{x}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\,dx}+\tfrac{a_{1}}{\varepsilon}\iint\limits_{Q_{T}}{h^{2}_{x}\,dxdt}$
(A.12)

for limiting process on $\delta\to 0$, and

$\int\limits_{\Omega}{G_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{m-n}h^{2}_{x}\,dxdt},\
\ h_{\varepsilon}>0$ (A.13)

for limiting process on $\varepsilon\to 0$. In the case of (A.13), we estimate
the right-hand using the following equality

$\int\limits_{\Omega}{v^{a}v^{2}_{x}\,dx}=\tfrac{1}{a+1}\int\limits_{\Omega}{v^{a+1}v_{xx}\,dx},\
\ v>0,\ a>-1.$ (A.14)

Really, for $h_{\varepsilon}>0$ we deduce by the Cauchy inequality and bound
(A.4) that

$a_{1}\iint\limits_{Q_{T}}{h^{m-n}h^{2}_{x}\,dxdt}=-\tfrac{a_{1}}{m-n+1}\iint\limits_{Q_{T}}{h^{m-n+1}h_{xx}\,dxdt}\\\
\leqslant\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\iint\limits_{Q_{T}}{h^{2(m-n+1)}\,dxdt}\\\
\leqslant\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+c_{8}\int\limits_{0}^{T}\Bigl{(}\int\limits_{\Omega}h_{x}^{2}\,dx\Bigr{)}^{m-n+1}\,dt+c_{9}T,$
(A.15)

where $c_{8}=\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}b_{4}$,
$c_{9}=\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}b_{5}M_{\varepsilon}^{2(m-n+1)}$,
and $m\geqslant n-1$. Thus, from (A.12) due to (A.15), we deduce

$\int\limits_{\Omega}{G_{\varepsilon}(h(x,T))\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+c_{8}\int\limits_{0}^{T}\Bigl{(}\int\limits_{\Omega}h_{x}^{2}\,dx\Bigr{)}^{m-n+1}dt+c_{9}T\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+c_{10}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{2}}\right\\}\;dt,$
(A.16)

where $\gamma_{2}=\max\\{3,m-n+1\\}$, $m\geqslant n-1$, $c_{10}=c_{8}+c_{9}$.

Further, from (A.11) and (A.12) we find for limiting process on $\delta\to 0$

$\displaystyle\int\limits_{\Omega}{h_{x}^{2}\,dx}+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\delta\varepsilon}(h(x,T))\;dx+a_{0}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
$\displaystyle\hskip
7.22743pt\leqslant\int\limits_{\Omega}{h_{0,\delta\varepsilon,x}}^{2}\;dx+\tfrac{2c_{2}}{a_{0}}\left(\int\limits_{\Omega}G_{\delta\varepsilon}(h(x,T))\;dx+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}h_{xx}^{2}\;dxdt\right)$
$\displaystyle\hskip
7.22743pt+2c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt\leqslant\int\limits_{\Omega}{h_{0,\delta\varepsilon,x}}^{2}\;dx$
$\displaystyle\hskip
7.22743pt+\tfrac{2c_{2}}{a_{0}}\left(\int\limits_{\Omega}G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\;dx+\tfrac{a_{1}}{\varepsilon}\iint\limits_{Q_{T}}{h^{2}_{x}\,dxdt}\right)$
$\displaystyle\hskip
7.22743pt+2c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt\leqslant\int\limits_{\Omega}{h_{0,\delta\varepsilon,x}}^{2}\;dx$
(A.17) $\displaystyle\hskip
7.22743pt+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\;dx+c_{11}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt$

where $c_{11}=\tfrac{2c_{2}a_{1}}{\varepsilon a_{0}}+2c_{7}$,
$\gamma_{1}=\max\\{3,2m-n\\}$, $m\geqslant\tfrac{n}{2}$. Similarly, from
(A.11) and (A.16) we find for limiting process on $\varepsilon\to 0$

$\int\limits_{\Omega}{h_{x}^{2}\,dx}+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\varepsilon}(h(x,T))\;dx+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0,\varepsilon,x}}^{2}\;dx\\\
+\tfrac{2c_{2}}{a_{0}}\left(\int\limits_{\Omega}G_{\varepsilon}(h_{0,\varepsilon})\;dx+c_{10}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{2}}\right\\}\;dt\right)\\\
+2c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt\leqslant\int\limits_{\Omega}{h_{0,\varepsilon,x}}^{2}\;dx\\\
+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\varepsilon}(h_{0,\varepsilon})\;dx+c_{11}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt,$
(A.18)

where $c_{11}=\tfrac{2c_{2}c_{10}}{\varepsilon a_{0}}+2c_{7}$,
$\gamma_{1}=\max\\{3,2m-n,m-n+1\\}$, $m\geqslant\max\\{\tfrac{n}{2},n-1\\}$.

Applying the nonlinear Grönwall lemma [15] to

$v(T)\leqslant
v(0)+c_{11}\int\limits_{0}^{T}\max\\{1,v^{\gamma_{1}}(t)\\}\;dt$

with
$v(t)=\int(h_{x}^{2}(x,t)+2c_{2}/a_{0}\>G_{\delta\varepsilon}(h(x,t)))\;dx$
yields

$v(t)\leqslant\begin{cases}\begin{cases}v(0)+c_{11}t&\mbox{if}\quad
t<t_{0}:=\tfrac{1-v(0)}{c_{11}}\\\
\left(1-c_{11}(\gamma_{1}-1)(t-t_{0})\right)^{-1/(\gamma_{1}-1)}&\mbox{if}\quad
t\geqslant t_{0}\end{cases}&\mbox{if}\quad v(0)<1\\\
\left(v(0)^{1-\gamma_{1}}-c_{11}(\gamma_{1}-1)t\right)^{-1/(\gamma_{1}-1)}&\mbox{if}\quad
v(0)\geqslant 1.\end{cases}$

From this,

$\displaystyle\int\limits_{\Omega}\\{h_{x}^{2}(x,t)+\tfrac{2c_{2}}{a_{0}}G_{\delta\varepsilon}(h(x,t))\\}\;dx$
(A.19) $\displaystyle\hskip 28.90755pt\leqslant
2^{\tfrac{1}{\gamma_{1}-1}}\max\left\\{1,\int\limits_{\Omega}({h_{0,\delta\varepsilon,x}}^{2}(x)+\tfrac{2c_{2}}{a_{0}}G_{\delta\varepsilon}(h_{0,\delta\varepsilon}(x)))\;dx\right\\}=K_{\delta\varepsilon}<\infty$

for all $t\in[0,T_{\delta\varepsilon,loc}]$ where

$T_{\delta\varepsilon,loc}:=\tfrac{1}{2c_{11}(\gamma_{1}-1)}\min\left\\{1,\left(\int\limits_{\Omega}({h_{0,\delta\varepsilon,x}}^{2}(x)+\tfrac{2c_{2}}{a_{0}}G_{\delta\varepsilon}(h_{0,\delta\varepsilon}(x)))\,dx\right)^{-(\gamma_{1}-1)}\right\\}.$
(A.20)

Using the $\delta\to 0,\varepsilon\to 0$ convergence of the initial data and
the choice of $\theta\in(0,2/5)$ (see (4.5)) as well as the assumption that
the initial data $h_{0}$ has finite entropy (3.2), the times
$T_{\delta\varepsilon,loc}$ converge to a positive limit and the upper bound
$K$ in (A.19) can be taken finite and independent of $\delta$ and
$\varepsilon$ for $\delta$ and $\varepsilon$ sufficiently small. (We refer the
reader to the end of the proof of Lemma 6.1 in this Appendix for a fuller
explanation of a similar case.) Therefore there exists $\delta_{0}>0$ and
$\varepsilon_{0}>0$ and $K$ such that the bound (A.19) holds for all
$0\leqslant\delta<\delta_{0}$ and $0<\varepsilon<\varepsilon_{0}$ with $K$
replacing $K_{\delta\varepsilon}$ and for all

$0\leqslant t\leqslant T_{loc}:=\tfrac{9}{10}\lim_{\varepsilon\to 0,\delta\to
0}T_{\delta\varepsilon,loc}.$ (A.21)

Using the uniform bound on $\int h_{x}^{2}$ that (A.19) provides, one can find
a uniform-in-$\delta$-and-$\varepsilon$ bound for the right-hand-side of
(A.17) yielding the desired a priori bound (4.7). Similarly, one can find a
uniform-in-$\delta$-and-$\varepsilon$ bound for the right-hand-side of (A.16)
yielding the desired a priori bound (4.8).

To prove the bound (4.9), multiply (4.1) by
$-a_{0}h_{xx}-a_{1}D^{\prime}_{\varepsilon}(h)$, integrate over $Q_{T}$,
integrate by parts, use the periodic boundary conditions (4.2), to find

$\mathcal{E}_{\delta\varepsilon}(T)+\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})^{2}\,dxdt}=\mathcal{E}_{\delta\varepsilon}(0).$
(A.22)

The parameters $\delta_{0}$ and $\varepsilon_{0}$ are determined by $a_{0}$,
$a_{1}$, $|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, $\int h_{0}^{2-n}$, by
how quickly $M_{\delta\varepsilon}$ converges to $M$, and by how quickly the
approximate initial data (4.5), $h_{0,\delta\varepsilon}$, converge to $h_{0}$
in $H^{1}(\Omega)$.

The time $T_{loc}$ and the constants $K_{1}$, and $K_{2}$ are determined by
$\delta_{0}$, $\varepsilon_{0}$, $a_{0}$, $a_{1}$, $|\Omega|$, $\int h_{0}$,
$\|h_{0x}\|_{2}$, and $\int h_{0}^{2-n}$. ∎

###### Sketch of Proof of Lemma 4.2.

In the following, we denote the positive, classical solution $h_{\varepsilon}$
by $h$ whenever there is no chance of confusion.

Taking $\delta\to 0$ in (A.9) yields

$\displaystyle\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
$\displaystyle\leqslant\tfrac{1}{2}\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{|h|^{2m-n}h^{2}_{x}\,dxdt}$
$\displaystyle\leqslant\tfrac{1}{2}\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{2a_{0}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{2m-n}\;\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\,dt.$
(A.23)

Applying the nonlinear Grönwall lemma [15] to

$v(T)\leqslant v(0)+\int\limits_{0}^{T}A_{1}(t)\;v(t)\;dt$

with $v(t)=\int h_{x}^{2}(x,t)\;dx$,
$A_{1}(t)=a_{1}^{2}/a_{0}\|h(\cdot,t)\|_{\infty}^{2m-n}$ yields

$v(T)\leqslant v(0)\;e^{B_{1}(T)},\text{ where
}B_{1}(t)=\int\limits_{0}^{t}{A_{1}(s)\;ds}.$

Similarly, taking $\delta\to 0$ in (A.9) and (A.12), due to (A.14) and (A.6),
yield

$\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}{h^{2m-n}h^{2}_{x}\,dxdt}\\\
=\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}-\tfrac{a_{1}^{2}}{a_{0}(2m-n+1)}\iint\limits_{Q_{T}}{h^{2m-n+1}h_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\iint\limits_{Q_{T}}{h^{2(2m-n+1)}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{4m-n}\;\int\limits_{\Omega}G_{\varepsilon}(h(x,t))\;dx\,dt,$
(A.24)
$\int\limits_{\Omega}{G_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{m-n}h^{2}_{x}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}-\tfrac{a_{1}}{m-n+1}\iint\limits_{Q_{T}}{h^{m-n+1}h_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\iint\limits_{Q_{T}}{h^{2(m-n+1)}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{2m-n}\;\int\limits_{\Omega}G_{\varepsilon}(h(x,t))\;dx\,dt.$
(A.25)

Summing (A.24) and (A.25), we find that

$\int\limits_{\Omega}{\\{h_{x}^{2}(x,T)+G_{\varepsilon}(h(x,T))\\}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{\\{h_{0\varepsilon,x}^{2}+G_{\varepsilon}(h_{0,\varepsilon})\\}\,dx}+\int\limits_{0}^{T}A_{2}(t)\;\int\limits_{\Omega}\\{h_{x}^{2}(x,t)+G_{\varepsilon}(h(x,t))\\}\;dx\,dt,$
(A.26)

where
$A_{2}(t)=\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\|h(\cdot,t)\|_{\infty}^{4m-n}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\|h(\cdot,t)\|_{\infty}^{2m-n}$.
Applying the nonlinear Grönwall lemma [15] to

$v(T)\leqslant v(0)+\int\limits_{0}^{T}A_{2}(t)\;v(t)\;dt$

with $v(t)=\int\\{h_{x}^{2}(x,t)+G_{\varepsilon}(h(x,t))\\}\;dx$, yields

$v(T)\leqslant v(0)\;e^{B_{2}(T)},\text{ where
}B_{2}(t)=\int\limits_{0}^{t}{A_{2}(s)\;ds}.$

∎

###### Sketch of Proof of Lemma 4.3.

In the following, we denote the positive, classical solution $h_{\varepsilon}$
by $h$ whenever there is no chance of confusion.

Multiplying (4.1) with $\delta=0$ by $G^{\prime(\alpha)}_{\varepsilon}(h)$,
integrating over $Q_{T}$, taking $\delta\to 0$, and using the periodic
boundary conditions (4.2), yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}+a_{0}\tfrac{\alpha(1-\alpha)}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
(A.27)
$\displaystyle=\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{\alpha}D^{\prime\prime}_{\varepsilon}(h)h^{2}_{x}\,dxdt}\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}.$

Case 1: $0<\alpha<1$. The coefficient multiplying $\iint
h^{\alpha-2}h_{x}^{4}$ in (A.27) is positive and can therefore be used to
control the term $\iint h^{\alpha+m-n}h_{x}^{2}$ on the right–hand side of
(A.27). Specifically, using the Cauchy-Schwartz inequality and the Cauchy
inequality,

$a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}\leqslant
a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}\leqslant\\\
\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}+\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha+2(m-n+1)}\,dxdt}.$
(A.28)

Using the bound (A.28) in (A.27) yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}+a_{0}\tfrac{\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
(A.29)
$\displaystyle\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha+2(m-n+1)}\,dxdt}.$
(A.30)

By (A.4),

$\iint\limits_{Q_{T}}{h^{\alpha+2(m-n+1)}\,dxdt}\leqslant
b_{4}\int\limits_{0}^{T}{\left(\int\limits_{\Omega}{h^{2}_{x}\,dx}\right)^{\tfrac{\alpha}{2}+m-n+1}dt}+b_{5}M_{\varepsilon}^{\alpha+2(m-n+1)}\;T.$
(A.31)

Using (A.31) in (A.29) yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}+a_{0}\tfrac{\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leqslant$
$\displaystyle\hskip
14.45377pt\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+d_{1}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\;dt+d_{2}\;T$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+d_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}\;dt$
(A.32)

where

$d_{1}=b_{4}\;\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)},\quad
d_{2}=b_{5}\;\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)}\;M_{\varepsilon}^{\alpha+2(m-n+1)},\quad
d_{3}=d_{1}+d_{2}.$

Taking $\delta\to 0$ in (A.9) yields

$\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}{h^{2m-n}h^{2}_{x}\,dxdt}.$
(A.33)

Applying the Cauchy inequality,

$\displaystyle\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}h^{2m-n}h_{x}^{2}\;dxdt$
(A.34) $\displaystyle\hskip
14.45377pt\leqslant\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt+\tfrac{3a_{1}^{4}}{2a_{0}^{3}\alpha(1-\alpha)}\iint\limits_{Q_{T}}h^{2(2m-n+1)-\alpha}\;dxdt.$

By (A.4),

$\iint\limits_{Q_{T}}{h^{2(2m-n+1)-\alpha}\,dxdt}\leqslant
b_{4}\int\limits_{0}^{T}{\left(\int\limits_{\Omega}{h^{2}_{x}\,dx}\right)^{2m-n+1-\tfrac{\alpha}{2}}dt}+b_{5}M_{\varepsilon}^{2(2m-n+1)-\alpha}\;T.$
(A.35)

Using (A.34) and (A.35) in (A.33) yields

$\displaystyle\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
$\displaystyle\hskip
14.45377pt+d_{4}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\;dt+d_{5}\;T\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}$
(A.36) $\displaystyle\hskip
14.45377pt+\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}+d_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$

where

$d_{4}=\tfrac{3a_{1}^{4}}{2a_{0}^{3}\alpha(1-\alpha)}\;b_{4},\qquad
d_{5}=b_{5}\tfrac{3a_{1}^{4}}{2a_{0}^{3}\alpha(1-\alpha)}\;M_{\varepsilon}^{2(2m-n+1)-\alpha},\qquad
d_{6}=d_{4}+d_{5}.$

Add

$\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}$

to both sides of (A.36) and add

$a_{0}\iint\limits_{Q_{T}}h^{\alpha}h_{xx}^{2}\;dxdt$

to the right–hand side of the resulting inequality. Using (A.32) yields

$\displaystyle\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
(A.37) $\displaystyle\hskip
14.45377pt\leq\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+d_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}$
$\displaystyle\hskip
28.90755pt+d_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+d_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{3}}\right\\}$

where $d_{7}=d_{3}+d_{6}$,
$\gamma_{3}=\max\\{\alpha/2+m-n+1,2m-n+1-\alpha/2\\}$.

Applying the nonlinear Grönwall lemma [15] to

$v(T)\leqslant v(0)+d_{7}\int\limits_{0}^{T}\max\\{1,v^{\gamma_{3}}(t)\\}\;dt$

with $v(T)=\int h_{x}^{2}(x,T)+\>G_{\varepsilon}^{(\alpha)}(h(x,T))\;dx$
yields

$v(t)\leq\begin{cases}\begin{cases}v(0)+d_{7}t&\mbox{if}\quad
t<t_{0}:=\tfrac{1-v(0)}{d_{9}}\\\
\left(1-d_{7}(\gamma_{3}-1)(t-t_{0})\right)^{-\tfrac{1}{\gamma_{3}-1}}&\mbox{if}\quad
t\geqslant t_{0}\end{cases}&\mbox{if}\quad v(0)<1\\\
\left(v(0)^{1-\gamma_{3}}-d_{7}(\gamma_{3}-1)t\right)^{-\tfrac{1}{\gamma_{3}-1}}&\mbox{if}\quad
v(0)\geqslant 1\end{cases}$

From this,

$\displaystyle\int\limits_{\Omega}(h_{x}^{2}(x,T)+G_{\varepsilon}^{(\alpha)}(h(x,T)))\;dx$
(A.38) $\displaystyle\hskip 28.90755pt\leqslant
2^{\tfrac{1}{\gamma_{3}-1}}\max\left\\{1,\int\limits_{\Omega}({h_{0,\varepsilon}}_{x}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right\\}=K_{\varepsilon}<\infty$

for all

$0\leqslant T\leqslant
T_{\varepsilon,loc}^{(\alpha)}:=\tfrac{1}{2d_{7}(\gamma_{3}-1)}\min\left\\{1,\left(\int\limits_{\Omega}({h_{0,\varepsilon}}_{x}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right)^{-(\gamma_{3}-1)}\right\\}.$

The bound (A.38) holds for all $0<\varepsilon<\varepsilon_{0}$ where
$\varepsilon_{0}$ is from Lemma 4.1 and for all
$t\leq\min\\{T_{loc},T_{\varepsilon,loc}^{(\alpha)}\\}$ where $T_{loc}$ is
from Lemma 4.1.

Using the $\delta\to 0,\varepsilon\to 0$ convergence of the initial data and
the choice of $\theta\in(0,2/5)$ (see (4.5)) as well as the assumption that
the initial data $h_{0}$ has finite $\alpha$-entropy (3.11), the times
$T_{\varepsilon,loc}^{(\alpha)}$ converge to a positive limit and the upper
bound $K_{\varepsilon}$ in (A.38) can be taken finite and independent of
$\varepsilon$. (We refer the reader to the end of the proof of Lemma 6.1 in
this Appendix for a fuller explanation of a similar case.) Therefore there
exists $\varepsilon_{0}^{(\alpha)}$ and $K$ such that the bound (A.38) holds
for all $0<\varepsilon<\varepsilon_{0}^{(\alpha)}$ with $K$ replacing
$K_{\varepsilon}$ and for all

$0\leqslant t\leqslant
T_{loc}^{(\alpha)}:=\min\left\\{T_{loc},\tfrac{9}{10}\lim_{\varepsilon\to
0}T_{\varepsilon,loc}^{(\alpha)}\right\\}$ (A.39)

where $T_{loc}$ is the time from Lemma 4.1. Also, without loss of generality,
$\varepsilon_{0}^{(\alpha)}$ can be taken to be less than or equal to the
$\varepsilon_{0}$ from Lemma 4.1.

Using the uniform bound on $\int h_{x}^{2}$ that (A.38) provides, one can find
a uniform-in-$\varepsilon$ bound for the right-hand-side of (A.32) yielding
the desired bound

$\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}}\,dxdt+a_{0}\tfrac{\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leqslant
K_{1}$ (A.40)

which holds for all $0<\varepsilon<\varepsilon_{0}^{(\alpha)}$ and all
$0\leqslant T\leqslant T_{loc}^{(\alpha)}$.

It remains to argue that (A.40) implies that for all
$0<\varepsilon<\varepsilon_{0}^{(\alpha)}$ that $h_{\varepsilon}^{\alpha/2+1}$
and $h_{\varepsilon}^{\alpha/4+1/2}$ are contained in balls in
$L^{2}(0,T;H^{2}(\Omega))$ and $L^{2}(0,T;W^{1}_{4}(\Omega))$ respectively. It
suffices to show that

$\iint\limits_{Q_{T}}\left(h_{\varepsilon}^{\alpha/2+1}\right)^{2}_{xx}\;dxdt\leqslant
K,\qquad\iint\limits_{Q_{T}}\left(h_{\varepsilon}^{\alpha/4+1/2}\right)^{4}_{x}\;dxdt\leqslant
K$

for some $K$ that is independent of $\varepsilon$ and $T$. The integral
$\iint(h_{\varepsilon}^{\alpha/2+1})_{xx}^{2}$ is a linear combination of
$\iint h^{\alpha-2}h_{x}^{4}$, $\iint h^{\alpha-1}h_{x}^{2}h_{xx}$, and $\iint
h^{\alpha}h_{xx}^{2}$. Integration by parts and the periodic boundary
conditions imply

$\tfrac{1-\alpha}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}=\iint\limits_{Q_{T}}{h^{\alpha-1}h^{2}_{x}h_{xx}\,dxdt}$
(A.41)

Hence $\iint(h_{\varepsilon}^{\alpha/2+1})_{xx}^{2}$ is a linear combination
of $\iint h^{\alpha-2}h_{x}^{4}$, and $\iint h^{\alpha}h_{xx}^{2}$. By (A.40),
the two integrals are uniformly bounded independent of $\varepsilon$ and $T$
hence $\iint(h_{\varepsilon}^{\alpha/2+1})_{xx}^{2}$ is as well, yielding the
first part of (4.14).

The uniform bound of $\iint(h_{\varepsilon}^{\alpha/4+1/2})_{x}^{4}$ follows
immediately from the uniform bound of $\iint h^{\alpha-2}h_{x}^{4}$, yielding
the second part of (4.14).

Case 2: $-\tfrac{1}{2}<\alpha<0$. For $\alpha<0$ the coefficient multiplying
$\iint h^{\alpha-2}h_{x}^{4}$ in (A.27) is negative. However, we will show
that if $\alpha>-1/2$ then one can replace this coefficient with a positive
coefficient while also controlling the term $\iint h^{\alpha}h_{x}^{2}$ on the
right-hand side of (A.27).

Applying the Cauchy-Schwartz inequality to the right–hand side of (A.41),
dividing by $\sqrt{\iint h^{\alpha-2}h_{x}^{4}}$, and squaring both sides of
the resulting inequality yields

$\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leq\tfrac{9}{(1-\alpha)^{2}}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}\qquad\forall\alpha<1.$
(A.42)

Using (A.42) in (A.27) yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\tfrac{1+2\alpha}{1-\alpha}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
(A.43) $\displaystyle\hskip