# New dissipated energy for the unstable thin film equation

Marina Chugunova, Roman M. Taranets

###### Abstract

The fluid thin film equation
$h_{t}=-(h^{n}h_{xxx})_{x}-a_{1}\,(h^{m}h_{x})_{x}$ is known to conserve mass
$\int\,h\,dx$, and in the case of $a_{1}\leq~{}0$, to dissipate entropy
$\int\,h^{3/2-n}\,dx$ (see [8]) and the $L^{2}$-norm of the gradient
$\int\,h_{x}^{2}\,dx$ (see [3]). For the special case of $a_{1}=0$ a new
dissipated quantity $\int\,h^{\alpha}\,h_{x}^{2}\,dx$ was recently discovered
for positive classical solutions by Laugesen (see [15]). We extend it in two
ways. First, we prove that Laugesen’s functional dissipates strong nonnegative
generalized solutions. Second, we prove the full $\alpha$-energy
$\int\,\bigl{(}\tfrac{1}{2}\,h^{\alpha}\,h_{x}^{2}\,-$
$\tfrac{a_{1}\,h^{\alpha+m-n+2}}{(\alpha+m-n+1)(\alpha+m-n+2)}\bigr{)}\,dx$
dissipation for strong nonnegative generalized solutions in the case of the
unstable porous media perturbation $a_{1}>0$ and the critical exponent
$m=n+2$.

2000 MSC: 35K55, 35K35, 35Q35, 76D08

keywords: fourth-order degenerate parabolic equations, thin liquid films,
energy, entropy

##  1 Introduction

It is well known that analysis of the existence, uniqueness and regularity of
weak solutions for nonlinear evolution equations relies heavily on a priori
estimates. Often, the physical energy or entropy which originate from the
related model can provide non-increasing in time quantities. Unfortunately, it
is far from obvious how to construct new non-increasing Lyapunov type
functionals. A general algebraic approach to the construction of entropies in
higher-order nonlinear PDEs can be found in [14] and can be applied to analyse
thin film equations with stabilizing porus media type perturbations. In this
paper, inspired by Laugesen’s result [15] on dissipation, we prove that the
energy functional introduced in [15] dissipates strong nonnegative generalized
solutions. However, our method of the proof is only applicable to some subset
of the Laugsen’s dissipation region [15] (see the shaded area on Figure 1).

We study the longwave-unstable generalized thin film equation

$h_{t}=-(h^{n}\,h_{xxx})_{x}-a_{1}\,(h^{m}\,h_{x})_{x},$ (1.1)

where $h(x,t)$ gives the height of the evolving free-surface. The exponent $n$
plays a stabilizing role due to fourth-order forward diffusion term and the
exponent $m$ plays a destabilizing role due to backward second-order diffusion
term for the case when $a_{1}>0$. This class of equations originates from many
physical/industrial applications involving air-fluid interface. For example:
the case $n=1,\quad m=1$ describes a thin jet in a Hele-Shaw cell [10], the
case $n=3,\quad m=-1$ describes Van der Waals driven rupture of thin films
[19], the case $m=n=3$ describes shape of fluid droplets hanging from a
ceiling [11], and the case $n=0,\quad m=1$ describes solidification of a
hyper-cooled melt (this is a modified Kuramoto-Sivashinsky equation) [4].

To prove that the nonnegativity property is preserved in nonlinear thin film
equation $h_{t}=-(h^{n}\,h_{xxx})_{x}$ for $n\geq 1$ (case $a_{1}=0$) Bernis
and Friedman [3] used set of dissipated and conserved quantities: mass
conservation $\int\,h\,dx=M$, surface energy dissipation
$\frac{d}{dt}\int{h_{x}^{2}\,dx}\leq 0$, and entropy dissipation
$\frac{d}{dt}\int{h^{2-n}\,dx}\leq 0$. The new so-called $\beta$-entropy
$\int\,h^{2-n+\beta}\,dx$ was introduced by Bertozzi and Pugh [5] and
independently and simultaneously by Beretta, Bertsch, Dal Passo [1] to extend
this result to $n>0$. They also successfully used this new entropy to obtain
exponential with respect to the $L^{\infty}$-norm convergence toward the mean
value steady state solution. To analyse this convergence rate in $H^{1}$-norm
for the special case $n=1,\ a_{1}=0$ Carlen and Ulusoy [9] used the dissipated
energy $\int{h^{\alpha}\,h_{x}^{2}\,dx}$ constructed by Laugesen [15] for
classical positive solutions. Exponential asymptotic convergence toward the
mean value was also studied by Tudorascu in [18]. This list of connections
between new properties of solutions in thin film PDEs proved by means of newly
discovered dissipated quantities is far from complete.

In this paper we prove that there exists a subinterval $I$ of $-1<\alpha<~{}1$
($I$ depends on $n$ only) and a nonnegative strong generalized solution such
that for any $\alpha\in I$ the full $\alpha$-energy

$\mathcal{E}_{0}^{(\alpha)}(t)=\int\limits_{\Omega}{\left(\tfrac{1}{2}\,h^{\alpha}\,h_{x}^{2}\,-\tfrac{a_{1}\,h^{\alpha+m-n+2}}{(\alpha+m-n+1)(\alpha+m-n+2)}\right)\,dx}$

dissipates. For the unstable porus media perturbation case $a_{1}>0$ this
dissipation is proven under the assumptions that the total mass of the
solution is less than or equal to the critical one, $m=n+2$ and domain
$\Omega$ is unbounded or $h$ is compactly supported. For the stable case
$a_{1}\leq 0$ no such assumptions are needed.

We proceed as follows. First, we show the dissipation for the classical
solutions of the regularized problem and then we take this dissipation to the
limit. We prove dissipation of the full $\alpha$-energy for positive classical
solutions of the regularized problem for any value of the coefficient $a_{1}$
and without any additional assumptions about the total mass of the solution or
its support. However our method of taking the dissipation to the limit due to
the Bernis-Friedman method of regularization requires additional conditions
for the case $a_{1}>0$.

##  2 Auxiliary results to generalized weak solutions

We consider nonnegative weak solutions to the following initial–boundary
problem:

$\displaystyle(\textup{P})$ $\displaystyle
h_{t}+\left({h^{n}h_{xxx}+a_{1}h^{m}h_{x}}\right)_{x}=0\text{ in
}Q_{T},\hfill$ (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 $h=h(t,x)$, $\Omega=(-a,a)$, $Q_{T}=(0,T)\times\Omega$, $n>0$, $m>0$,
and $a_{1}\in\mathbb{R}^{1}$. We define a generalized weak solution in the
Bernis-Friedman sense (see, e. g. [1, 3]).

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

Let $n>0$, $m>0$, and $a_{1}\in\mathbb{R}^{1}$. A generalized weak solution of
problem $($P$)$ is a function $h$ satisfying

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

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

$\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt-\iint\limits_{\mathcal{P}}{h^{n}(h_{xxx}+a_{1}h^{m-n}h_{x})\phi_{x}\,dxdt}\
=0$ (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 [3]; there, the first
term was the integral of $h\phi_{t}$. The proof of the existence of
generalized weak solutions follows the ideas of [3, 1, 5, 6, 7, 17].

Let

$G_{0}^{(\beta)}(z):=\left\\{\begin{gathered}\tfrac{z^{\beta-n+2}}{(\beta-n+2)(\beta-n+1)}\text{
if }\beta-n\neq\\{-1,-2\\},\\\ z\ln z-z\text{ if }\beta-n=-1,\\\ -\ln z\text{
if }\beta-n=-2,\end{gathered}\right.$ (2.10)

$(G^{(\beta)}_{0}(z))^{\prime\prime}=z^{\beta-n}$, and
$G_{0}(z):=G_{0}^{(0)}(z)$.

######  Theorem 1.

Let $a_{1}\in\mathbb{R}^{1}$, $n>0$; $m\geqslant n/2$ for $a_{1}>0$, and $m>0$
for $a_{1}\leq 0$.

(a) [Existence.] Let the nonnegative initial data $h_{0}\in H^{1}(\Omega)$
satisfy

$\int\limits_{\Omega}{G_{0}(h_{0}(x))\,dx}<\infty,$ (2.11)

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,

$h\in L^{2}(0,T_{loc};H^{2}(\Omega)).$ (2.12)

Let

$\mathcal{E}_{0}(T):=\int\limits_{\Omega}{\\{\tfrac{1}{2}h_{x}^{2}(x,T)-a_{1}D_{0}(h(x,T))\\}\,dx},$
(2.13)

where $D_{0}(z):=\frac{z^{m-n+2}}{(m-n+1)(m-n+2)}$. Then the weak solution
satisfies

$\mathcal{E}_{0}(T)+\iint\limits_{\\{h>0\\}}{h^{n}(h_{xxx}+a_{1}h^{m-n}h_{x})^{2}\,dxdt}\leqslant\mathcal{E}_{0}(0),$
(2.14)
$\iint\limits_{\\{h>0\\}}{h^{n}h^{2}_{xxx}\,dxdt}\leqslant\,\,const<\infty.$
(2.15)

for all $T\leq T_{loc}$. The time of existence, $T_{loc}$, is determined by
$a_{1}$, $|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, and $\int G_{0}(h_{0})$.
Moreover, $T_{loc}=+\infty$ for $a_{1}\leq 0$.

(b) [Regularity.] If the initial data from (a) also satisfies

$\int\limits_{\Omega}{G^{(\beta)}_{0}(h_{0}(x))\,dx}<\infty$

for some $-1/2<\beta<1,\ \beta\neq 0$ then there exists
$0<T_{loc}^{(\beta)}\leq T_{loc}$ such that the nonnegative generalized weak
solution has the extra regularity

$h^{\tfrac{\beta+2}{2}}\in L^{2}(0,T_{loc}^{(\beta)};H^{2}(\Omega))\text{ and
}h^{\tfrac{\beta+2}{4}}\in L^{2}(0,T_{loc}^{(\beta)};W^{1}_{4}(\Omega)).$
(2.16)

The time of existence, $T_{loc}^{(\beta)}$, is determined by $a_{1}$,
$|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, and $\int G^{(\beta)}_{0}(h_{0})$.
Moreover, $T_{loc}^{(\beta)}=+\infty$ for $a_{1}\leq 0$.

There is nothing special about the time $T_{loc}$ in the Theorem 1. In the
case $a_{1}>0$ and $n/2\leq m<n+2$ (or $m=n+2$ and $M\leq M_{c}$), 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 [3] also holds: there exists a nonnegative weak
solution with the integral representation

$\displaystyle\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt+\iint\limits_{Q_{T}}(nh^{n-1}h_{x}h_{xx}\phi_{x}+h^{n}h_{xx}\phi_{xx})\;dxdt$
(2.17) $\displaystyle\hskip 108.405pt-
a_{1}\iint\limits_{Q_{T}}{h^{m}h_{x}\phi_{x}\;dxdt}=0.$

##  3 Dissipation of energy for nonnegative weak solutions

The main result of the present paper is the following

######  Theorem 2.

Let $a_{1}\in\mathbb{R}^{1}$, $1/2<n<3$; $m\geqslant n/2$ for $a_{1}>0$, and
$m>0$ for $a_{1}\leq 0$, and

$\mathcal{E}_{0}^{(\alpha)}(T):=\int\limits_{\Omega}{\\{\tfrac{1}{2}h^{\alpha}h_{x}^{2}(x,T)-a_{1}\tilde{D}_{0}(h(x,T))\\}\,dx},$
(3.1)

where
$\tilde{D}_{0}(z):=\frac{z^{\alpha+m-n+2}}{(\alpha+m-n+1)(\alpha+m-n+2)}$, and
$\mathcal{E}_{0}^{(0)}(T)=\mathcal{E}_{0}(T)$. Then there exists a non-empty
subinterval $I$ (see [15] for the explicit form of the $I$) of $0\leq\alpha<1$
for $\frac{1}{2}<n<3$, and of $\frac{3}{2}-n<\alpha<0$ for $\frac{3}{2}<n<3$
such that for any $\alpha\in I$ the nonnegative weak solution from Theorem 1
satisfies the following estimates:

(i) if $a_{1}\leqslant 0$ then

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0);$ (3.2)

(ii) if $a_{1}>0$ then

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{1}\iint\limits_{Q_{T}}{h^{\alpha+3m-2n+2}dxdt}\text{
for }m>n+2;$ (3.3)
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+T(C_{1}M^{\frac{2\alpha+5m-3n+4}{n+2-m}}+C_{2}M^{\alpha+3m-2n+2})$
(3.4)

for $m<n+2$ and $\alpha>2n-3m-1$;

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{3}T\,M^{\alpha+3m-2n+2}$
(3.5)

for $m<n+2$ and $2n-3m-2<\alpha\leqslant 2n-3m-1$;

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{2}T\,M^{\alpha+n+8}\text{
for }m=n+2\text{ and }0<M\leqslant M_{c}.$ (3.6)

Here $C_{2}=0$ if $\Omega$ is unbounded or $h$ has compact support.

######  Remark 3.1 (Extra Regularity).

In particular, the extra regularity $h^{\frac{\alpha+2}{2}}\in
L^{\infty}(0,T;H^{1}(\Omega))$ follows directly from Theorem 2. Hence,
$h^{\frac{\alpha+2}{2}}(\cdot,T)\in H^{1}(\Omega)$ for almost all
$T\in[0,T_{loc}^{(\beta)}]$ and therefore $h^{\frac{\alpha+2}{2}}(\cdot,T)\in
C^{1/2}(\overline{\Omega})$ for almost all $T\in[0,T_{loc}^{(\beta)}]$. Assume
that $T_{0}$ is chosen such that $h^{\frac{\alpha+2}{2}}(\cdot,T_{0})\in
C^{1/2}(\overline{\Omega})$ and $h(x_{0},T_{0})=0$ at some
$x_{0}\in\bar{\Omega}$. Then there exists a constant $L$ such that

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

Hence $h(x,T_{0})\leq L^{\frac{2}{\alpha+2}}|x-x_{0}|^{\frac{1}{\alpha+2}}$,
i. e. $h(.,T)\in C^{\frac{1}{\alpha+2}}(\bar{\Omega})$ for almost every
$T\in[0,T_{loc}^{(\beta)}]$.

######  Remark 3.2 (Rate of decrease).

For $a_{1}\leq 0$ and $1/2<n<3$ we can generalize the results from [9, Theorem
1.1] in the following way:

$\int\limits_{\Omega}{h^{\alpha}h_{x}^{2}(x,t)\,dx}\leq
C(1+t)^{-\frac{1}{2}}\text{ for }\tfrac{n-4}{2}\leq\alpha<0,$

whence $\|h-\bar{h}\|_{\infty}\leq C(1+t)^{-\frac{1}{4}}$ for any nonnegative
strong solution $h$. Here
$C=C(a_{1},\alpha,n,\bar{h},\mathcal{E}^{(\alpha)}_{0}(0))$, and
$\bar{h}=\frac{1}{|\Omega|}\|h_{0}\|_{1}$. The proof is similar to [9].

###  3.1 Regularized Problem

Given $\delta,\varepsilon>0$, a regularized parabolic problem, similar to that
of Bernis and Friedman [3], is considered:

$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle
h_{t}+\left({f_{\delta\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})}\right)_{x}=0,\qquad\hfill$
(3.7) $\displaystyle(\textup{P}_{\delta,\varepsilon})$
$\displaystyle\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)\text{ for
}t>0,\,i=\overline{0,3},\hfill$ (3.8)
$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle\qquad\qquad
h(x,0)=h_{0,\delta\varepsilon}(x),\hfill$ (3.9)

where

$f_{\delta\varepsilon}(z):=f_{\varepsilon}(z)+\delta=\tfrac{|z|^{s+n}}{|z|^{s}+\varepsilon|z|^{n}}+\delta,\
D^{\prime\prime}_{\varepsilon}(z):=\tfrac{|z|^{m-n}}{1+\varepsilon|z|^{m-n}}$
(3.10)

$\forall\,z\in\mathbb{R}^{1},\ \varepsilon>0,\ s\geqslant 4$. The $\delta>0$
in (3.10) makes the problem (3.7) 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 $f_{\varepsilon}(h)\sim
h^{s}$. The nonnegative initial data, $h_{0}$, is approximated via

$\begin{gathered}h_{0,\delta\varepsilon}\in C^{4+\gamma}(\Omega),\
h_{0,\delta\varepsilon}\geqslant h_{0,\delta}+\varepsilon^{\theta}\text{ for
some }0<\theta<\tfrac{2}{2s-3},\\\
\tfrac{\partial^{i}h_{0,\delta\varepsilon}}{\partial
x^{i}}(-a)=\tfrac{\partial^{i}h_{0,\delta\varepsilon}}{\partial
x^{i}}(a)\text{ for }i=\overline{0,3},\\\ h_{0,\delta\varepsilon}\to
h_{0}\text{ strongly in }H^{1}(\Omega)\text{ as }\delta,\varepsilon\to
0.\end{gathered}$ (3.11)

The $\varepsilon$ term in (3.11) ‘‘lifts’’ the initial data so that it will be
positive even if $\delta=0$ and the $\delta$ is involved in smoothing the
initial data from $H^{1}(\Omega)$ to $C^{4+\gamma}(\Omega)$.

_Sketch of Proof_ : By Eĭdelman [12, Theorem 6.3, p.302], the regularized
problem has the 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 values of $\delta$ and
$\varepsilon$, by Eĭdelman [12, Theorem 9.3, p.316] if one can prove a uniform
in time an a priori bound $|h_{\delta\varepsilon}(x,t)|\leq
A_{\delta\varepsilon}<\infty$ 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 [12, 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 1) $T_{loc,\delta\varepsilon}\to
T_{loc}>0$, 2) the solutions $h_{\delta\varepsilon}$ converge to a
(nonnegative) limit, $h$, which is a generalized weak solution, and 3) $h$
inherits certain a priori bounds. This is done by proving various a priori
estimates for $h_{\delta\varepsilon}$ 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 $\varepsilon\to 0$ will then result in the desired
generalized weak solution $h$. This last step is where the possibility of non-
unique weak solutions arise; see [1] for simple examples of how such
constructions applied to $h_{t}=-(|h|^{n}h_{xxx})_{x}$ can result in two
different solutions arising from the same initial data.

###  3.2 Dissipation of energy for positive solutions

Figure 1:  The dissipation region computed numerically by Matlab: $\alpha$
versus $n$. The dashed line corresponds to $\alpha=3/2-n$.

######  Lemma 3.1.

Let $\alpha$ belong to the full domain shown on Figure 1, and

$\mathcal{E}_{\varepsilon}^{(\alpha)}(T):=\int\limits_{\Omega}{\\{\tfrac{1}{2}h^{\alpha}h_{x}^{2}(x,T)-a_{1}\tilde{D}_{\varepsilon}(h(x,T))\\}\,dx},$
(3.12)

where
$\tilde{D}^{\prime\prime}_{\varepsilon}(z):=z^{\alpha}D^{\prime\prime}_{\varepsilon}(z)$.
Then the unique positive classical solution $h_{\varepsilon}$ of the problem
($P_{0,\varepsilon}$) satisfies

$\mathcal{E}^{(\alpha)}_{\varepsilon}(T)\leqslant\mathcal{E}^{(\alpha)}_{\varepsilon}(0)+\mu\iint\limits_{Q_{T}}{h^{\alpha-2}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}^{4}dxdt}+\\\
\varepsilon\,k_{1}\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}+\varepsilon^{2}k_{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx},$
(3.13)

where $k_{i}=k_{i}(\alpha,n,s)$ are constants, and $\mu=\mu(\alpha,a_{1})$
such that $\mu(0,a_{1})=0$ and $\mu(\alpha,a_{1})\leqslant 0$ for
$a_{1}\leqslant 0$.

Note that, although we use the same convenient notations introduced in [15],
the proof of Lemma 3.1 has essential differences from the proof of Theorem 1
of [15]. Indeed, we introduce new ideas in order to estimate the lower-order
term in the equation (3.7). In particular, the new quantity $N$ is introduced,
the quantity $R$ is modified, and so are the terms involving the
regularization parameter $\varepsilon$ in (3.19).

###### Proof of Lemma 3.1.

To prove the bound (3.13), multiply (3.7) with $\delta=0$ by
$-\frac{\alpha}{2}h^{\alpha-1}h_{x}^{2}-h^{\alpha}h_{xx}-a_{1}\tilde{D}^{\prime}_{\varepsilon}(h)$,
integrate over $\Omega$, use integration by parts, apply the periodic boundary
conditions (3.8), to find

$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)=-\int\limits_{\Omega}{h^{\alpha}f_{\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})^{2}\,dx}-\\\
\tfrac{\alpha}{2}(\alpha-1)\int\limits_{\Omega}{h^{\alpha-2}h_{x}^{3}f_{\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dx}-\\\
2\alpha\int\limits_{\Omega}{h^{\alpha-1}h_{x}h_{xx}f_{\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dx}.$
(3.14)

The equality (3.14) can be rewritten as

$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)=-R^{2}-2\alpha\,RS-\tfrac{\alpha}{2}(\alpha-1)\,RL,$
(3.15)

where the quantities

$R:=\langle(h^{\alpha}f_{\varepsilon}(h))^{1/2}(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})|=|(h^{\alpha}f_{\varepsilon}(h))^{1/2}(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\rangle,$
$S:=\langle(h^{\alpha-2}f_{\varepsilon}(h))^{1/2}h_{x}h_{xx}|=|(h^{\alpha-2}f_{\varepsilon}(h))^{1/2}h_{x}h_{xx}\rangle,$
$L:=\langle(h^{\alpha-4}f_{\varepsilon}(h))^{1/2}h_{x}^{3}|=|(h^{\alpha-4}f_{\varepsilon}(h))^{1/2}h_{x}^{3}\rangle,$
$N:=\langle(h^{\alpha-2}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h))^{1/2}h_{x}^{2}|=|(h^{\alpha-2}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h))^{1/2}h_{x}^{2}\rangle,$

each represent half of an inner product in $L^{2}(\Omega)$. We will need the
following integration by parts formulas

$SL=-\tfrac{1}{5}(\alpha-3)\int\limits_{\Omega}{h^{\alpha-4}f_{\varepsilon}(h)h_{x}^{6}\,dx}-\tfrac{1}{5}\int\limits_{\Omega}{h^{\alpha-3}f^{\prime}_{\varepsilon}(h)h_{x}^{6}\,dx}=\\\
-\tfrac{1}{5}(\alpha+n-3)L^{2}-\tfrac{1}{5}\varepsilon(s-n)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx},$
(3.16)
$RL=-(\alpha+n-2)SL-\varepsilon(s-n)\int\limits_{\Omega}{h^{\alpha-s-3}f_{\varepsilon}^{2}(h)h_{x}^{4}h_{xx}\,dx}-3S^{2}+a_{1}N^{2}=\\\
\tfrac{1}{5}(\alpha+n-2)(\alpha+n-3)L^{2}-3S^{2}+a_{1}N^{2}-\varepsilon(s-n)\int\limits_{\Omega}{h^{\alpha-s-3}f_{\varepsilon}^{2}(h)h_{x}^{4}h_{xx}\,dx}+\\\
\tfrac{1}{5}\varepsilon(s-n)(\alpha+n-2)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}=\tfrac{1}{5}(\alpha+n-2)(\alpha+n-3)L^{2}-3S^{2}+\\\
a_{1}N^{2}+\tfrac{1}{5}\varepsilon(s-n)(2\alpha+3n-s-5)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}+\\\
\tfrac{2}{5}\varepsilon^{2}(s-n)^{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx}.$
(3.17)

Here we use the auxiliary equality
$f^{\prime}_{\varepsilon}(z)=nz^{-1}f_{\varepsilon}(z)+\varepsilon(s-n)z^{-(s+1)}f_{\varepsilon}^{2}(z)$.
Thus, from (3.15) we have

$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)+\tfrac{\varepsilon^{2}}{5}\alpha(\alpha-1)(s-n)^{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx}+\tfrac{a_{1}}{2}\alpha(\alpha-1)N^{2}=\\\
-R^{2}-2\alpha\,RS-\tfrac{\alpha}{10}(\alpha-1)(\alpha+n-2)(\alpha+n-3)L^{2}+\tfrac{3\alpha}{2}(\alpha-1)S^{2}+\\\
\tfrac{\varepsilon}{10}\alpha(\alpha-1)(s-n)(s-2\alpha-3n+5)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}.$
(3.18)

Our next step is to express (3.18) as the negative of a sum of squares to
obtain the energy dissipation. To achieve this, we use (3.16) and (3.17) to
deduce that for all $\kappa\in\mathbb{R}^{1}$,

$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)=-(R+\alpha\,S+\kappa\,L)^{2}+\beta(S+\tfrac{1}{5}(\alpha+n-3)\,L)^{2}+\gamma\,L^{2}+\mu\,N^{2}+\\\
\varepsilon\,k_{1}\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}+\varepsilon^{2}k_{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx},$
(3.19)

where

$k_{1}=\tfrac{2}{25}(s-n)\biggl{(}5\kappa(\alpha-s+3n-5)+\tfrac{5\alpha}{4}(\alpha-1)(s-2\alpha-3n+5)+\\\
(\alpha+n-3)(\tfrac{\alpha}{2}(5\alpha-3)-6\kappa)\biggr{)},\
k_{2}=\tfrac{1}{5}(s-n)^{2}\biggl{(}4\kappa-\alpha(\alpha-1)\biggr{)},$
$\beta=\tfrac{\alpha}{2}(5\alpha-3)-6\kappa,\,\
\mu=a_{1}\biggl{(}2\kappa-\tfrac{\alpha}{2}(\alpha-1)\biggr{)},$
$\gamma=\kappa^{2}-\tfrac{2}{25}\kappa(\alpha+n-3)\bigl{(}5(2-n)+3(\alpha+n-3)\bigr{)}-\\\
\tfrac{\alpha}{50}(\alpha+n-3)\bigl{(}5(\alpha-1)(\alpha+n-2)-(5\alpha-3)(\alpha+n-3)\bigr{)}=\\\
\kappa^{2}-\tfrac{6}{25}\kappa(\alpha+n-3)\bigl{(}\alpha-\tfrac{2n-1}{3}\bigr{)}-\tfrac{3}{50}\alpha(\alpha+n-3)\bigl{(}\alpha-\tfrac{2n-1}{3}\bigr{)}.$

Now we have to choose the parameter $\kappa$ in such a way that
$\beta\leqslant 0$ and $\gamma\leqslant 0$. In this case, the parameter
$\mu>0$ for $a_{1}>0$, and $\mu\leqslant 0$ for $a_{1}\leqslant 0$. According
to [15], we can find $\kappa$ such that $\beta\leqslant 0$, and
$\gamma\leqslant 0$ when $1/2<n<3$, see also Figure 1 where this region was
computed numerically by Matlab (see [15] for the explicit form of the domain).

###  3.3 Limit process in (3.13)

Rewrite the integral $\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}$
in the form

$\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}=\iint\limits_{Q_{T}}{\tfrac{\varepsilon\,h_{\varepsilon}^{\alpha+s-n}}{(h_{\varepsilon}^{s-n}+\varepsilon)^{2}}h_{\varepsilon}^{n-4}h_{\varepsilon,x}^{6}\,dxdt}.$

Using the Young’s inequality

$ab\leqslant\tfrac{a^{p}}{p}+\tfrac{b^{q}}{q}\Rightarrow p\,ab\leqslant
a^{p}+(p-1)b^{q},\ \tfrac{1}{p}+\tfrac{1}{q}=1,$ (3.20)

with $a=z^{\frac{s-n}{p}}$ and
$b=\bigl{(}\frac{\varepsilon}{p-1}\bigr{)}^{\frac{1}{q}}$, we deduce

$\varepsilon\tfrac{z^{\alpha+s-n}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\varepsilon\tfrac{z^{\alpha}}{z^{s-n}+\varepsilon}\leqslant\tfrac{(p-1)^{\frac{1}{q}}}{p}\tfrac{\varepsilon\,z^{\alpha}}{z^{\frac{s-n}{p}}\varepsilon^{\frac{1}{q}}}=\tfrac{(p-1)^{\frac{1}{q}}}{p}\varepsilon^{\frac{q-1}{q}}z^{\frac{p\alpha-s+n}{p}},$

choosing $p=\frac{s-n}{\alpha}>1$ and $q=\frac{s-n}{s-n-\alpha}>1$
($\Rightarrow 0<\alpha<s-n$), we find

$\varepsilon\tfrac{z^{\alpha+s-n}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\tfrac{\alpha}{s-n}\bigl{(}\tfrac{s-n-\alpha}{\alpha}\bigr{)}^{\frac{s-n-\alpha}{s-n}}\varepsilon^{\frac{\alpha}{s-n}}.$

Similarly, we deal with the integral
$\varepsilon^{2}\int\limits_{\Omega}{h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{\varepsilon,x}^{6}\,dx}$.
Due to Lemma A.1 and (2.15),
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{n-4}h_{\varepsilon,x}^{6}\,dxdt}$ is
uniformly bounded then

$\Bigl{|}\iint\limits_{Q_{T}}{(k_{1}\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})+k_{2}\varepsilon^{2}h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h_{\varepsilon}))h_{\varepsilon,x}^{6}\,dxdt}\Bigr{|}\leqslant\\\
C\,\varepsilon^{\frac{\alpha}{s-n}}\iint\limits_{Q_{T}}{h_{\varepsilon}^{n-4}h_{\varepsilon,x}^{6}\,dxdt}\leqslant
C\,\varepsilon^{\frac{\alpha}{s-n}},$ (3.21)

where the positive constant $C$ is independent of $\varepsilon$. Letting
$\varepsilon\to 0$, from (3.21) we obtain

$(k_{1}\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})+k_{2}\varepsilon^{2}h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h_{\varepsilon}))h_{\varepsilon,x}^{6}\to
0\text{ in }L^{1}(Q_{T})$ (3.22)

for $0<\alpha<s-n$.

Now, we show (3.22) for the case of $\alpha<0$. Rewrite the integral
$\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}$
in the form

$\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}=\iint\limits_{Q_{T}}{\tfrac{\varepsilon\,h_{\varepsilon}^{s-2}}{(h_{\varepsilon}^{s-n}+\varepsilon)^{2}}h_{\varepsilon}^{\alpha-2}h_{\varepsilon,x}^{6}\,dxdt}.$

Using the inequality (3.20) with $a=z^{\frac{s-n}{p}}$ and
$b=\bigl{(}\frac{\varepsilon}{p-1}\bigr{)}^{\frac{1}{q}}$, we obtain

$\varepsilon\tfrac{z^{s-2}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\tfrac{(p-1)^{\frac{2}{q}}}{p^{2}}\tfrac{\varepsilon\,z^{s-2}}{z^{\frac{2(s-n)}{p}}\varepsilon^{\frac{2}{q}}}=\tfrac{(p-1)^{\frac{2}{q}}}{p^{2}}\varepsilon^{\frac{q-2}{q}}z^{\frac{p(s-2)-2(s-n)}{p}},$

choosing $p=\frac{2(s-n)}{s-2+\alpha-\beta}<2$ and
$q=\frac{2(s-n)}{2(s-n)-s+2-\alpha+\beta}>2$ ($\Rightarrow n>2-\alpha+\beta$),
we find

$\varepsilon\tfrac{z^{s-2}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\bigl{(}\tfrac{s-2+\alpha-\beta}{2(s-n)}\bigr{)}^{2}\bigl{(}\tfrac{s-2(n-1)-\alpha+\beta}{s-2+\alpha-\beta}\bigr{)}^{\frac{s-2(n-1)-\alpha+\beta}{s-n}}\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}}z^{\beta-\alpha},$

where $\beta\in(-1/2,1)$ follows from (2.16). Similarly, we deal with the
integral
$\varepsilon^{2}\int\limits_{\Omega}{h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{\varepsilon,x}^{6}\,dx}$.
Due to $h\in L^{\infty}(0,T;H^{1}(\Omega))$ and (2.16),
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{\alpha-2}h_{\varepsilon,x}^{6}\,dxdt}$
is uniformly bounded then

$\Bigl{|}\iint\limits_{Q_{T}}{(k_{1}\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})+k_{2}\varepsilon^{2}h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h_{\varepsilon}))h_{\varepsilon,x}^{6}\,dxdt}\Bigr{|}\leqslant\\\
C\,\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}}\iint\limits_{Q_{T}}{h_{\varepsilon}^{\beta-2}h_{\varepsilon,x}^{6}\,dxdt}\leqslant
C\,\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}}\iint\limits_{Q_{T}}{h_{\varepsilon}^{\beta-2}h_{\varepsilon,x}^{4}\,dxdt}\leqslant\\\
C\,\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}},$ (3.23)

where the positive constant $C$ is independent of $\varepsilon$. Letting
$\varepsilon\to 0$, we obtain (3.22) for $\frac{3}{2}-n<\alpha<0$ and
$\frac{3}{2}<n<3$. In view of the Lebesgue’s theorem, we have

$\iint\limits_{Q_{T}}{h_{\varepsilon}^{\alpha-2}f_{\varepsilon}(h_{\varepsilon})D^{\prime\prime}_{\varepsilon}(h_{\varepsilon})h_{\varepsilon,x}^{4}\,dxdt}\to\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}\,dxdt}$
(3.24)

if $m>0$ and $\alpha>-\frac{1}{2}-m$, due to $h\in
L^{\infty}(0,T;H^{1}(\Omega))$ and (2.16).

Integrating (3.19) over the time interval, and letting $\varepsilon\to 0$, in
view of (3.22) and (3.24), we obtain (3.13) for some subinterval $I$ for
$0\leq\alpha<1$ and $\frac{1}{2}<n<3$ or for $-1<\alpha<0$ and
$\frac{3}{2}<n<3$. Note that, the convergence on the left-hand side follows
from Fatou’s lemma and from the corresponding a priori estimate (see, for
example, [3, 5, 7, 17]). ∎

###  3.4 Proof of Theorem 2

Taking the limit $\varepsilon\to 0$ we obtain

$\mathcal{E}^{(\alpha)}_{0}(T)+\gamma\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+\mu\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}.$
(3.25)

Now, we estimate $\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}$. Using
the Hölder inequality, we obtain

$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant\int\limits_{0}^{T}{\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{2}{3}}\Bigl{(}\int\limits_{\Omega}{h^{\alpha+3m-2n+2}dx}\Bigr{)}^{\frac{1}{3}}dt}.$
(3.26)

Applying Lemma A.2 to $v=h^{\frac{\alpha+n+2}{6}}$ with
$a=\frac{6(\alpha+3m-2n+2)}{\alpha+n+2}$, $d=6$, $b=\frac{6}{\alpha+n+2}<a\
(\Rightarrow\alpha>2n-3m-1)$, $i=0$, and $j=1$, we deduce

$\int\limits_{\Omega}{h^{\alpha+3m-2n+2}dx}\leqslant
d_{1}\Bigl{(}\int\limits_{\Omega}{v_{x}^{6}dx}\Bigr{)}^{\frac{\alpha+3m-2n+1}{\alpha+n+7}}\Bigl{(}\int\limits_{\Omega}{h\,dx}\Bigr{)}^{\frac{3(2\alpha+5m-3n+4)}{\alpha+n+7}}+\\\
d_{2}\Bigl{(}\int\limits_{\Omega}{h\,dx}\Bigr{)}^{\alpha+3m-2n+2}\leqslant
c_{1}M^{\frac{3(2\alpha+5m-3n+4)}{\alpha+n+7}}\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{\alpha+3m-2n+1}{\alpha+n+7}}+\\\
c_{2}M^{\alpha+3m-2n+2}.$ (3.27)

Substituting (3.27) in (3.26), we find

$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant
c_{1}M^{\frac{2\alpha+5m-3n+4}{\alpha+n+7}}\int\limits_{0}^{T}{\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{\alpha+m+5}{\alpha+n+7}}dt}+\\\
c_{2}M^{\frac{\alpha+3m-2n+2}{3}}\int\limits_{0}^{T}{\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{2}{3}}dt}.$
(3.28)

If $m<n+2$ then, using Young’s inequality, from (3.28) we arrive at

$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant\epsilon\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+\\\
C(\epsilon)T(c_{1}M^{\frac{2\alpha+5m-3n+4}{n+2-m}}+c_{2}M^{\alpha+3m-2n+2}).$
(3.29)

Substituting (3.29) in (3.25), and choosing $\epsilon$ small enough, we obtain

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+T(C_{1}M^{\frac{2\alpha+5m-3n+4}{n+2-m}}+C_{2}M^{\alpha+3m-2n+2})$
(3.30)

for $\alpha>2n-3m-1$ and $m<n+2$. Here $C_{2}=0$ if $\Omega$ is unbounded or
$h$ is compactly supported. In particular, if $0<\alpha+3m-2n+2\leqslant 1$,
i. e. $2n-3m-2<\alpha\leqslant 2n-3m-1$ then, using the Hölder inequality and
applying Young’s inequality, from (3.26) we obtain

$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant\epsilon\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+\\\
C(\epsilon)|\Omega|^{2n-3m-1-\alpha}T\,M^{\alpha+3m-2n+2}$ (3.31)

for $2n-3m-2<\alpha\leqslant 2n-3m-1$. Substituting (3.31) in (3.25), and
choosing $\epsilon$ small enough, we obtain

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{3}T\,M^{\alpha+3m-2n+2}.$
(3.32)

If $m=n+2$ then, using Young’s inequality, from (3.28) we deduce

$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant
c_{1}M^{2}\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+\\\
\epsilon\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+C(\epsilon)T\,M^{\alpha+n+8}.$
(3.33)

Substituting (3.33) in (3.25), and choosing $\epsilon$ enough small, we obtain

$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{2}T\,M^{\alpha+n+8}$
(3.34)

for $\alpha>-n-7$, $m=n+2$ and $M\leqslant M_{c}$. Here $C_{2}=0$ if $\Omega$
is unbounded or $h$ is compactly supported.

Acknowledgement. The authors thank R. S. Laugesen and A. Burchard for useful
comments and discussions. The research of M. Chugunova is supported by the
NSERC Postdoctoral Fellowship. R. M. Taranets would like to thank M. C. Pugh
for the hospitality of the University of Toronto.

## Appendix A

######  Lemma A.1.

([13, 2]) Let $\Omega\subset\mathbb{R}^{N},\ N<6$, be a bounded convex domain
with smooth boundary, and let
$n\in\bigl{(}2-\sqrt{1-\tfrac{N}{N+8}},3\bigr{)}$ for $N>1$, and
$\frac{1}{2}<n<3$ for $N=1$. Then the following estimate holds for any
positive functions $v\in H^{2}(\Omega)$ such that $\nabla v\cdot\vec{n}=0$ on
$\partial\Omega$ and $\int\limits_{\Omega}{v^{n}|\nabla\Delta v|^{2}}<\infty$:

$\int\limits_{\Omega}{\varphi^{6}\\{v^{n-4}|\nabla
v|^{6}+v^{n-2}|D^{2}v|^{2}|\nabla v|^{2}\\}}\leqslant\\\
c\Bigl{\\{}\int\limits_{\Omega}{\varphi^{6}v^{n}|\nabla\Delta
v|^{2}}+\int\limits_{\\{\varphi>0\\}}{v^{n+2}|\nabla\varphi|^{6}}\Bigr{\\}},$

where $\varphi\in C^{2}(\Omega)$ is an arbitrary nonnegative function such
that the tangential component of $\nabla\varphi$ is equal to zero on
$\partial\Omega$, and the constant $c>0$ is independent of $v$.

######  Lemma A.2.

([16]) If $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with piecewise-
smooth boundary, $a>1$, $b\in(0,a),\ d>1,$ and $0\leqslant i<j,\
i,j\in\mathbb{N}$, then there exist positive constants $d_{1}$ and $d_{2}$
$(d_{2}=0\text{ if }\Omega$ is unbounded$)$ depending only on $\Omega,\ d,\
j,\ b,$ and $N$ such that the following inequality is valid for every $v(x)\in
W^{j,d}(\Omega)\cap L^{b}(\Omega)$:

$\left\|{D^{i}v}\right\|_{L^{a}(\Omega)}\leqslant
d_{1}\left\|{D^{j}v}\right\|_{L^{d}(\Omega)}^{\theta}\left\|v\right\|_{L^{b}(\Omega)}^{1-\theta}+d_{2}\left\|v\right\|_{L^{b}(\Omega)},\
\theta=\frac{{\tfrac{1}{b}+\tfrac{i}{N}-\tfrac{1}{a}}}{{\tfrac{1}{b}+\tfrac{j}{N}-\tfrac{1}{d}}}\in\left[{\tfrac{i}{j},1}\right)\\!\\!.$

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