\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 81, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/81\hfil Convergence to equilibrium]
{Convergence to equilibrium of relatively compact solutions
to evolution equations}

\author[T. B\'arta \hfil EJDE-2014/81\hfilneg]
{Tom\'a\v{s} B\'arta}  % in alphabetical order

\address{Tom\'a\v{s} B\'arta  \newline
Department of Mathematical Analysis,
Faculty of Mathematics and Physics,
Charles University,
Sokolovsk\'a 83, 186 75 Prague 8, Czech Republic}
\email{barta@karlin.mff.cuni.cz}

\thanks{Submitted Oactober 29, 2013. Published March 21, 2014.}
\subjclass[2000]{35R20, 35B40, 34D05, 34G20}
\keywords{Convergence to equilibrium; gradient system; 
\hfil\newline\indent Kurdyka-{\L}ojasiewicz
gradient inequality;  gradient-like system}

\begin{abstract}
 We prove convergence to equilibrium for relatively compact solutions
 to an  abstract evolution equation satisfying energy estimates near
 the omega-limit set. These energy estimates generalize {\L}ojasiewicz
 and Kurdyka-{\L}ojasiewicz-Simon gradient inequalities.
 We apply the abstract results to several ODEs and PDEs of first and
 second order.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

Convergence results of the type ``if $\varphi$ is in the omega-limit set of 
$u:\mathbb{R}_+\to X$ and a condition (C) holds, then $\lim_{t\to+\infty} u(t) = \varphi$'' 
have been extensively studied (see, e.g., Haraux and Jendoubi \cite{HJ98}, 
Albis et al.\ \cite{AMA05}, Chill et al.\ \cite{CHJ09}, Lageman \cite{Lag07}, 
Chergui \cite{Cher08,Cher09}, B\'arta et al.\ \cite{BCF12}).
 Each of the proofs of these results can be split into two parts: 
 the first part shows the key estimate
\begin{equation}	\label{key_est}
-\frac{d}{dt}\mathbb{E}(u(t)) \ge c\|\dot u(t)\|
\end{equation}
for some function $\mathbb{E}: X\to \mathbb{R}$ and the second part proves convergence 
with help of this estimate.

The second part of the proofs is always the same 
(see proof of Theorem~\ref{main_generalized} below or corresponding 
parts of proofs in the articles mentioned above). 
The first part follows from condition (C). 
Examples of condition (C) are the {\L}ojasiewicz inequality
\begin{equation}			\label{loj}
|\mathbb{E}(u)-\mathbb{E}(\varphi)|^{1-\theta}\le c \|\mathbb{E}'(u)\| \quad\text{for all $u$ near $\varphi$}
\end{equation}
or the more general Kurdyka-{\L}ojasiewicz-Simon inequality
\begin{equation}			\label{kur-loj}
\Theta(|\mathbb{E}(u)-\mathbb{E}(\varphi)|)\le c \|\mathbb{E}'(u)\| \quad\text{for all $u$ near $\varphi$}.
\end{equation}


If $u$ is a solution to the ordinary differential equation
\begin{equation}		\label{ODE}
\dot u + F(u) = 0,
\end{equation}
one can write
\begin{equation}	\label{orbital_der}
-\frac{d}{dt}\mathbb{E}(u(t)) = - \langle \mathbb{E}'(u(t)), \dot u(t)\rangle = \langle \mathbb{E}'(u(t)), F(u(t))\rangle.
\end{equation}
In many important examples (e.g. if \eqref{ODE} is a gradient system 
with $F=\nabla \mathbb{E}$) one can continue with
\begin{equation}			\label{AC}
\langle \mathbb{E}'(u(t)), F(u(t))\rangle \ge c \|\mathbb{E}'(u(t))\|\cdot \|F(u(t))\|.
\end{equation}
This inequality is known as angle condition and it plays an important 
role in proving \eqref{key_est}.

For partial differential equations, the situation is more complicated since 
we usually have $\mathbb{E}':V\to V'$ and $\dot u$ has values in $V'$. 
So, already the first equality in \eqref{orbital_der} is often unclear, 
since the expression on the right-hand side has no meaning.

Therefore, it seems to be a good idea to formulate a general convergence 
result assuming that \eqref{key_est} holds and then study, under which 
conditions \eqref{key_est} holds. Another reason for this splitting is 
that \eqref{key_est} is equivalent to the fact that $u$ has finite length 
(and all the mentioned convergence results are based on proving that $u$ has 
finite length).

Let us mention that another approach to convergence of (weak) 
solutions of first and second order evolution equations with maximal monotone 
operators can be found in the works by Djafari Rouhani and his co-workers, 
see \cite{RH10} and references therein.

In Section 2 we formulate and prove general convergence results assuming 
that \eqref{key_est} holds. In Sections 3 and 4 we give several applications 
to first and second order equations, respectively. Although the results 
in Sections 3 and 4 are known, we present some proofs to illustrate the 
applicability of the results in Section 2.

\section{General convergence results}

Before we formulate and prove the main results, we introduce some notations. 
Let $V$, $H$, be Hilbert spaces such that $V\hookrightarrow H\hookrightarrow V'$. 
Then $\|\cdot\|$,  $\|\cdot\|_V$,  $\|\cdot\|_*$ will be the norms 
in $H$, $V$, and $V'$, respectively. Corresponding scalar products will be 
denoted by the same subscripts. The open ball in $V$ of radius $r$ 
centered at $\phi\in V$ is denoted by $B_V(\phi,r)$.

If $u:\mathbb{R}_+\to V$ then the omega-limit set of $u$ in $V$ is
$$
\omega_V(u):=\{\phi\in V:\ \exists t_n\nearrow +\infty\text{ such that } 
\|u(t_n)-\phi\|_V\to 0\}.
$$
We say that $u\in C^1(\mathbb{R}_+,H)$ \textit{has finite length} in $H$ if
 $\int_0^{+\infty}\|\dot u(s)\|\,\mathrm{d} s<+\infty$.

We say that a function $\mathbb{E}$ \textit{satisfies {\L}ojasiewicz 
(or Simon-{\L}ojasiewicz) inequality on a neighborhood of $\varphi$}, 
if there exists $\theta\in (0,1/2]$ and $c>0$ such that \eqref{loj} holds 
(`$u$ near $\phi$' means $u\in B_V(\phi,\varepsilon)$ for some $\varepsilon>0$).
 We say that $\mathbb{E}$ \textit{satisfies Kurdyka-{\L}ojasiewicz-Simon inequality
  on a neighborhood of $\varphi$}, if there exists $c>0$ and a function 
  $\Theta\in C([0,+\infty))$ satisfying $\Theta(s)>0$ for all $s>0$, 
$1/\Theta\in L^1_{loc}([0,+\infty))$ and condition~\eqref{kur-loj}. 
We will call functions $\Theta$ with the above properties 
\textit{Kurdyka functions}. Taking $\Theta(s)=s^{1-\theta}$ yields that 
{\L}ojasiewicz inequality is a special case of Kurdyka-{\L}ojasiewicz-Simon
 inequality. If $\Theta$ is a Kurdyka function, we define 
$\Phi_{\Theta}(t):=\int_0^t 1/\Theta(s)\,\mathrm{d} s$.

The following are well known results.

\begin{lemma}
If $u$ has finite length in $H$, then it has a limit in $H$.
\end{lemma}

\begin{lemma}		\label{l-two}
Let $u:\mathbb{R}_+\to V$. If $\lim_{t\to+\infty} u(t)= \psi$ in $H$ and $u$ 
has precompact range in $V$, then $\lim_{t\to+\infty}u(t)= \psi$ in $V$.
\end{lemma}

\begin{lemma}
Let $u:\mathbb{R}_+\to V$. If $u$ has finite length in $H$ and precompact range in $V$, 
then it converges in $V$ (as $t\to+\infty$).
\end{lemma}

We formulate the general convergence result proposed in the introduction. 
Its proof follows immediately from Theorem \ref{main_generalized}. 
Let us emphasize that $H$ can be an arbitrarily large space. 
So, in the applications, it is sufficient to verify \eqref{key_est} with
 a very weak norm on the right-hand side.

\begin{theorem}      \label{main}
Let $u\in C(\mathbb{R}_+,V)\cap C^1(\mathbb{R}_+,H)$ with $V$-precompact range and 
$\varphi\in \omega_V(u)$. Let $\rho>0$ and
$\mathbb{E}\in C(V,\mathbb{R})$ be such that $t\mapsto \mathbb{E}(u(t))$ is nonincreasing on $\mathbb{R}_+$ and
\eqref{key_est} holds for almost every 
$t\in \{s\in\mathbb{R}_+:\ u(s)\in B:=B_V(\varphi, \rho)\}$. 
Then $\lim_{t\to+\infty} \|u(t)-\varphi\|_V =0$.
\end{theorem}

\begin{remark} \rm
By the previous Lemmas, it is sufficient to show that $u$ has finite 
length in $H$. One can see from the proof of Theorem \ref{main_generalized} below, 
that the theorem remains valid if $\mathbb{E}$ is only defined on the closure 
of the range of $u$ and continuous in $V$-norm on this set. Moreover, 
if $u$ is injective, then this weaker condition is not only sufficient
 but also necessary for $u$ to have finite length in $H$. In fact, one 
can define $\mathbb{E}(u(t)):=\int_t^{+\infty} \|\dot u(s)\|\,\mathrm{d} s$, 
then \eqref{key_est} holds on $\mathbb{R}_+$, so $t\mapsto \mathbb{E}(u(t))$ is
 nonincreasing on $\mathbb{R}_+$ and continuity of $\mathbb{E}$ also holds.
\end{remark}

Theorem \ref{main} does not speak about differential equations but it can 
be applied immediately to a solution of a first order equation
$$
\dot u(t) + F(u)=0
$$
if $\mathbb{E}$ is nonicreasing along the solution (e.g. a Lyapunov function) 
and \eqref{key_est} holds. Here $F$ may be an unbounded nonlinear operator. 
Second order equations
$$
\ddot u(t) + F(u(t),\dot u(t)) + M(u(t)) = 0
$$
can be reformulated as a first order equation on a product space denoting 
$v:=\dot u$. But then the energy or Lyapunov function typically depends on 
$u$ and $v$ but we are interested
in convergence of the first coordinate $u$ only 
(the second coordinate converges to zero ``automatically'' 
--- see Theorem \ref{DerConvergence}). So, we will formulate 
Theorem \ref{main_generalized} suitable for this situation. 
It is easy to see that Theorem \ref{main} follows immediately from 
Theorem \ref{main_generalized} (take $V_2=\{0\}=H_2$ and 
$V:=V_1\times V_2$, $H:=H_1\times H_2$), so we will not prove it.

\begin{theorem}      \label{main_generalized}
Let $u = (u_1,u_2)$ satisfy $u_1\in C(\mathbb{R}_+,V_1) \cap C^1(\mathbb{R}_+,H_1)$ and
$u_2\in C(\mathbb{R}_+,V_2)\cap C^1(\mathbb{R}_+,H_2)$ with $V_1\hookrightarrow H_1$, and let 
$(u_1(\cdot), u_2(\cdot))$ have a precompact range in $V_1\times V_2$.
 Let $\varphi\in \omega_{V_1}(u_1)$, $\rho>0$ and
$\mathbb{E}\in C(V_1\times V_2,\mathbb{R})$ be such that $t\mapsto \mathbb{E}(u(t))$ is nonincreasing
 on $\mathbb{R}_+$ and
\begin{equation} 	\label{key_est_two}
-\frac{d}{dt} \mathbb{E}(u(t))\ge \|\dot u_1(t)\|_{H_1}
\end{equation}
for almost every $t\in \{s\in \mathbb{R}_+:\ u_1(s)\in B:=B_{V_1}(\varphi, \rho)\}$.
Then $\lim_{t\to+\infty} \|u_1(t)-\varphi\|_{V_1} =0$.
\end{theorem}

\begin{remark}		\label{bigt} \rm
(i) It will be clear from the proof that Theorem \ref{main_generalized} 
remains valid if \eqref{key_est_two} holds only for almost every 
$t\in \{s\in [T,+\infty):\ u_1(s)\in B:=B_{V_1}(\varphi, \rho)\}$ for some $T>0$.
\end{remark}


\begin{proof}[Proof of Theorem \ref{main_generalized}]
Let $t_n\nearrow +\infty$ be an increasing sequence such that 
$\|u_1(t_n)-\varphi\|_{V_1} \to 0$. By precompactness of the range we may 
assume that $\|u_2(t_n)-\psi\|_{V_2} \to 0$ for some $\psi\in V_2$ 
(passing to a subsequence of $t_n$ if necessary).

Since $t\mapsto \mathbb{E}(u(t))$ is nonincreasing it has a limit for $t\to +\infty$. 
Since it is continuous, we have $\lim_{t\to +\infty} \mathbb{E}(u(t))=\mathbb{E}(\varphi,\psi)$ 
and we can assume without loss of generality $\mathbb{E}(\varphi,\psi)=0$ and 
$\mathbb{E}(u(t))\ge 0$ for all $t\in\mathbb{R}_+$ (redefining $\mathbb{E}(u):=\mathbb{E}(u)-\mathbb{E}(\varphi,\psi)$).

Since $\|u_1(t_n)-\varphi\|_{V_1} \to 0$, we have $u_1(t_n)\in B$ for all 
$n\ge n_0$. Let us denote $s_n:=\inf_{s\ge t_n}\{u_1(s)\not\in B\}$ and assume 
for contradiction that $s_n<+\infty$ for all $n$. From continuity of $u$ we 
have $s_n>t_n$ and $\|u_1(s_n)-\varphi\|_{V_1}=\rho$.

For $t\in (t_n,s_n)$ inequality \eqref{key_est_two} holds, so
$$
\mathbb{E}(u(t_n))-\mathbb{E}(u(t))\ge \int_{t_n}^t \|\dot u_1(s)\|_{H_1} \,\mathrm{d} s.
$$
So, we can estimate
\begin{align*}
\|u_1(t)-\varphi\|_{H_1}
& \le \|u_1(t)-u_1(t_n)\|_{H_1} + \|u_1(t_n)-\varphi\|_{H_1}\\
& \le \int_{t_n}^t \|\dot u_1(s)\|_{H_1} \,\mathrm{d} s + \|u_1(t_n)-\varphi\|_{H_1}\\
& \le \mathbb{E}(u(t_n))-\mathbb{E}(u(t)) +  \|u_1(t_n)-\varphi\|_{H_1} \\
& \le \mathbb{E}(u(t_n)) +  \|u_1(t_n)-\varphi\|_{H_1}
\end{align*}
and by continuity of $u$ this inequality holds for $t=s_n$. Hence,
$\|u_1(s_n)-\varphi\|_{H_1} \le \mathbb{E}(u(t_n)) +  \|u_1(t_n)-\varphi\|_{H_1} \to 0$
 as $n\to \infty$ (since $V_1\hookrightarrow H_1$).

On the other hand, by continuity of $u$ we have 
$\|u_1(s_n) - \varphi\|_{V_1} = \rho$ for all $n\in\mathbb{N}$. So, there is a 
subsequence of $u_1(s_n)$ converging to some $\tilde\varphi\in V_1$ 
(by precompactness of the range), $\tilde\varphi\ne \varphi$, which is a 
contradiction with $\|u_1(s_n)-\varphi\|_{H_1} \to 0$.

Hence, $s_n=+\infty$ for some $n$. Hence, $\dot u_1\in L^1(\mathbb{R}_+,H_1)$, 
it has finite length in $H_1$ and converges to $\phi$ in the norm of $V_1$ 
by Lemma \ref{l-two}.
\end{proof}

In case of second order equations, if a solution has a limit then its 
derivative usually tends to zero. However, convergence of the derivative 
often needs much weaker assumptions (or different assumptions) and it is 
helpful to know the convergence of the derivative a-priori, before one 
shows convergence of the function itself. Therefore, we formulate the 
following theorem.

\begin{theorem}		\label{DerConvergence}
Let $V\hookrightarrow H\hookrightarrow V'$ be Hilbert spaces, $F\in C(V\times H,V')$, 
$E\in C^1(V,R)$ and $M=E':V\to V'$.
Assume that there exists a nondecreasing function $g:(0,+\infty)\to (0,+\infty)$ 
such that
$$
\langle F(u,v),v \rangle_{V',V} \ge g(\|v\|_*)
$$
for all $u$, $v\in V$.
If $u\in C^1(\mathbb{R}_+,V)\cap C^2(\mathbb{R}_+,H)$ is a classical solution of
\begin{equation} \label{SOPVeryGen}
\begin{gathered}
\ddot u(t) + F(u(t),\dot u(t)) + M(u(t)) = 0, \\
u(0) =u_0\in V,\ \dot u(0)=u_1\in H
\end{gathered}
\end{equation}
such that $(u,\dot u)$ is precompact in $V\times H$,
then $\lim_{t\to+\infty} \|\dot u(t)\|=0$.
\end{theorem}

\begin{proof}
Since range of $(u,\dot u)$ is precompact in $V\times H$, range of 
$F(u, \dot u)+M(u)$ is bounded in $V'$. Hence, range of $\ddot u$ is bounded 
in $V'$ and $\dot u$ is Lipschitz continuous in $V'$.
Moreover, we have
\begin{align*}
- \frac{d}{dt} \frac12 \|\dot u(t)\|^2 
&= - \langle \ddot u(t), \dot u \rangle_{V',V} \\
&= \langle F(u(t),\dot u(t)), \dot u\rangle_{V',V} + \frac{d}{dt} E(u(t)) \\
&\ge  g(\|\dot u(t)\|_{*}) + \frac{d}{dt} E(u(t)).
\end{align*}
Since $|E(u(s))|\le K$ for some $K>0$ and all $s\ge 0$, integrating on  $[t_0, t]$,
\begin{equation}
\begin{aligned}
\int_{t_0}^t g(\|\dot u(s)\|_{*}) \,\mathrm{d} s 
& \le \frac12(- \|\dot u(t)\| + \|\dot u(t_0)\|) - E(u(t)) + E(u(t_0)) \\
& \le \frac12 \|\dot u(t_0)\|+2K.
\end{aligned}
\end{equation}
Hence, $s\mapsto g(\|\dot u(s)\|_{*})\in L^1((0,+\infty))$ 
and due to Lipschitz continuity we have $\lim_{t\to+\infty}\|\dot u(t)\|_{*}=0$. 
Since range of $\dot u$ is precompact in $H$,   
$\lim_{t\to+\infty}\|\dot u(t)\| =0$.
\end{proof}

\begin{corollary} \label{SO-corollary}
Let the assumptions of Theorem \ref{DerConvergence} 
be satisfied and let there exist $\rho>0$ and $\mathbb{E}\in C(V\times H,\mathbb{R})$ 
such that $t\mapsto \mathbb{E}(u(t),\dot u(t))$ is nonincreasing 
on $(0,+\infty)$ and
\begin{equation}		\label{key_est_three}
-\frac{d}{dt} \mathbb{E}(u(t),\dot u(t))\ge c \|\dot u(t)\|_*
\end{equation}
for almost every 
$t\in \{s\in\mathbb{R}_+:\ u(s)\in B_{V}(\varphi,\rho)\times B_{H}(0,\varepsilon)\}$ 
where $\varepsilon>0$ is arbitrary.
Then $\lim_{t\to+\infty} \|u(t)-\varphi\|_{V} + \|\dot u(t)\|=0$.
\end{corollary}

\begin{proof}
The derivative converges to 0 by Theorem \ref{DerConvergence}. 
Then $\dot u(t)\in B_{H}(0,\varepsilon)$ for all $t\ge T$. Then \eqref{key_est_two} 
is satisfied for $t\in [T,+\infty)$ and applying Theorem \ref{main_generalized} 
with $H_1=V'$ (see Remark \ref{bigt}) we obtain convergence of $u(t)$.
\end{proof}

\begin{remark} \rm
We can see that the $*$-norm on the right-hand side of \eqref{key_est_three} 
can be replaced by any other norm weaker than $H$-norm.
\end{remark}

\section{Applications to first order equations}

In this section, we show several known results that are covered by 
Theorem \ref{main}.

\subsection{{\L}ojasiewicz convergence result}

We start with the classical convergence result by {\L}ojasiewicz. 
Let us remark that the following Proposition speaks about ordinary differential 
equations (then $u$ has values in a finite-dimensional space $H=V$ 
and $E\in C^1(H)$) and also about partial differential equations 
(then $V\hookrightarrow H$ are Hilbert spaces, $u\in C(\mathbb{R}_+,V)\cap C^1(\mathbb{R}_+,H)$ 
and $E\in C^1(V)$).


\begin{proposition}
Let $u$ be a solution to the gradient system $\dot u + \nabla E (u) = 0$, 
$\varphi\in \omega_V(u)$ and let $E$ satisfy the {\L}ojasiewicz or 
Kurdyka-{\L}ojasiewicz-Simon inequality on a neighborhood of $\varphi$. 
Then there exists a function $\mathbb{E}$ such that $t\mapsto \mathbb{E}(u(t))$ is 
nonincreasing and \eqref{key_est} holds on a neighborhood of $\varphi$.
\end{proposition}

\begin{proof}
It is sufficient to define $\mathbb{E}(u):=E(u)^{\theta}$ in case 
of {\L}ojasiewicz inequality and $\mathbb{E}(u):=\Phi_{\Theta}(E(u))$ in 
case of Kurdyka-{\L}ojasiewicz-Simon inequality.
\end{proof}

\subsection{Convergence result by Chill, Haraux, Jendoubi and its corollaries}

Theorem 1 in  \cite{CHJ09} is another corollary of Theorem \ref{main}. 
If we replace {\L}ojasiewicz inequality by the more general
 Kurdyka-{\L}ojasiewicz-Simon inequality, then the theorem in \cite{CHJ09} 
reads as follows.

\begin{theorem}[{\cite[Theorem 1]{CHJ09}}] \label{CHJ}
Let $u\in C(\mathbb{R}_+,V)\cap C^1(\mathbb{R}_+,H)$ with $V$-precompact range and 
$\varphi\in \omega_V(u)$. Let $\rho>0$, $c>0$ and
$E\in C^2(V,\mathbb{R})$ be such that $t\mapsto \mathbb{E}(u(t))$ is differentiable almost 
everywhere and
$$
-\frac{d}{dt}E(u(t)) \ge c \|E'(u(t))\|_*\|\dot u(t)\|_*
$$
for almost every $t\in \mathbb{R}_+$ with $u(t)\in B_V(\varphi,\rho)$. 
Assume in addition that
$$
\text{if $E(u(\cdot))$ is constant for $t\ge t_0$, then $u$ is constant 
for $t\ge t_0$}
$$
and that $E$ satisfies the Kurdyka-{\L}ojasiewicz-Simon gradient inequality
with a Kurdyka function $\Theta$.
Then $\lim_{t\to+\infty} \|u(t)-\varphi\|_V =0$.
\end{theorem}

\begin{proof}
We can assume that $E(\varphi)=0$. If $E(u(t))=0$ for some $t_0$, 
then $u$ is constant for all $t>t_0$ and the assertion holds. 
Otherwise, $E(u(t))>0$ for all $t\in \mathbb{R}_+$. In this case, 
let us define $\mathbb{E}(u):=\Phi_{\Theta}(E(u))$. Then
$$
-\frac{d}{dt}\mathbb{E}(u(t))  \ge  \frac1{\Theta(E(u(t)))}\cdot
 c \|E'(u(t))\|_*\|\dot u(t)\|_*\ge c \|\dot u(t)\|_*.
$$
So, assumptions of Theorem \ref{main} hold and $\|u(t)-\varphi\|_V\to 0$.
\end{proof}

For many applications and corollaries of Theorem \ref{CHJ} see \cite{CHJ09}.

\subsection{Convergence result by B\'arta, Chill, Fa\v{s}angov\'a}


In \cite{BCF12}, B\'arta, Chill and Fa\v{s}angov\'a proved a convergence
theorem formulated on manifolds. If we reformulate it for $\mathbb{R}^N$, 
it becomes a corollary of Theorem \ref{main}.

\begin{theorem}[{\cite[Theorem 3]{BCF12}}]     \label{BCF_result}
Let $F\in C(\mathbb{R}^N,\mathbb{R}^N)$, $u :\mathbb{R}_+ \to \mathbb{R}^N$ be a global solution of the 
ordinary differential equation
\begin{equation}   \label{ode}
\dot u(t) + F(u(t)) = 0
\end{equation}
and let $E : \mathbb{R}^N\to\mathbb{R}$ be a continuously differentiable, 
strict Lyapunov function for \eqref{ode}. Assume that there exist 
a Kurdyka function $\Theta$, $\varphi\in \omega(u)$ and a neighbourhood 
$U$ of $\varphi$ such that for every $v\in U$ we have $F(v)\ne 0$ and
\begin{equation} \label{newL}
\Theta (|E (v)-E (\varphi )|) \le \langle E'(v), \frac{F(v)}{\|F(v)\|}\rangle .
\end{equation}
Then $u$ has finite length and, in particular, $\lim_{t\to+\infty} u(t)=\varphi$.
\end{theorem}


\begin{proof}
Let us recall that $E$ is \textit{a strict Lyapunov function} for \eqref{ode},
 if $\langle E'(u), F(u)\rangle>0$, whenever $u\in \mathbb{R}^N$, $F(u)\ne 0$.
Since $E(u(\cdot))$ is nonincreasing and continuous, it has a limit which 
is equal to $E(\varphi)$. We can assume that $\mathbb{E} (\varphi )= 0$, so that 
$E (u(t))\geq 0$ for all $t\in\mathbb{R}_+$. If $E (u(t_0)) = 0$ for some $t_0\geq 0$, 
then $E (u(t))=0$ for every $t\geq t_0$, and therefore, since $E$ is a 
strict Lyapunov function, the function $u$ is constant for $t\geq t_0$. 
In this case, there remains nothing to prove.

Hence, we may assume that $E (u(t)) >0$ for every $t\geq 0$ and 
define $\mathbb{E}(u):=\Phi_{\Theta}(E(u))$.
Then
\begin{align*}
- \frac{d}{dt} \mathbb{E}(u(t)) 
& = \frac1{\Theta (E (u(t))} \, \big( -\frac{d}{dt} E(u(t)) \big) \\
& = \frac1{\Theta (E (u(t))}\, \langle E'(u(t)), F(u(t)) \rangle \\
&\ge \| F(u(t))\| = \| \dot u (t))\|
\end{align*}
in a neighborhood of $\varphi$. Hence the assumptions of Theorem \ref{main} 
are satisfied and $\lim_{t\to\infty} u(t) =\varphi$.
\end{proof}


\section{Applications to second order equations}


\subsection{Second order ODE with weak nonlinear damping}

The equation
$$
\ddot u(t) + |\dot u(t)|^{\alpha} \dot u(t) + \nabla E((u(t))) = 0
$$
with $\alpha>0$ was studied by Chergui in \cite{Cher08} and the 
convergence result was then extended to more general dampings
\begin{equation}   \label{SO_ODE}
\ddot u(t) + G(u(t),\dot u(t)) + \nabla E((u(t))) = 0
\end{equation}
by B\'arta, Chill and Fa\v{s}angov\'a \cite{BCF12}, where 
$G\in C^2(\mathbb{R}^N\times \mathbb{R}^N)$ and for every $u$, $v\in\mathbb{R}^N$ it holds that
\begin{equation} \label{gcond}
\begin{gathered}
 \langle G(u, v), v\rangle \ge g(\|v\|)\, \|v\|^2 , \\
\|G(u,v)\|\le c g(\|v\|)\, \|v\| , \\
 \|\nabla G(u,v)\| \le c \, g(\|v\|) ,
\end{gathered}
\end{equation}
where $c\geq 0$ is a constant and $g:\mathbb{R}_+\to\mathbb{R}_+$ is a nonnegative, concave, 
nondecreasing function, $g(s)>0$ for $s>0$.

Under these assumptions we have
$$
\langle G(u, v), v\rangle \ge g(\|v\|)\, \|v\|^2 
= g(\|v\|_*)\, \|v\|_*^2 =: \tilde g(\|v\|_*),
$$
so assumptions of Theorem \ref{DerConvergence} hold with $\tilde g$.
By Corollary \ref{SO-corollary}, it is sufficient to prove that
$$
\mathbb{E}((u,v)):=\Phi_{\Theta}\left(\frac12\|v\|^2+E(u)
+\varepsilon \langle G(u,\nabla E(u)),v \rangle\right)
$$
satisfies the key estimate \eqref{key_est_three},
which needs some work (see \cite{BCF12} for details).

\subsection{A semilinear wave equation with nonlinear damping}

The following problem was studied by Chergui in \cite{Cher09}. 
Consider the equation
\begin{equation} \label{SOPC}
u_{tt}+ |u_t|^{\alpha}u_t = \Delta u + f(x,u)
\end{equation}
in $\mathbb{R}_+\times \Omega$ with Dirichlet boundary conditions and initial values
$$
u(0,\cdot)=u_0\in H_0^1(\Omega),\quad u_t(0,\cdot)=u_1\in L^2(\Omega).
$$
Function $f:\Omega\times \mathbb{R}\to \mathbb{R}$ satisfies
\begin{itemize}
\item If $N=1$: $f$, $\partial_2 f$ are bounded in $\Omega\times[-r,r]$ 
for all $r>0$,
\item If $N\ge 2$: $f(\cdot,0)\in L^{\infty}(\Omega)$ and
 $|\partial_2 f(x,s)|\le c(1+|s|^{\gamma})$ on $\Omega\times \mathbb{R}$,
\end{itemize}
where $c\ge 0$, $\gamma\ge 0$ and $(N-2)\gamma<2$.

Then the main part of the proof of \cite[Theorem 1.4]{Cher09} can be 
interpreted as proving that (for appropriate $\alpha$ and $\theta$ and 
small $\varepsilon>0$)
\begin{align*}
&\mathbb{E}((u(t), \dot u(t)))\\
&:= \Big(\frac12\|\dot u(t)\|^2_2 + E(u(t)) 
 - \varepsilon \|\dot u(t)\|^{\alpha}_*\langle \Delta u(t)
  + f(x,u(t)),\dot u(t)\rangle_*\Big)^{\theta - (1-\theta)\alpha}
\end{align*}
satisfies estimate \eqref{key_est_three}, where
\begin{equation} \label{SOPC-defE}
E(u):=\frac12\|\nabla u\|_2^2 - \int_{\Omega} F(x,u)\,\mathrm{d} x,
\quad  F(x,u):=\int_0^u f(x,s)\,\mathrm{d} s.
\end{equation}
Let us mention that Corollary \ref{SO-corollary} can be applied in this case, 
if we consider classical solutions (the result in \cite{Cher09}
refers to weak solutions).


\subsection{Abstract wave equation with linear damping}

The following abstract second-order equation is studied in \cite{CHJ09}. 
We have $V\hookrightarrow H\hookrightarrow V'$, $\gamma\ne 0$, $E\in C^2(V)$, $M=E'$ 
and consider the equation
\begin{equation} \label{SOPCHJ}
u_{tt}+ \gamma u_t +M(u) = 0.
\end{equation}
Let us introduce the duality mapping $K:V'\to V$ given by
 $\langle u,v\rangle_* = \langle u,Kv\rangle_{V',V} = \langle u,Kv\rangle$ for $u\in H$, $v\in V'$.


\begin{theorem}[{\cite[Corollary 16]{CHJ09}}] 		\label{awe-theorem}
Assume that $\gamma>0$ and
\begin{itemize}
\item[(1)] for every $v\in V$, the operator $KM'(v)$ extends to a bounded operator 
on $H$  and $\sup_v \|KM'(v)\|_{L(H)}$ is finite when $v$ ranges over a compact 
subset of $V$, and

\item[(2)] $u\in C^1(\mathbb{R}_+,V)\cap C^2(\mathbb{R}_+,H)$ is a global solution to \eqref{SOPCHJ},
 $(u,\dot  u)$ has precompact range in $V\times H$
and there exist $\varphi\in \omega(u)$, $C>0$, $\rho>0$ and a sublinear Kurdyka 
function $\Theta$, such that
$E$ satisfies Kurdyka-{\L}ojasiewicz-Simon gradient inequality in
 $B_V(\varphi,\rho)$.
\end{itemize}
Then $\lim_{t\to+\infty} \|u(t)-\varphi\|_V =0$.
\end{theorem}


Since
$$
\langle \gamma \dot u, \dot u \rangle \ge \gamma c \|\dot u\|^2_* =: g(\|\dot u\|_*),
$$
the assumptions of Theorem \ref{DerConvergence} are satisfied and 
$\|\dot u\|\to 0$. It is not difficult to show that function
 $\mathbb{E}(u,\dot u):= \Phi_{\Theta}(\Psi(u,\dot u))$ satisfies the key estimate 
\eqref{key_est_three}, where
$$
\Psi(u,\dot u):=\frac12 \|\dot u\|^2 + E(u) + \varepsilon\langle M(u),\dot u\rangle_*
$$
and $\varepsilon>0$ is small enough.
Then Corollary \ref{SO-corollary} proves the assertion.


\subsection*{Acknowledgements}
This work is supported by GACR 201/09/0917. The author is a researcher 
in the University Centre for Mathematical Modeling,
Applied Analysis and Computational Mathematics (Math MAC) and a
 member of the Ne\v{c}as Center for Mathematical Modeling.


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\end{document}