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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 112, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/112\hfil Controllability and periodic solutions]
{Controllability and periodic solutions of nonlinear wave equations}

\author[B. A. Ton \hfil EJDE-2017/112\hfilneg]
{Bui An Ton}

\address{Bui An Ton \\
Department of Mathematics,
University of British Columbia,
Bancouver, B.C. V6T 1Z2, Canada}
\email{bui@math.ubc.ca}

\dedicatory{Communicated by Jesus Ildefonso Diaz \break
Dedicated to the memory of Felix E. Browder whose guidance is gratefully
acknowledged}

\thanks{Submitted July, 25, 2016. Published April 26, 2017.}
\subjclass[2010]{35L05, 35L70, 93B05}
\keywords{Exact controllability; time-periodic solution; control;
\hfill\break\indent 3D nonlinear wave equation}

\begin{abstract}
 The controllability of time-periodic solutions of a $n$-dimensional
 nonlinear wave equation is established with $ n=2,3$.
 The result is used to establish the existence of time-periodic solutions of
 a nonlinear wave equation.
\end{abstract}

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


\section{Introduction}\label{S:intro}

 The purpose of the article is to establish the existence of time-periodic
solutions of a nonlinear wave equation in bounded domains of $\mathbb{R}^n$ with $ n=2, 3$,
using controllability.
Following the pioneering work of Rabinowitz \cite{r1,r2} on time-periodic solutions
of the one-dimensional nonlinear wave equation, extensive studies of the problem
were done by Berti-Bolle \cite{b1,b2}, Brezis-Nirenberg \cite{b3} and others.
Controllability and fictitious domains were used by Glowinski and his
collaborators \cite{b5}, Glowinski-Rossi \cite{g1} to treat numerically 
the existence of
time-periodic solutions of the linear wave equation in cylindrical domains.
For higher spatial dimensions, Berti and Polle \cite{b3} used The Nash-Moser
iteration to study  T-periodic solutions of the problem
\begin{gather*}
 u''-\Delta u + mu = \varepsilon F(\omega t,x,u) \\
 u(t,x)= u(t,x+2k\pi) \quad \forall k\in Z^{n}
 \end{gather*}
where $F$ is $ 2\pi/\omega$ periodic in time and $2\pi$-periodic in $ x_{j}$,
$j=1,\dots ,n$.

 In \cite{t1,t2} the author established the existence of time-periodic solutions
of a nonlinear wave equation in non-cylindrical domains of $ \mathbb{R}^n$,
$n=2,3$ with the forcing term in a non-empty subset of $K^{\perp}$ with
\[
 K=\{ v: v \in L^{2}(0,T;L^{2}(G)),\, \int^{T}_0v(\cdot,t)dt=0 \}
 \]
In this paper we shall show that for any $f$ in $ K^{\perp}$ there exists
a time-periodic solution of a nonlinear wave equation in cylindrical domains.
The proof is carried out in Section 5.Notations and the basic assumption
of the paper are given in Section 2.

Given $f$ in $ K^{\perp} $ and $ u_0$ in $H^{1}_0(G)\cap L^{p}(G)$ we shall
establish the existence of a control $ g_{f}(u_0)$ in
$(H^{1}_0(G)\cap L^{p}(G))^{*}$ and a time-periodic solution of the nonlinear
wave equation
\begin{gather*}
 u''-\Delta u +| u|^{p-2}u  = f- g_{f}(u_0) \quad\text{in }G\times (0,T),\\
 u=0 \text{ on }\partial G \times (0,T) ,\quad
 \{ u, u'\}\big|_{t=0}=\{u, u'\}\big|_{t=T}=\{u_0,0\}
 \end{gather*}
The solution and its derivative take prescribed values at $t=0$ and at $t=T$.

 In Section 4 we consider a semi-exact controllability problem.
Given $ f$ in $K^{\perp}$ and $ u_0 $ in $ H^{1}_0(G) \cap L^{p}(G)$,
we shall prove the existence of
(i) a control $g_{f}(u_0) $ and
(ii) a time-periodic solution of the problem
\begin{gather*}
 u''-\Delta u +| u |^{p-2}u = f - g_{f}(u_0)\quad\text{in }
 G\times (0,T),\\
 u=0 \text{ on }\partial G\times (0,T),\quad
 u(0)=u_0=u(T),\quad  u'(0)=u'(T).
 \end{gather*}
As the solution $u$ takes a prescribed common value at $t=0$ and at
$ t=T $, its derivative $u'$ is not required to take a specific value
at the two end points, we shall call it a semi-exact controllability problem.

 \subsection*{Notation}
Let $G$ be a bounded open subset of $\mathbb{R}^n$ with $ n=2,3$, and let
\[
K = \{ v: v\in L^{2}(0,T;L^{2}(G)),\; \int^{T}_0v(\,.,s)ds =0 \}.
\]
The set $K$ is a closed convex subset of $L^{2}(0,T;L^{2}(G)) $and let $ J$,
 be the duality mapping of $L^{2}(0,T;L^{2}(G))$ into $L^{2}(0,T;L^{2}(G))$
with gauge function $ \Phi(r)=r$. The penalty function
\[
 \beta(v) = J(v-P_{K}v)
\]
where $P_{K}$ is the projection of $K$ onto $L^{2}(0,T;L^{2}(G))$,
is well-defined. For a given $ u $ in $L^{2}(0,T;L^{2}(G))$ there exists
a unique $P_{K}u$ in $K$ such that
\[
 \|u-P_{K}u\|_{L^{2}(0,T;L^{2}(G))} \leq \| u-k\|_{L^{2}(0,T;L^{2}(G))} \quad
\forall k \in K.
\]
In this article, we denote by $ (\cdot,\cdot)$ the various pairings
between $L^{2}(G),L^{p}(G)$ and their duals.

\subsection*{Assumption} 
We assume that $ 2 \leq p <\infty $ if $ G \subset R^{2}$ and
$ 2 \leq p \leq 4 $ if  $ G\subset R^{3}$.


\section{Exact controllability time periodic problem}\label{S:exact}

 The main result of the section is the following theorem

 \begin{theorem} \label{thm3.1} %\label{exact1}
Let $ \{f , u_0\} $ be in $ K^{\perp} \times \{H^{1}_0(G) \cap L^{p}(G)\}$
then there exist:
\begin{itemize}
\item[(i)] $ g_{f}(u_0) $ in $[H^{1}_0(G)\cap L^{p}(G)]^*$


\item[(ii)] $ \{u, u'\} $ in $L^{\infty}(0,T; H^{1}_0(G)\cap L^{p}(G))
\times L^{\infty}(0,T;L^{2}(G))$, solution of the problem
\begin{equation}\label{exact1}
\begin{gathered}
 u''- \Delta u + | u|^{p-2} u  = f- g_{f}(u_0) \quad \text{in }
 G \times (0,T) \\
 u=0 \text{ on } \partial G \times (0,T),\quad
\{ u, u'\}\big|_{t=0} =\{u, u'\}\big|_{t=T}= \{ u_0,0\}
\end{gathered}
 \end{equation}
\end{itemize}
\end{theorem}


We consider the initial boundary-value problem
\begin{equation}\label{exact2}
\begin{gathered}
 u''_{\varepsilon} -\varepsilon \Delta u'_{\varepsilon}-\Delta u_{\varepsilon}
+| u_{\varepsilon}|^{p-2}u_{\varepsilon}
+\varepsilon^{-1} \beta(u'_{\varepsilon})
= f\quad \text{in } G\times (0,T),\\
 u_{\varepsilon} = u'_{\varepsilon}=0 \text{ on } \partial G\times (0,T) ,\quad
 \{u_{\varepsilon}, u'_{\varepsilon}\}\big|_{t=0}=\{u_0, u_1\}\,.
\end{gathered}
 \end{equation}

\begin{lemma}\label{lem3.1} % exact1
 Let $ \{ f, u_0, u_1\}$ be in
$K^{\perp}\times [H^{1}_0(G)\cap L^{p}(G)]\times L^{2}(G)$
then there exists a unique solution $ u_{\varepsilon}$ of \eqref{exact2}. Moreover
\begin{align*}
&\|u'_{\varepsilon}(t)\|^{2}_{L^{2}(G)}
 + 2\varepsilon\|\nabla u'_{\varepsilon}\|^{2}_{L^{2}(0,t;L^{2}(G))}
 +\|\nabla u_{\varepsilon}(t)\|^{2}_{L^{2}(G)}\\
&+ 2 p^{-1}\|u_{\varepsilon}(t)\|^{p}_{L^{p}(G)}
 + 2\varepsilon^{-1}\int^{t}_0(\beta(u'_{\varepsilon}), u'_{\varepsilon})ds\\
&\leq \|u_1\|^{2}_{L^{2}(G)}+ \|\nabla u_0\|^{2}_{L^{2}(G)}
+ 2p^{-1}\|u_0\|^{p}_{L^{p}(G)} +2\int^{t}_0(f, u'_{\varepsilon})ds
 \end{align*}
\end{lemma}

The standard Galerkin approximation method gives the existence of a unique
solution of \eqref{exact2} with the stated estimate. We shall not reproduce
the proof.

 \begin{lemma}\label{lem3.2} % exact2
Let $ u_{\varepsilon}$ be as in Lemma \ref{lem3.1} then there exists a subsequence
such that
 \[
 \{ u_{\varepsilon}, u'_{\varepsilon}, \beta(u'_{\varepsilon}) \}
\to \{ u, u', \, 0\}
 \]
 in the space
\begin{align*}
&\Big\{C(0,T;L^{2}(G))\cap [L^{\infty}(0,T; H^{1}_0(G)\cap L^{p}(G))]_{weak^{*}}
\Big\} \\
&\times [L^{\infty}(0,T;L^{2}(G))]_{weak^{*}}
\times [L^{2}(0,T;L^{2}(G))]_{\rm weak}.
 \end{align*}
Furthermore $\beta(u')=0$, i.e.\ $u' $ in $K $ and thus,
$ u(\cdot,0)=u(\cdot,T)=u_0$.
 \end{lemma}

 \begin{proof}
(1) From the estimate of Lemma \ref{lem3.1} and  the Gronwalls lemma, 
there exists a subsequence
such that $ \{ u_{\varepsilon}, u'_{\varepsilon}\} \to \{ u, u'\}$
in
\[
 C(0,T;L^{2}(G)) \cap [L^{\infty}(0,T;H^{1}_0(G)\cap L^{p}(G))]_{weak^{*}}
\times [L^{\infty}(0,T;L^{2}(G))]_{weak^{*}}
 \]
We have
\begin{align*}
\|\beta(u'_{\varepsilon})\|_{L^{2}(0,T;L^{2}(G))}
&= \|J(u'_{\varepsilon}-P_{K}u'_{\varepsilon})\|_{L^{2}(0,T;L^{2}(G))}\\
&=\Phi(\|u'_{\varepsilon}-P_{K}u'_{\varepsilon}\|_{L^{2}(0,T;L^{2}(G))})\\
&=\|u'_{\varepsilon}-P_{K}u'_{\varepsilon}\|_{L^{2}(0,T;L^{2}(G))}\\
&\leq \|u'_{\varepsilon}\|_{L^{2}(0,T;L^{2}(G))}
 +\|P_{K}u'_{\varepsilon}-P_{K}0\|_{L^{2}(0,T;L^{2}(G))}\\
&\leq 2 \|u'_{\varepsilon}\|_{L^{2}(0,T;L^{2}(G))}
 \leq  M
\end{align*}
Thus,
 \[
 \beta(u'_{\varepsilon}) \to \chi \quad \text{in } (L^{2}(0,T;L^{2}(G)))_{\rm weak}.
 \]

(2) We now show that $ \chi =0$. From \eqref{exact2} we have
 \begin{align*}
&- \varepsilon \int^{T}_0( u'_{\varepsilon},\varphi')dt
+\varepsilon^{2}\int^{T}_0( \nabla u'_{\varepsilon},
 \nabla \varphi)dt + \varepsilon \int^{T}_0( \nabla u_{\varepsilon},
\nabla \varphi)dt\\
&+ \varepsilon \int^{T}_0( | u_{\varepsilon}|^{p-2} u_{\varepsilon}, \varphi)dt
+ \int^{T}_0(\beta(u'_{\varepsilon}), \varphi)dt\\
&= \varepsilon\int^{T}_0 (f, \varphi)dt \quad \forall
\varphi \in C^{\infty}_0(0,T;H^{1}_0(G) \cap L^{p}(G))
 \end{align*}
Thus,
\[
\int^{T}_0( \beta(u'_{\varepsilon}), \varphi)dt \to 0 \quad
\forall \varphi \in C^{\infty}_0(0,T; H^{1}_0(G)\cap L^{p}(G))
\]
Since $ \beta(u'_{\varepsilon}) \to \chi $ in
$[ L^{2}(0,T;L^{2}(G)]_{\rm weak}$, we deduce that $\chi =0$.


(3) We now show that $ \beta(u') =0$. Since $ \beta $ is monotone in
$L^{2}(0,T;L^{2}(G))$ we have
\[
 \int^{T}_0 ( \beta(u'_{\varepsilon})-\beta(v'), u'_{\varepsilon}- v') dt \geq 0 \,\,\forall v'\in
 L^{2}(0,T;L^{2}(G)),
\]
in particular for all $v$ with
 \[
 v=\int^{t}_0 \varphi(\,.,s) ds,\,\,\,\varphi \in L^{2}(0,T;L^{2}(G)).
 \]
Thus,
\[
 \int^{T}_0 ( \beta(u'_{\varepsilon})-\beta(\varphi) ,
u'_{\varepsilon}- \varphi) dt \geq 0 \quad \forall \varphi \in L^{2}(0,T;L^{2}(G)).
 \]
From the estimate of Lemma \ref{lem3.1} and from the above we have
\[
 \lim_{\varepsilon \to 0} \int^{T}_0( \beta(u'_{\varepsilon}), u'_{\varepsilon})dt
= 0 =\lim_{\varepsilon}\int^{T}_0( \beta(u'_{\varepsilon}), \varphi)dt.
 \]
Hence
\[
 -\int^{T}_0 ( \beta(\varphi), u'- \varphi)dt \geq 0 \quad
\forall \varphi \in L^{2}(0,T;L^{2}(G)).
\]
Take $\varphi = u'+ \lambda w$, $\lambda >0 $ and $ w $ in
$L^{2}(0,T;L^{2}(G))$. We have
\[
 \int^{T}_0( \beta(u'+\lambda w),\, w) dt \geq 0 \quad
\forall w \in L^{2}(0,T;L^{2}(G)).
\]
Letting $ \lambda \to 0$  we obtain
 \[
 \int^{T}_0( \beta(u'), w) dt \geq 0 \quad \forall w \in L^{2}(0,T;L^{2}(G)).
 \]
Changing $ w $ to $ -w$ and we deduce that
$ \beta(u') =0$  i.e.\ $u'\in K$  and $u(\cdot,0)= u(\cdot,T)= u_0$.
\end{proof} 

\begin{lemma}\label{lem3.3} %exact3
Let $ \{ u_{\varepsilon}, \,u\}$, be as in Lemmas \ref{lem3.1} and
\ref{lem3.2}.  There exists $ g_{f}(u_0,u_1)$ in
$[H^{1}_0(G)\cap L^{p}(G)]^{*} $ and associated with $g_{f}(u_0,u_1)$,
a unique solution $u$, of the problem
\begin{equation}\label{exact3}
\begin{gathered}
u''- \Delta u + | u|^{p-2}u = f- g_{f}(u_0,u_1)\quad \text{in }
 G \times (0,T),\\
u=0 \text{ on } \partial G\times (0,T),\quad
 \{ u, u'\}\big|_{t=0}= \{u_0, u_1\}
= \{ u(\cdot,T), u_1\}
\end{gathered}
\end{equation}
with
\[
\int^{T}_0( g_{f}(u_0,u_1),\varphi) dt
=\lim_{\varepsilon \to 0} \varepsilon^{-1}
\int^{T}_0 ( \beta(u'_{\varepsilon}), \varphi)dt
\]
for all $\varphi \in C^{\infty}_0(0,T;H^{1}_0(G) \cap L^{p}(G))$.
Furthermore,
\begin{align*}
&\liminf\|u_{\varepsilon}'(t)\|^{2}_{L^{2}(G)}+ \|\nabla u(t)\|^{2}_{L^{2}(G)}
 + 2p^{-1}\|u(t)\|^{p}_{L^{p}(G)}\\
&\leq \|u_1\|^{2}_{L^{2}(G)}+ \|\nabla u_0\|^{2}_{L^{2}(G)}
+ 2p^{-1}\|u_0\|^{p}_{L^{p}(G)} +2\int^{t}_0( f, u')ds.
\end{align*}
\end{lemma}

\begin{proof} (1) Since
$u_{\varepsilon} \to u $  in
$ C(0,T;L^{2}(G)) \cap (L^{\infty}(0,T;L^{p}(G)))_{weak^{*}}$,
a standard argument gives
\[
| u_{\varepsilon}|^{p-2} u_{\varepsilon} \to | u|^{p-2}u \quad
\text{in } [L^{\infty}(0,T;L^{q}(G))]_{weak^{*}}.
\]

(2) Let $\varphi $ be in $ C^{\infty}_0(0,T;H^{1}_0(G)\cap L^{p}(G))$
then $\varphi' $ is in $K $ and we have
\[
\int^{T}_0( \beta(u'_{\varepsilon}) -\beta(\varphi'),
 u'_{\varepsilon} - \varphi') dt
= \int^{T}_0( \beta(u_{\varepsilon}'), u'_{\varepsilon}- \varphi') dt \geq 0.
\]
It follows from \eqref{exact2} that
\begin{equation}\label{exact4}
\begin{aligned}
&\int^{T}_0 (u''_{\varepsilon}, u'_{\varepsilon}- \varphi')dt
 + \int^{T}_0( \nabla( \varepsilon u'_{\varepsilon}+ u_{\varepsilon}) ,
 \nabla(u'_{\varepsilon}-\varphi')) dt\\
&+ \int^{T}_0( | u_{\varepsilon}|^{p-2}u_{\varepsilon},
  u'_{\varepsilon}- \varphi') dt+\varepsilon^{-1}\int^{T}_0
 ( \beta(u'_{\varepsilon}), u'_{\varepsilon}- \varphi') dt\\
&=\int^{T}_0( f, u'_{\varepsilon} -\varphi')dt
\end{aligned}
\end{equation}
Hence
\begin{align*}
&\|u'_{\varepsilon}(T)\|^{2}_{L^{2}(G)}
 + 2\varepsilon \|\nabla u'_{\varepsilon}\|^{2}_{L^{2}(0,T:L^{2}(G))}
 +\|\nabla u_{\varepsilon}(T)\|^{2}_{L^{2}(G)}
 + 2p^{-1}\|u_{\varepsilon}(T)\|^{p}_{L^{p}(G)}\\
&-2\int^{T}_0(f,u'_{\varepsilon})dt -\Big\{ \|u_1\|^{2}_{L^{2}(G)}
 +\|\nabla u_0\|^{2}_{L^{2}(G)}+ 2p^{-1}\|u_0\|^{p}_{L^{p}(G)}\Big\}\\
&\leq 2\int^{T}_0( u''_{\varepsilon}, \varphi')dt
 + 2\int^{T}_0(\nabla(\varepsilon u'_{\varepsilon}
 + u_{\varepsilon}), \nabla \varphi')dt
 + 2\int^{T}_0(| u_{\varepsilon}|^{p-2}u_{\varepsilon}- f,\varphi') dt
\end{align*}
Letting $\varepsilon \to 0$, we obtain
\begin{align*}
&\liminf \|u_{\varepsilon}'(T)\|^{2}_{L^{2}(G)}+ \|\nabla u(T)\|^{2}_{L^{2}(G)}
 + 2p^{-1}\|u(T)\|^{p}_{L^{p}(G)}\\
&-\{ \|u_1\|^{2}_{L^{2}(G)}+\|\nabla u_0\|^{2}_{L^{2}(G)}
 +2p^{-1}\|u_0\|^{p}_{L^{p}(G)}\} \\
&\leq 2 \int^{T}_0 < u'' -\Delta u + | u|^{p-2}u -f, \varphi' > dt
\end{align*}
for all $\varphi \in C^{\infty}_0(0,T;H^{1}_0(G)\cap L^{p}(G))$.
We have used the fact that $f\in K^{\perp} $ and that $ u' $ is in $K$. Set
\[
 \Phi(u, \varphi')=2 \int^{T}_0 < u''- \Delta u + | u|^{p-2}u -f, \varphi'> dt
 \]
and
\begin{align*}
 E(u) &= \liminf\|u_{\varepsilon}'(T)\|^{2}_{L^{2}(G)}
 + \|\nabla u(T)\|^{2}_{L^{2}(G)}+2p^{-1}\|u(T)\|^{p}_{L^{p}(G)}
 -\|u_1\|^{2}_{L^{2}(G)}\\
 &\quad -\|\nabla u_0\|^{2}_{L^{2}(G)}- 2p^{-1}\|u_0\|^{p}_{L^{p}(G)}
 \end{align*}
Then
\[
 E(u) \leq \Phi(u,\varphi') \quad
\forall \varphi \in C^{\infty}_0(0,T; H^{1}_0(G)\cap L^{p}(G)).
 \]
In particular
\[
 E(u) \leq \Phi(u, -\varphi') \quad \forall \varphi
\in C^{\infty}_0(0,T;H^{1}_0(G) \cap L^{p}(G))
 \]
Hence
\[
 E(u) \leq \Phi(u,\varphi') \leq - E(u) \quad
\forall \varphi \in C^{\infty}_0(0,T;H^{1}_0(G) \cap L^{p}(G))
 \]
Let $\lambda >0 $ then $ \lambda^{-1} \varphi $ is in
$ C^{\infty}_0(0,T; H^{1}_0(G) \cap L^{p}(G))$ and we have
\[
 \lambda E(u) \leq \Phi(u,\varphi') \leq - \lambda E(u)
 \]
Letting $\lambda \to 0$ we obtain
 \[
 \Phi(u,\,\varphi') = \int^{T}_0\langle u''-\Delta u +| u|^{p-2}u -f,
\varphi'\rangle dt = 0
 \]
for all $\,\varphi \in C^{\infty}_0(0,T;H^{1}_0(G)\cap L^{p}(G))$.
Therefore
\[
 \{ u''-\Delta u + | u|^{p-2}u -f \}'= 0 \quad \text{in }
{\mathcal D}'(0,T;[H^{1}_0(G)\cap L^{p}(G)]^{*}).
 \]
It follows that
\begin{equation}\label{exact5}
 u'' -\Delta u + | u|^{p-2}- f = g_{f}(u_0,u_1) \quad \text{in }
{\mathcal D}'(0,T; [H^{1}_0(G)\cap L^{p}(G)]^{*})
 \end{equation}
for any $ g_{f}(u_0,u_1)$ in $ [H^{1}_0(G)\cap L^{p}(G)]^{*}$.

(3) We now show that $ g_{f}(u_0,u_1)$ is uniquely defined.
From \eqref{exact3} we have
\begin{align*}
&-\int^{T}_0( u'_{\varepsilon}, \varphi')dt
 + \int^{T}_0( \nabla(\varepsilon u'_{\varepsilon}+ u_{\varepsilon}),
 \nabla \varphi)dt
 +\int^{T}_0( | u_{\varepsilon}|^{p-2} u_{\varepsilon}, \varphi) dt\\
&+\varepsilon^{-1} \int^{T}_0( \beta(u'_{\varepsilon}), \varphi)dt
 -\int^{T}_0(f, \varphi) dt =0
\end{align*}
for all $ \varphi \in C^{\infty}_0(0,T; H^{1}_0(G) \cap L^{p}(G))$.

Letting $ \varepsilon \to 0$  we obtain
\begin{align*}
&-\int^{T}_0( u', \varphi')dt +\int^{T}_0( \nabla u, \nabla \varphi) dt \\
&+\int^{T}_0( | u|^{p-2},\,\varphi)dt
 + \lim_{\varepsilon \to 0}\varepsilon^{-1}\int^{T}_0( \beta(u'_{\varepsilon}),
 \varphi) dt \\
&= \int^{T}_0(f,\,\varphi)dt
 \end{align*}
for all $\varphi \in C^{\infty}_0(0,T; H^{1}_0(G) \cap L^{p}(G))$.
Thus,
\[
u'' -\Delta u +| u|^{p-2}u + \lim_{\varepsilon \to 0}\varepsilon^{-1}
\beta(u'_{\varepsilon}) =f \quad \text{in }
{\mathcal D}'(0,T;[H^{1}_0(G)\cap L^{p}(G)]^{*})
 \]
Comparing with \eqref{exact4} and we have
\[
 \lim_{\varepsilon \to 0}\varepsilon^{-1} \beta(u'_{\varepsilon})
= g_{f}(u_0,u_1) \quad \text{in } {\mathcal D}'(0,T;[H^{1}_0(G)
\cap L^{p}(G)]^{*})
 \]
It is clear that if $ h$ is any other element of $ (H^{1}_0(G)\cap L^{p}(G))^{*}$
in \eqref{exact5} then
\[
 h=g_{f}(u_0,u_1) =\lim_{\varepsilon \to 0}
\varepsilon^{-1}\beta(u'_{\varepsilon})\quad \text{in }
 {\mathcal D}'(0,T;[H^{1}_0\cap L^{p}(G)]^{*})
 \]

(4) Suppose that $ v$ is a solution of the problem
\begin{gather*}
v''- \Delta v + | v|^{p-2}v + g_{f}(u_0,u_1) =f \quad \text{in} G\times (0,T),\\
v =0 \text{ on } \partial G\times (0,T),\quad v(\cdot,0)=u_0,\quad v'(\cdot,0)=u_1
 \end{gather*}
Then an argument as in Lions \cite[p.14-15]{t2},  shows that $u=v$ and
completes the proof.
\end{proof}

\begin{lemma}\label{lem3.4}
Let $ g_{f}(u_0,u_1)$ be as in Lemma \ref{lem3.3} then
\begin{align*}
&\|g_{f}(u_0,u_1)\|_{[H^{1}_0(G)\cap L^{p}(G)]^{*}}\\
& \leq C\{ 1+ \|u_0\|^{p-1}_{H^{1}_0(G)}+\|u_1\|^{p-1}_{L^{2}(G)}
 +\|u_0\|^{p-1}_{L^{p}(G)}+ \|f\|_{L^{2}(0,T;L^{2}(G))}\}
 \end{align*}
\end{lemma}

\begin{proof}
Let $ h $ be in $H^{1}_0(G)\cap L^{p}(G)$ and let $ \zeta $ be in
$C^{\infty}_0(0,T) $ with $\zeta \geq 0$. From Lemma \ref{lem3.3} we have
\begin{align*}
 \int^{T}_0 \zeta ( g_{f}(u_0,u_1), h)
&= \int^{T}_0( f,\zeta h) dt + \int^{T}_0( u', \zeta' h)
 -\int^{T}_0(\nabla u,\,\zeta \nabla h) dt\\
&\quad - \int^{T}_0 ( | u|^{p-2}u, \zeta h)dt
 \end{align*}
Hence
\begin{align*}
 \alpha |(g_{f}(u_0,u_1),h)|
&\leq C\Big\{ \|f\|_{L^{2}(0,T;L^{2}(G))}+ \|u'\|_{L^{2}(0,T;L^{2}(G))}
 + \|\nabla u\|_{L^{2}(0,T;L^{2}(G))}\\
&\quad +\|u\|^{p-1}_{L^{\infty}(0,T;L^{p}(G))} \Big\} \|h\|_{H^{1}_0(G)}
\end{align*}
for all $h$ in $H^{1}_0(G)\cap L^{p}(G) $ and where
 \[
 \alpha =\int^{T}_0 \zeta dt >0.
 \]
Since $ 2\leq p $, it follows from the estimate of Lemma \ref{lem3.3} that
\begin{align*}
&\|g_{f}(u_0,u_1)\|_{[H^{1}_0(G)\cap L^{p}(G)]^{*}} \\
&\leq C \Big\{ 1+ \|u_0\|_{H^{1}_0(G)}+\|u_1\|_{L^{2}(G)}+\|u_0\|^{p-1}_{L^{p}(G)}
 +\|f\|_{L^{2}(0,T;L^{2}(G))}\Big\}
 \end{align*}
The proof is complete.
\end{proof}

 \begin{lemma}\label{lem3.5} %L:exact 6
 Let $ u''_{\varepsilon}$ be as in Lemma \ref{lem3.1}. Then
\[
 \|u''_{\varepsilon}\|_{L^{2}(0,T;[H^{1}_0(G) \cap L^{p}(G)]^{*})}\leq C
 \]
 where $C$ is independent of $\varepsilon$. Moreover
\begin{gather*}
 u'_{\varepsilon} \to u'\quad\text{in } C(0,T;[H^{1}_0(G)\cap L^{p}(G)]^{*})
\cap [L^{\infty}(0,T;L^{2}(G))]_{\rm weak*}, \\
 \|u'(T)\|_{L^{2}(G)}\leq \liminf \|u'_{\varepsilon}(T)\|_{L^{2}(G)}
 \end{gather*}
\end{lemma}

 \begin{proof}
Let $\varphi $ be in $C^{\infty}_0(0,T;H^{1}_0(G)\cap L^{p}(G))$ and set
\[
 \gamma_{\varepsilon}(\varphi) =\int^{T}_0( u''_{\varepsilon}, \varphi)dt.
 \]

$\bullet$ Case 1: $ \gamma_{\varepsilon}(\varphi) \geq 0$. We have
\begin{align*}
&\lim | \int^{T}_0( u''_{\varepsilon}, \varphi)dt| \\
&= \lim \int^{T}_0(u''_{\varepsilon}, \varphi)dt\\
&= -\int^{T}_0(\nabla u ,\nabla \varphi)dt-\int^{T}_0(| u|^{p-2}u,\varphi)dt
 -\lim \varepsilon^{-1} \int^{T}_0(\beta(u'_{\varepsilon}),\varphi)dt
 +\int^{T}_0(f,\varphi)dt\\
&= -\int^{T}_0(\nabla u, \nabla \varphi)dt-\int^{T}_0(| u|^{p-2}u,\varphi)dt
-\int^{T}_0(g_{f}(u_0,u_1),\varphi)dt
 +\int^{T}_0(f, \varphi)dt\\
&\leq C\{\|u\|_{L^{2}(0,T;H^{1}_0(G))}
 + \|u\|^{p-1}_{L^{\infty}(0,T;L^{p}(G))}+\|f\|_{L^{2}(0,T;L^{2}(G))}\}\\
&\quad \times \|\varphi\|_{L^{2}(0,T;H^{1}_0(G)\cap L^{p}(G))}
 \end{align*}



 $\bullet$ Case 2: $ \gamma_{\varepsilon}(\varphi) \leq 0$. Then we have
\begin{align*}
&\lim | \int^{T}_0(u''_{\varepsilon}, \varphi)dt| \\
&= \lim -\int^{T}_0(u''_{\varepsilon},\varphi)dt\\
&= \int^{T}_0(\nabla u,\nabla \varphi)dt +\int^{T}_0(| u|^{p-2}u,\varphi)dt
+\int^{T}_0(g_{f}(u_0,u_1),\varphi)dt
-\int^{T}_0(f, \varphi)dt\\
&\leq  C\{\|u\|_{L^{2}(0,T;H^{1}_0(G))}+\|u\|^{p-1}_{L^{\infty}(0,T;L^{p}(G))}
+\|f\|_{L^{2}(0,T;L^{2}(G))}\} \\
&\quad \times \|\varphi\|_{L^{2}(0,T;H^{1}_0(G)\cap L^{p}(G))}
\end{align*}
Hence
\[
 \lim \big| \int^{T}_0(u''_{\varepsilon}, \varphi)dt|
\leq M\|\varphi\|_{L^{2}(0,T;H^{1}_0(G)\cap L^{p}(G))}\quad
 \forall \varphi \in C^{\infty}_0(0,T;H^{1}_0(G)\cap L^{p}(G)).
\]
Since  $ C^{\infty}_0(0,T;H^{1}_0(G)\cap L^{p}(G))$ is dense in
$ L^{2}(0,T;H^{1}_0(G)\cap L^{p}(G))$, we have
\[
 \|u''_{\varepsilon}\|_{L^{2}(0,T;[H^{1}_0(G)\cap L^{p}(G)]^{*})}\leq M
 \]
The other assertions of the lemma are trivial to verify.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm3.1}]
 Taking $ u_1=0 $, from Lemma \ref{lem3.3} there exists $g_{f}(u_0)$ in
$[H^{1}_0(G) \cap L^{p}(G)]^{*}$ and
\[
 \{ u , u'\} \in L^{\infty}(0,T; H^{1}_0(G) \cap L^{p}(G))
\times L^{\infty}(0,T;L^{2}(G)),
\]
solution of the problem
\begin{gather*}
 u'' -\Delta u +| u|^{p-2}u= f-g_{f}(u_0) \quad \text{in } G\times (0,T), \\
 u=0 \text{ on }\partial G \times (0,T),\quad  u(\cdot,0)=u(\cdot,T)=u_0,\quad
 u'(\cdot,0)=0.
 \end{gather*}
From the estimate in Lemma \ref{lem3.3} we obtain
\[
 \|u'(T)\|^{2}_{L^{2}(G)} \leq 0
\]
as $ f $ is in $K^{\perp} $ and $ u'$ is in $K$. Therefore

\[
 u'(\cdot,0)=0= u'(\cdot,T).
\]
The proof is complete.
\end{proof}

 \section{Semi exact controllability}

 In this section we shall establish the existence of time-periodic solutions
 of a nonlinear wave equation with the solution taking a prescribed value
at $ t=0$.

 \begin{theorem} \label{thm4.1} %\label{control}
Let $\{ f, u_0\} $ be in $K^{\perp} \times \{H^{1}_0(G)\cap L^{p}(G)\} $.
There exists
\begin{itemize}
\item[(i)] $ g_{f}(u_0)$ in $[H^{1}_0(G)\cap L^{p}(G)]^{*} $

\item[(ii)] a solution $u$ of the problem
\begin{equation}\label{control1}
\begin{gathered}
 u''- \Delta u + | u |^{p-2}u = f -g_{f}(u_0) \quad \text{in } G\times (0,T),\\
 u=0 \text{ on } \partial G\times (0,T),\quad
 \{ u, u'\}\big|_{t=0}=\{ u, u'\}\big|_{t=T}=
 \{ u_0, u'(0)\}
\end{gathered}
 \end{equation}
with $ \{u, u'\}$ in $ L^{\infty}(0,T;H^{1}_0(G)\cap L^{p}(G))
\times L^{\infty}(0,T;L^{2}(G))$.
\end{itemize}
\end{theorem}

 As $ u'(\cdot,0) $ and $ u'(\cdot,T) $ are not required to take a
prescribed value and are allowed to take the same value derived from
the equation, we have only half of the exact controllability condition.

 A simple corollary of the theorem yields the existence of time-periodic
solutions of linear wave equations.

 \begin{corollary}\label{coro1}
Let $ f $ be in $K^{\perp} $ then there exists
$ \{\tilde{u}, \tilde{u}'\} $ in
$L^{\infty}(0,T;H^{1}_0(G))\times L^{\infty}(0,T;L^{2}(G))$,
 solution of the problem
\begin{equation}\label{control2}
\begin{gathered}
 \tilde{u}'' - \Delta \tilde{u} + \tilde{u}=f \quad \text{in } G \times (0,T),\\
 \tilde{u}=0 \text{ on }\partial G\times (0,T),\quad
\{\tilde{u} ,\tilde{u}'\}\big|_{t=0}= \{ \tilde{u}, \tilde{u}'\}\big|_{t=T}
\end{gathered}
 \end{equation}
\end{corollary}

 \begin{proof}
Given $ f $ in $K^{\perp}$ and a $ u_0$ in $H^{1}_0(G)$ it follows from
the theorem that there exists $ g_{f}(u_0)$ in $H^{-1}(G)$ and associated
with it a solution $ u$ of the problem
\begin{gather*}
 u'' -\Delta u + u + g_{f}(u_0) = f \quad \text{in }  G\times (0,T),\\
 u=0 \text{ on }\partial G\times (0,T), \quad
 \{ u,u'\}\big|_{t=0} = \{ u,u'\}\big|_{t=T}=\{u_0, u'(0)\}
 \end{gather*}
Consider the elliptic boundary problem
\[
 -\Delta \hat{u} + \hat{u}= g_{f}(u_0) \text{ in } G,\quad
 \hat{u} =0 \text{ on } \partial G.
 \]
There exists a unique solution $\hat{u}$ in $H^{1}_0(G)$ of the problem.
Set $ \tilde{u}= u+ \hat{u}$ and the corollary is proved
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.1}] 
(1) Let
\[
 \{f, u_0, u_1\}\in K^{\perp}\times \{H^{1}_0(G)\cap L^{p}(G) \}\times L^{2}(G)
\]
then there exists $ g_{f}(u_0,u_1)$ in $ [H^{1}_0(G)\cap L^{p}(G)]^{*}$
and associated with it, a unique solution $u$ of the problem
\begin{equation}\label{control3}
\begin{gathered}
 u''-\Delta u + | u |^{p-2}u + g_{f}(u_0,u_1)
= f \quad \text{in } G\times (0,T),\\
 u=0 \text{ on }\partial G\times (0,T),\quad  u(\cdot,0)=u_0=u(\cdot,T),\quad
 u'(\cdot,0)=u_1
\end{gathered}
 \end{equation}
Moreover Lemmas \ref{lem3.4} and \ref{lem3.5} show that
\[
 \|u'(T)\|^{2}_{L^{2}(G)} \leq \|u_1\|^{2}_{L^{2}(G)}
 \]


(2) Let
$ {\mathcal B}=\{ v:\, \|v\|_{L^{2}(G)}\leq 1 \}$.
Then it is clear that ${\mathcal B}$ is a compact convex subset of
$ [H^{1}_0(G)\cap L^{p}(G)]^{*}$. Denote by ${\mathcal A}$ the mapping
of ${\mathcal B} $ into ${\mathcal B}$ given by
\begin{equation}\label{control4}
 {\mathcal A}(u_1) = u'(T)
 \end{equation}
as $ f \in K^{\perp}$ and $ u' $ is in $K$. The mapping is well-defined
and takes ${\mathcal B}$ into ${\mathcal B}$.

We now show that ${\mathcal A}$ is a $[H^{1}_0(G)\cap L^{p}(G)]^{*}$-continuous
mapping. Let $ u_{1,n} $ in ${\mathcal B}$, then corresponding to
$ \{ f, u_0, u_{1,n}\} $, there exists $ g_{f}(u_0,u_{1,n}) $ in
$[H^{1}_0(G)\cap L^{p}(G)]^{*}$ and $ u_{n} $, solution of the problem
\begin{gather*}
 u''_{n}-\Delta u_{n} +| u_{n}|^{p-2} + g_{f}(u_0,u_{1,n})
= f\quad \text{in } G \times (0,T),\\
 u_{n}=0 \text{ on }\partial G\times (0,T), \quad
 u_{n}(0)=u_0=u_{n}(T),\quad u'_{n}(0)=u_{1,n}
 \end{gather*}

From Lemmas \ref{lem3.3}--\ref{lem3.5} we get
\[
 \|g_{f}(u_0,u_{1,n})\|_{[H^{1}_0(G)\cap L^{p}(G)]^{*}}+ \|u_{n}\|_{L^{\infty}(0,T;H^{1}_0(G)\cap L^{p}(G))}+ \|u'_{n}\|_{L^{\infty}(0,T;L^{2}(G))}\leq C
\]
We have a subsequence such that
\[
 \{ u_{n}, u'_{n}, g_{f}(u_0,u_{1,n})\} \to \{ u, u', g_{f}(u_0, u_1) \}
 \]
in
\[
 [L^{\infty}(0,T; H^{1}_0(G)\cap L^{p}(G))]_{weak^{*}}\times [L^{\infty}(0,T;L^{2}(G)]_{weak^{*}}\times [H^{1}_0(G)\cap L^{p}(G)]_{weak^{*}}
 \]
It is clear that
$ \{u_{n}, \, u'_{n} \} \to \{ u , u'\}$ in
$ C(0,T;L^{2}(G)) \times C(0,T;[H^{1}_0(G)\cap L^{p}(G)]^{*})$,
 and therefore
\[
 \{u_{n}(0), u'_{n}(0), u_{n}'(T) \} \to \{ u(0), u'(0), u'(T)\}
 \]
in
$ L^{2}(G) \times [H^{1}_0(G)\cap L^{p}(G)]^{*}
\times [H^{1}_0(G)\cap L^{p}(G)]^{*}$.
 Hence $ u(0)=u_0=u(T) $ and $ u'(0)=u_1$. A standard argument shows that
\[
 | u_{n}|^{p-2}u_{n} \to | u |^{p-2}u \quad \text{in }[L^{q}(0,T;L^{q}(G)]_{\rm weak}
 \]
and thus,
\begin{gather*}
 u''- \Delta u +| u|^{p-2}u + g_{f}(u_0,u_1) = f \quad \text{in } G\times(0,T),\\
 u=0 \text{ on }\partial G\times (0,T) ,\quad  u(0)=u_0=u(T),quad  u'(0)=u_1
 \end{gather*}
It follows that $ {\mathcal A}(u_1) = u'(T)$.


An application of the Schauder fixed point theorem yields the existence of
$ \hat{u}_1$ in ${\mathcal B}$ such that ${\mathcal A}(\hat{u}_1)=\hat{u}_1$.
With $ u_0$ given and with the fixed point $ \hat{u}_1$, there exists as
in Lemma \ref{lem3.3} a control $g_{f}(u_0, \hat{u}_1)=\hat{g}_{f}(u_0) $
in $[H^{1}_0(G)\cap L^{p}(G)]^{*}$ and associated with the control, a solution of
\begin{gather*}
 \hat{u}''- \Delta \hat{u}+ | \hat{u}|^{p-2}\hat{u}
= f- \hat{g}_{f}(u_0) \quad \text{in }G\times (0,T),\\
 \hat{u}=0 \text{ on }\partial G\times (0,T), \quad
 \{\hat{u},\hat{u}'\}\big|_{t=0}=\{\hat{u},\hat{u}'\}\big|_{t=T}
 \end{gather*}
with $ \hat{u}(0)=\hat{u}(T)= u_0$. The theorem is proved.
\end{proof}

 \section{Periodic solutions}

 In this section we shall use $ u_0$ of Theorem \ref{thm4.1} as a control to show 
that for any given $ f\in K^{\perp}$, there exists
 \[
 \{ \tilde{f}, \tilde{u}_0, g_{\tilde{f}}(\tilde{u}_0)\}
\in K^{\perp}\times H^{1}_0(G)\cap L^{p}(G) \times [H^{1}_0(G)\cap L^{p}(G)]^{*}
 \]
such that $ f = \tilde{f}- g_{\tilde{f}}(\tilde{u}_0) $.
The main result of the section and of this article is the following theorem.

 \begin{theorem}\label{period} 
Let $ f $ be in $K^{\perp}$. Then there exists a solution $\{u,u'\}$ in the space
$ L^{\infty}(0,T;H^{1}_0(G)\cap L^{p}(G))\times L^{\infty}(0,T; L^{2}(G)) $ 
for the problem
 \begin{equation}\label{period1}
\begin{gathered}
 u'' -\Delta u + | u|^{p-2}u = f \quad \text{in } G \times (0,T),\\
 u=0 \text{ on }\partial G\times (0,T) ,\quad 
 \{ u, u'\}\big|_{t=0}= \{ u, u'\}\big|_{t=T}.
\end{gathered}
 \end{equation}
 \end{theorem}

 \begin{proof}
First we consider the initial boundary-value problem
\begin{equation}\label{period2}
\begin{gathered}
 w'' -\Delta w + | w|^{p-2}w = f \quad \text{in }G\times (0,T),\\
 w=0 \text{ on }\partial G\times (0,T),\quad  \{w, w'\}\big|_{t=0}=\{ u_0,u_1\}
\end{gathered}
 \end{equation}
It is known that for a given
\[
 \{f, u_0, u_1\} \in L^{2}(0,T;L^{2}(G))\times 
\{H^{1}_0(G)\cap L^{p}(G)\times L^{2}(G)\},
 \]
there exists a unique solution of \eqref{period2} with
\begin{align*}
 &\|w'(t)\|^{2}_{L^{2}(G)}+ \|\nabla w(t)\|^{2}_{L^{2}(G)}
 + 2/p\|w(t)\|^{p}_{L^{p}(G)}\\
 &\leq e^{t} \{ \|u_1\|^{2}_{L^{2}(G)}+ \|\nabla u_0\|^{2}_{L^{2}(G)}
+ 2/p \|u_0\|^{p}_{L^{p}(G)}+ \|f\|^{2}_{L^{2}(0,T;L^{2}(G))}\}
 \end{align*}
Consider the optimization problem
\begin{equation}\label{period3}
\begin{aligned}
 \alpha(f)&= \inf \Big\{ \|u(0)-u(T)\|_{L^{2}(G)}
+ \|u'(0)-u'(T)\|_{L^{2}(G)}: u \text{ is the solution of \eqref{period2}} \\
 &\quad  \forall \{u_0, u_1\} \text{ with } \|u_0\|_{H^{1}_0(G)\cap L^{p}(G)}
 + \|u_1\|_{L^{2}(G)}\leq R \Big\}
\end{aligned}
 \end{equation}
From Theorem \ref{thm4.1} we know that for each $ u_0$ in $H^{1}_0(G) \cap L^{p}(G)$, 
for a given $f$ in $K^{\perp}$ there exists $g_{f}(u_0) $ in 
$[H^{1}_0(G)\cap L^{p}(G)]^{*}$ and a solution $ u$ of
\begin{gather*}
 u''-\Delta u + | u|^{p-2}u = f-g_{f}(u_0) \quad \text{in }G\times (0,T),\\
 u=0 \text{ on }\partial G\times (0,T) ,\quad  u(0)=u_0=u(T),\quad u'(0)=u'(T).
 \end{gather*}
Let
\[
 S =\cup_{f\in K^{\perp}}\big\{ f \oplus \{ -g_{f}(u_0):
 u_0 \in H^{1}_0(G)\cap L^{p}(G)\}\big\},
 \]
where $ g_{f}(u_0) $ is as in Theorem \ref{thm4.1} and thus, $ \alpha(f-g_{f}(u_0)) =0$.

The set $S$ is non-empty and $ L^{2}(G) =L^{2}(G)\oplus 0 \subset S $.
Indeed $ L^{2}(G)\subset K^{\perp}$ as the stationary solution of the 
elliptic boundary problem
\[
 -\Delta w +| w|^{p-2}w = f(x) \text{ in }  G,\quad
 w=0 \text{ on }\partial G
\]
is time-periodic. Thus
$ \alpha(f)=0 =\alpha( f- g_{f})$ and $ g_{f}=0$, 
and hence $ f $ is in $S$.

We have
\[
 S \subset K^{\perp}\oplus \cup_{h\in K^{\perp}} 
\{ -g_{h}(u_0): u_0\in H^{1}_0(G)\cap L^{p}(G)\}
\]
Thus,
\begin{align*}
 L^{2}(G) &= \{L^{2}(G) \oplus 0\} \cap \{K^{\perp}\oplus 0\}\\
 &\subset S \cap \{K^{\perp}\oplus 0\}\\
 &\subset \big\{ K^{\perp}\oplus\cup_{h\in K^{\perp}}\{- g_{h}(u_0):
 u_0\in H^{1}_0(G)\cap L^{p}(G)\}\big\}\cap \{ K^{\perp}\oplus 0\}\\
 & \subset K^{\perp}\oplus 0.
 \end{align*}
Indeed
\[
 0 \in \cup_{h\in K^{\perp}} \{ -g_{h}(u_0): u_0\in H^{1}_0(G) \cap L^{p}(G)\}
 \]
as $\alpha(\hat{f})=0 = g_{\hat{f}} $ for $ \hat{f}\in L^{2}(G) $.
Hence $ \{K^{\perp}\oplus 0\} \subset S $.

Let $ f $ in $\{K^{\perp}\oplus 0\} $ then there exists
$h$ in $K^{\perp}$ and $ g_{h}(u_0)$ for some $ u_0$ in 
$H^{1}_0(G)\cap L^{p}(G) $ such that
\[
 f = h- g_{h}(u_0),\quad \alpha(h-g_{h}(u_0)) =0
\]
and therefore $ \alpha(f) =0$. Thus for $ f\in K^{\perp} $ there exists 
$ \tilde{u}$, solution of the problem
\begin{gather*}
 \tilde{u}''-\Delta \tilde{u}+ | \tilde{u}|^{p-2}\tilde{u}= f \quad
\text{in }G\times (0,T),\\
 \tilde{u}=0 \text{ on }\partial G \times (0,T),\quad
\{ \tilde{u},\tilde{u}'\}\big|_{t=0}=\{\tilde{u},
 \tilde{u}'\}\big|_{t=T}
 \end{gather*}
The proof is complete.
\end{proof}

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\end{document}