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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 55, pp. 1--7.\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-2017/55\hfil Nonlinear nonlocal problems in elasticity]
{Global well-posedness for nonlinear nonlocal Cauchy problems arising in elasticity}

\author[H. Bae, S. Ulusoy  \hfil EJDE-2017/55\hfilneg]
{Hantaek Bae, S\"uleyman Ulusoy}

\address{Hantaek Bae \newline
Department of Mathematical Sciences,
Ulsan National Institute of Science and Technology (UNIST), Korea}
\email{hantaek@unist.ac.kr}


\address{S\"uleyman Ulusoy \newline
Department Of Mathematics and Natural Sciences,
American University of Ras al Khaimah,
PO Box 10021,Ras Al Khaimah,  UAE}
\email{suleyman.ulusoy@aurak.ac.ae, suleymanulusoy@yahoo.com}

\dedicatory{Communicated by Jerry Bona}

\thanks{Submitted May 24, 2016. Published February 22, 2017.}
\subjclass[2010]{35Q74, 35L15, 74B20}
\keywords{Nonlinear nonlocal wave equations; kernel function; global solution}

\begin{abstract}
 In this article, we prove global well-posedness for a family of one
 dimensional nonlinear nonlocal Cauchy problems arising in elasticity.
 We consider the equation
 \[
 u_{tt}-\delta Lu_{xx}=\big(\beta \ast [(1-\delta)u+u^{2n+1}]\big)_{xx}\,,
 \]
 where $L$ is a differential operator, $\beta$ is an integral operator,
 and $\delta =0$ or 1. (Here, the case $\delta=1$ represents
 the additional doubly dispersive effect.)
 We prove the global well-posedness of the equation in energy spaces.
\end{abstract}

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


\section{Introduction}\label{sec:1}

There is a new trend on nonlinear nonlocal differential equations ,
because there exists a large class of problems in classical physics and
continuum mechanics (classical field theories) that fall outside their
traditional domain of application. The nonlocal effect is closely connected
to length scales. If the external length scales (e.g. crack length, wavelength)
are close to the internal scales (e.g. granular distance, lattice parameter),
local theories fail and we need to rely on nonlocal theories that can account
for the long-range interatomic attractions; for example,
(i) the energy balance law is postulated to remain in global form,
and (ii) a material point of the body is considered to be attracted by all
points of the body, at all past times \cite{ER}.

In this article, we study a family of nonlinear nonlocal equations arising
in elasticity. Let $u=u(t,x)=\partial_{x}X(t,x)$ of the displacement
$X(t,x)$ in one-dimensional, homogeneous, nonlinear and nonlocal elastic
infinite medium. In general, $u(t, x)$ satisfies
\[
u_{tt}=\text{second derivative of the stress } S(u).
\]
We now introduce the scalar function $\beta$, which is the attenuation or
influence function aimed to inject in the constitutive law the nonlocal
effect at the field $x$ produced by the local strain at $x'$:
\[
S(u):=\int_{\mathbb{R}}\beta(|x-x'|)\sigma(u(x',t)) dx'.
\]
By decomposing the (classical) local stress $\sigma(u)$ into its linear
and nonlinear parts,
 \[
\sigma(u)= u+g(u), \quad g(0)=0,
 \]
we have the following model \cite{HU}:
\begin{subequations}\label{eq:1.1}
\begin{gather}
 u_{tt}=\big(\beta \ast ( u + g(u))\big)_{xx}, \quad x \in \mathbb{R}, \; t>0,
 \label{eq:1.1 a}\\
 u(0,x)=\phi(x), \\
 u_t (0,x)=\psi(x),
\end{gather}
\end{subequations}
where the subscripts denote partial derivatives and the symbol $\ast$
denotes convolution in the spatial variable. It is clear from \eqref{eq:1.1 a}
that the well-posedness of the Cauchy problem depends crucially on the
structure of the kernel $\beta$. The choice of appropriate kernel functions
remains an interesting and hard open problem in the nonlocal theory
of elasticity, but some well known forms of kernel functions, such as
 triangular or Dirac delta functions, are currently in use for engineering problems,
see \cite{HU, ER, Pi} for examples of frequently used kernel functions.
For further motivations on the consideration of nonlinear nonlocal
equations in elasticity see the papers \cite{AG, BLL, CLE, HU, ER, E2, H, P, S}.
More recently, the global well-posedness of \eqref{eq:1.1} investigated
in \cite{HU} with the kernel $\beta$ of the form
\begin{equation} \label{eq:1.2}
0< \widehat{\beta}(\xi) \leq (1+|\xi|^2)^{-r/2}, \quad r>3
\end{equation}
and $g(x)$ is a super-linear function satisfying a certain growth condition.


In this article, we provide another set of $r$ and the nonlinearity $g$
that provides the global existence of solutions.
More precisely, we assume the following conditions:
\begin{equation} \label{beta}
 \begin{split}
 g(x)=x^{2n+1} \text{ for some $n\in \mathbb{N}$},\quad
\frac{6n+7}{2n+3}<r\leq \frac{6n+2}{2n+1}.
 \end{split}
\end{equation}
The range of $r$ will be computed in Section \ref{sec:2}. We note that
$g$ is defocusing in the sense
\begin{equation} \label{eq:1.5}
 G(x)=\int_0^{x} g(s)ds=\frac{x^{2n+2}}{2n+2}\ge 0
\end{equation}
so that all the terms in the left-hand side of \eqref{eq:2.6} are non-negative.
 Before stating our result, we define the operator $\mathcal{P}$:
\begin{equation} \label{eq:1.3}
\widehat{Pu}(\xi)=|\xi|^{-1}\big(\widehat{\beta}(\xi)\big)^{-1/2}\widehat{u}(\xi).
\end{equation}


\begin{theorem}\label{thm:1.1}
Suppose $r$ and $g$ satisfy the conditions in \eqref{beta}.
For any initial data $\phi, \psi \in H^{1}(\mathbb{R})$ with
additionally satisfying
\[
E_0:=\| \mathcal{P}\psi\|^2_{L^2(\mathbb{R})}+\|\phi\|^2_{L^2(\mathbb{R})}
+2\int_{\mathbb{R}} G(\phi(x))dx<\infty,
\]
there exists a unique global-in-time solution
$u\in C^{1}([0,\infty); H^{1}(\mathbb{R}))$ of \eqref{eq:1.1}.
\end{theorem}

\begin{remark} \rm
Let us compare our result with the global result in \cite{HU}.
Although the condition \eqref{eq:1.5} is stronger than the condition
$G(u)\ge -ku^2$ in \cite{HU}, our result improves the condition
of $r$ from $r>3$ in \cite{HU} to the condition in \eqref{beta}
 which is less than $3$.
\end{remark}

We next consider a general class of doubly dispersive nonlinear nonlocal
model \cite{BHA}:
\begin{subequations} \label{eq:1.7}
\begin{gather}
 u_{tt}-Lu_{xx}=(\beta\ast g(u))_{xx}, \quad x \in \mathbb{R}, \; t>0, \label{eq:1.7 a} \\
 u(0,x) = \phi(x), \\
 u_t (0,x) = \psi(x),
\end{gather}
\end{subequations}
where the symbols of the operators $L$ and $\beta$ are given by
\begin{equation} \label{eq:3.2}
\widehat{L}(\xi) =(1+|\xi|^2)^{\rho/2},\quad
0< \widehat{\beta}(\xi) \leq (1+|\xi|^2)^{-r/2}.
\end{equation}
The operator $L$ is a differential operator which provides a regularity
to the linear term, while $\beta$ is an integral operator for the
smoothness of the nonlinear term. This equation models the bi-directional
propagation of dispersive waves in a one dimensional, homogeneous,
nonlinearly and non-locally elastic infinite medium. For a background
of this equation, see \cite{BHA} and some of the references cited therein.
In \cite[[Theorem 6.3]{BHA}, the authors proved the global existence of a solution
under the conditions $\rho+r>1$ and $\rho+2r\ge 2$ using the representation
formula of $u$ in the Fourier variable.

In this article, we provide another set of $\rho$, $r$ and the nonlinearity $g$
 that provides the global existence of solutions. We assume that $\rho$,
$r$ and $g$ satisfy the conditions
\begin{equation} \label{beta2}
 \begin{split}
 g(x)=x^{2n+1} \text{ for some $n\in \mathbb{N}$},\quad \rho +r>1.
 \end{split}
\end{equation}

We fix the parameters as follows:
\[
k\in \mathbb{N}, \quad 2k+1+\rho\ge \frac{\rho+r}{2} +1, \quad
2k\ge\frac{r}{2}, \quad \frac{1}{2}<2k+\rho<2n+1.
\]

\begin{theorem}\label{thm:1.2}
Suppose $\rho$, $r$ and $g$ satisfy the conditions in \eqref{beta2}.
 Then for any initial data $\phi\in H^{2k+1+\rho}(\mathbb{R})$ and
$\psi \in H^{2k}(\mathbb{R})$ with
\begin{equation} \label{eq:1.10}
E_0:=\| \mathcal{P}\psi\|^2_{L^2(\mathbb{R})}
+\|\sqrt{L\beta^{-1}}\phi\|^2_{L^2(\mathbb{R})}
+2\int_{\mathbb{R}} G(\phi(x))dx<\infty,
\end{equation}
there exists a unique global-in-time solution
\[
u\in C([0,\infty); H^{2k+1+\rho}(\mathbb{R}))
\cap C^{1}([0,\infty); H^{2k}(\mathbb{R}))
\]
such that
\begin{equation} \label{eq:1.11}
 \begin{gathered}
 \| \mathcal{P}u_{t}(t)\|^2_{L^2(\mathbb{R})}
+\|\sqrt{L\beta^{-1}}u(t)\|^2_{L^2(\mathbb{R})}
+2\int_{\mathbb{R}} G(u(t,x))dx=E_0,\\
 \|u_{t}(t)\|^2_{H^{2k}(\mathbb{R})} +\|u(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})}
\leq C(E_0, \phi, \psi,t).
 \end{gathered}
\end{equation}
\end{theorem}

\subsection*{Notation}
 $\widehat{f}(\xi)$ is the Fourier transform of $f$. $H^{s}$ is the energy
space whose norm is given by
\[
 \|u\|^2_{H^{s}(\mathbb{R})}=\int_{\mathbb{R}}
(1+|\xi|^2)^{s}|\widehat{u}(\xi)|^2d\xi.
\]
All constants will be denoted by $C$ that is a generic constant depending
only on the quantities specified in the context.


\section{Proof of main restuls} \label{sec:2}

We begin with a lemma dealing with the effect of composition by a smooth functions.

\begin{lemma}[\cite{Runst}] \label{lem:2.2}
For $g(x)=x^{2n+1}$ and $0\leq s<2n+1$, we have
\begin{gather*} %\label{eq:2.1}
 \|g(u)\|_{H^{s}(\mathbb{R})}\leq C(n, \| u\|_{L^{\infty}(\mathbb{R})})
 \|u\|_{H^{1}(\mathbb{R})}, \\
 \|g(u)-g(v)\|_{H^{s}(\mathbb{R})}
\leq C(\| u\|_{L^{\infty}(\mathbb{R})}, \|v\|_{L^{\infty}(\mathbb{R})},
 \| u\|_{H^{s}(\mathbb{R})}, \| v\|_{H^{s}(\mathbb{R})})\|u-v\|_{H^{s}(\mathbb{R})}.
\end{gather*}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm:1.1}]
Our analysis starts by rewriting \eqref{eq:1.1} as a $H^{1}-$valued
ordinary differential equations:
\begin{subequations}\label{eq:2.2}
\begin{gather}
 u_{t}= v, \ u(0,x)=\phi(x),\\
 v_{t}= K\ast f(u), \ v(0,x)=\psi(x),
\end{gather}
\end{subequations}
where $f(u)=u+g(u)$ and $K=\beta_{xx}$. We first note that
$\widehat{K}(\xi) \in L^{\infty}(\mathbb{R})$ from the condition \eqref{beta}
with $r>2$. So, $K=\beta_{xx}$ is a bounded operator in $H^{1}(\mathbb{R})$.
 Moreover, Lemma \ref{lem:2.2} implies that $K\ast f(u)$ is locally Lipschitz
on $H^{1}(\mathbb{R})$. Therefore, the local well-posedness with initial
data in $H^{1}(\mathbb{R})$ follows from the well-posedness of the system of
ordinary differential equations. Moreover, it is shown in \cite[Lemma 3.9]{HU}
 that there exists a global solution in $H^{1}(\mathbb{R})$ if and only if for
any $T>0$
\[
\limsup_{t\to T^{-}}\|u(t)\|_{L^{\infty}(\mathbb{R})}<\infty.
\]
Therefore, we focus on the $L^{\infty}$ norm of $u$. To this end, we
rewrite \eqref{eq:1.1 a} as
\begin{equation} \label{eq:2.5}
\mathcal{P}^2u_{tt}+u+g(u)=0.
\end{equation}
where $\mathcal{P}$ is defined in \eqref{eq:1.3}. Multiplying \eqref{eq:2.5}
by $2u_{t}$ and integrating in $x$, we have
\begin{equation} \label{eq:2.6}
\| \mathcal{P}u_{t}(t)\|^2_{L^2(\mathbb{R})}
+\|u(t)\|^2_{L^2(\mathbb{R})}+2\int_{\mathbb{R}} G(u(t,x))dx=E_0.
\end{equation}
As shown in \cite{HU}, the first term on the left-hand side of
\eqref{eq:2.6} implies that
\[
\|u(t)\|_{H^{\frac{r}{2}-1}(\mathbb{R})}\leq C(E_0).
\]
By the Sobolev embedding, we have
\begin{equation} \label{Sobolev}
\| u(t)\|_{L^{\frac{2}{3-r}}(\mathbb{R})} \leq C(E_0) \quad \text{for $r<3$}.
\end{equation}
Using this, we estimate the nonlinear term $K\ast g(u)$.
By Hausdorff-Young inequality,
\[
\| K\ast g(u)\|_{L^{\infty}(\mathbb{R})}
\leq C \| K\|_{L^{q'}(\mathbb{R})} \|g(u) \|_{L^{q}(\mathbb{R})}
\leq C \| \widehat{K}\|_{L^{q}(\mathbb{R})} \|g(u) \|_{L^{q}(\mathbb{R})},
\quad 1<q\leq 2.
\]
To bound $\| \widehat{K}\|_{L^{q}(\mathbb{R})}
=\| |\xi|^2\widehat{\beta}\|_{L^{q}(\mathbb{R})}$, we need
\begin{equation} \label{r1}
(r-2)q>1.
\end{equation}
On the other hand, to bound $\|g(u) \|_{L^{q}(\mathbb{R})}$ using
 \eqref{Sobolev}, we also need
\begin{equation}\label{r2}
(2n+1)q\leq \frac{2}{3-r}.
\end{equation}
By solving \eqref{r1} and \eqref{r2} for $r$, we have
\[
\frac{6n+7}{2n+3}<r.
\]
Moreover, by choosing
\[
r\leq \frac{6n+2}{2n+1}
\]
we have $q\leq 2$ from \eqref{r2}. Combining these two inequalities for $r$,
we obtain that
\begin{equation} \label{range of r}
\frac{6n+7}{2n+3}<r\leq \frac{6n+2}{2n+1}
\end{equation}
and under this condition, we have
\begin{equation} \label{bound of nonlinear terms 1}
\| K\ast g(u)\|_{L^{\infty}(\mathbb{R})} \leq C(E_0)
\end{equation}
Moreover, since
\[
\frac{5}{2}<\frac{6n+7}{2n+3},
\]
$K\in L^2(\mathbb{R})$. Therefore,
\begin{equation}\label{bound of nonlinear terms 2}
 \| K\ast u\|_{L^{\infty}(\mathbb{R})}
\leq \| K\|_{L^2(\mathbb{R})} \| u\|_{L^2(\mathbb{R})} \leq C(E_0).
\end{equation}
We now bound $\|u\|_{L^{\infty}(\mathbb{R})}$ using
\eqref{bound of nonlinear terms 1} and \eqref{bound of nonlinear terms 2}.
Integrating \eqref{eq:1.1} twice in time, we have
\[
u(t,x)=\phi(x)+t\psi(x) +\int^{t}_0(t-s)(K\ast f(u(s)))ds.
\]
Since
\[
\|K\ast f(u)\|_{L^{\infty}(\mathbb{R})}
\leq \| K\ast u\|_{L^{\infty}(\mathbb{R})}
+ \| K\ast g(u)\|_{L^{\infty}(\mathbb{R})} \leq C(E_0),
\]
we conclude that
\begin{equation} \label{eq:2.3}
\|u(t)\|_{L^{\infty}(\mathbb{R})}
\leq \|\phi\|_{L^{\infty}(\mathbb{R})}+t\|\psi\|_{L^{\infty}(\mathbb{R})}
+C(E_0)t^2.
\end{equation}
Therefore, $\|u(t)\|_{L^{\infty}(\mathbb{R})}$ does not blow up in finite time.
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:1.2}]
We begin with the bounds of $\|u\|_{L^2(\mathbb{R})}$ and
$\|u\|_{L^{\infty}(\mathbb{R})}$. To this end, we rewrite \eqref{eq:1.7 a} as
\begin{equation} \label{eq:3.10}
\mathcal{P}^2u_{tt}+L\beta^{-1}u+g(u)=0.
\end{equation}
Multiplying \eqref{eq:3.10} by $2u_{t}$ and integrating in $x$, we have
\begin{equation} \label{eq:3.11}
\| \mathcal{P}u_{t}(t)\|^2_{L^2(\mathbb{R})}
+\|\sqrt{L\beta^{-1}}u(t)\|^2_{L^2(\mathbb{R})}
+2\int_{\mathbb{R}} G(u)(t,x)dx=E_0.
\end{equation}
Since
\[
\sqrt{\widehat{L}(\xi)} (\widehat{\beta}(\xi))^{-1/2}
\ge (1+|\xi|^2)^{\frac{\rho+r}{4}} \quad \text{and} \quad \rho+r>1
\]
Equality \eqref{eq:3.11} implies
\begin{equation} \label{eq:3.12}
\|u(t)\|_{L^2(\mathbb{R})} \leq E_0, \quad
\|u(t)\|_{L^{\infty}(\mathbb{R})} \leq E_0.
\end{equation}
We next obtain the energy estimates. Let $\sqrt{-\Delta}=\Lambda$. Then, we have
\begin{equation} \label{total energy}
\begin{split}
&\frac{d}{dt} \|u_{t}\|^2_{H^{2k}(\mathbb{R})}
 +\frac{d}{dt}\sum^{k}_{i=0}\|\Lambda^{2i}\sqrt{L}u_{x} \|^2_{L^2(\mathbb{R})} \\
&\leq \sum^{k}_{i=0}\| \Lambda^{2i}\beta_{xx}\ast g(u)\|^2_{L^2(\mathbb{R})}
 + \|u_{t}\|^2_{H^{2k}(\mathbb{R})}.
\end{split}
\end{equation}
Since for $\rho+r\ge 1$ and $i=0,1,2,\dots,k$
\[
(1+\left|\xi\right|)^{2i}|\xi|^2 (1+|\xi|^2)^{-\frac{r}{2}}
\leq C (1+\left|\xi\right|)^{2i}|\xi| (1+|\xi|^2)^{\frac{\rho}{2}}
\leq C (1+|\xi|)^{2k+1+\rho},
\]
by Lemma \ref{lem:2.2} we have
\begin{equation} \label{control of g term}
\begin{split}
\sum^{k}_{i=0}\| \Lambda^{2i}\beta_{xx}\ast g(u)\|^2_{L^2(\mathbb{R})}
&\leq C \| g(u)\|^2_{H^{2k+1+\rho}(\mathbb{R})} \\
&\leq C(\|u\|_{L^{\infty}(\mathbb{R})}, n)\|u\|^2_{H^{2k+1+\rho}(\mathbb{R})}.
\end{split}
\end{equation}
Therefore, by \eqref{total energy}, \eqref{control of g term},
\eqref{eq:3.11} and \eqref{eq:3.12},
\begin{equation}
\begin{split}
& \|u_{t}(t)\|^2_{H^{2k}(\mathbb{R})} +\|u(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})} \\
&\leq C(\phi, \psi, E_0) +\int^{t}_0 \|u_{t}(s)\|^2_{H^{2k}(\mathbb{R})}ds \\
&\leq C(\phi, \psi, E_0) +\int^{t}_0 [\|u_{t}(s)\|^2_{H^{2k}(\mathbb{R})}
+\|u(s)\|^2_{H^{2k+1+\rho}(\mathbb{R})}]ds
\end{split}
\end{equation}
and thus by Gronwall's inequality, we obtain
\begin{equation} \label{conclusion}
\|u_{t}(t)\|^2_{H^{2k}(\mathbb{R})} +\|u(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})}
\leq C(\phi, \psi, E_0)e^{t}.
\end{equation}
To obtain a unique solution $u$, we iterate the equation by defining $u^{l}$
as a solution of
\begin{subequations}
\begin{gather}
 u^{l}_{tt}-Lu^{l}_{xx}=\beta_{xx}\ast g(u^{l-1}), \\
 u^{l}(0,x) = \phi(x), \\
 u^{l}_t (0,x) = \psi(x).
\end{gather}
\end{subequations}
Let $\omega^{l}=u^{l}-u^{l-1}$. Then, $\omega^{l}$ satisfies
\begin{subequations}
\begin{gather}
 \omega^{l}_{tt}-L\omega^{l}_{xx}=\beta_{xx}\ast \left[g(u^{l-1}) - g(u^{l-2})\right], \\
 \omega^{l}(0,x) =0, \quad \omega^{l}_t (0,x) =0.
\end{gather}
\end{subequations}
By following the calculation above, we have
\begin{equation}
\frac{d}{dt}\| \mathcal{P}\omega^{l}_{t}(t)\|^2_{L^2(\mathbb{R})}
+\frac{d}{dt}\|\sqrt{L\beta^{-1}}\omega^{l}(t)\|^2_{L^2(\mathbb{R})}
\leq C(E_0) \| \omega^{l-1}\|^2_{L^2(\mathbb{R})}
+ \| \omega^{l}_{t}\|^2_{L^2(\mathbb{R})}
\end{equation}
and
\begin{equation}
\begin{split}
& \frac{d}{dt} \|\omega^{l}_{t}\|^2_{H^{2k}(\mathbb{R})}
 +\frac{d}{dt} \| \omega^{l}(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})} \\
&\leq C \| g(u^{l-1}) - g(u^{l-2})\|^2_{H^{2k+1+\rho}(\mathbb{R})}
 + \|\omega^{l}_{t}\|^2_{H^{2k}(\mathbb{R})} \\
&\leq C (\phi, \psi, E_0) \| \omega^{l-1}\|^2_{H^{2k+1+\rho}(\mathbb{R})}
 + \|\omega^{l}_{t}\|^2_{H^{2k}(\mathbb{R})}.
\end{split}
\end{equation}
This implies
\begin{equation}
\begin{split}
& \sup_{0\leq t\leq T} \big[\|\omega^{l}_{t}(t)\|^2_{H^{2k}(\mathbb{R})}
 +\|\omega^{l}(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})}\big] \\
& \leq C (\phi, \psi, E_0)T\sup_{0\leq t\leq T}
\big[\|\omega^{l-1}_{t}(t)\|^2_{H^{2k}(\mathbb{R})}
 +\|\omega^{l-1}(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})} \big]\\
& + T \sup_{0\leq t\leq T} \big[\|\omega^{l}_{t}(t)\|^2_{H^{2k}(\mathbb{R})}
+\|\omega^{l}(t)\|^2_{H^{2k+1+\rho}(\mathbb{R})}\big] .
\end{split}
\end{equation}
We take $T>0$ such that $T<1/2$ and $C (\phi, \psi, E_0)T<1/4$.
Then, $\{u^{l}(t): l\in \mathbb{N}\}$ is a Cauchy sequence in $E_{2k}(T)$ with
\[
\|u\|_{E_{2k}(T)}=\sup_{0\leq t\leq T}
\big[\|u_{t}(t)\|_{H^{2k}(\mathbb{R})} +\|u(t)\|_{H^{2k+1+\rho}(\mathbb{R})}\big].
\]
Therefore, there exists a unique local in-time solution in $E_{2k}(T)$.
Moreover, the solution does not blow up in finite time by the global bound
\eqref{conclusion}. This completes the proof.
\end{proof}


\subsection*{Concluding Remark}
The present article is part of a research program whose objective is
the investigation of general models arising in elasticity.
The well-posedness of nonlinear nonlocal elastic equations is of great
scientific interest, physically relevant and presents new challenges in
the analysis. In this paper, we have established the global well-posedness
of \eqref{eq:1.1} and \eqref{eq:1.7} for large data with defocusing
nonlinearity throughout our analysis. We believe that we can apply our method
to other models, such as peridynamics \cite{S}; this equation is a model
proposed to describe the dynamical response of an infinite homogeneous
elastic bar within the context of the peridynamic formulation of elasticity theory.



\subsection*{Acknowledgments}
H. Bae was supported by the 2014 Research Fund (Project Number 1.140076.01)
of UNIST(Ulsan National Institute of Science and Technology).
S. Ulusoy was partially supported by T\"{U}B\.{I}TAK grant 112T237.


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\end{document}