Existence local and global solution of multipoint Cauchy problem for nonlocal nonlinear equations

TL;DR

This paper investigates the existence and uniqueness of local and global solutions for multipoint Cauchy problems involving nonlocal nonlinear wave equations with convolution operators, under certain smoothness and growth conditions.

Contribution

It provides new results on the existence and uniqueness of solutions for a class of nonlocal nonlinear wave equations with general kernel functions.

Findings

Established local and global existence of solutions.

Proved uniqueness of solutions under specified conditions.

Extended the theory to equations with general convolution kernels.

Abstract

In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equat{\i}ons are studied.The equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data together with some growth conditions on the nonlinear term

Equations

$$u_{tt}-a\Delta u+b\ast u=\Delta \left[g\ast f\left(u\right)\right],x\in R^n,t\in \left(0,\infty \right),$$

$$u\left(x,0\right)=\phi \left(x\right)+\sum _{k=1}^m{\alpha }_{k}u\left(x,{\lambda }_k\right),\text{ for a.e. }x\in R^n,$$

$$u_t\left(x,0\right)=\psi \left(x\right)+\sum _{k=1}^m{\beta }_k{u}_t\left(x,{\lambda }_k\right),\text{ for a.e. }x\in R^n,$$

$$u_{tt}-a\Delta u+b\ast u=\Delta \left[g\ast f\left(u\right)\right],x\in R^n,t\in \left(0,\infty \right),$$

$$u\left(0,x\right)=\phi \left(x\right),u_t\left(0,x\right)=\psi \left(x\right)\text{ for a.e. }x\in R^n,$$

$${\Vert f\Vert }_p={\Vert f\Vert }_{L^p\left(\Omega ;E\right)}={\left(\underset{\Omega }{\int }{\Vert f\left(x\right)\Vert }_E^{p}dx\right)}^{\frac{1}{p}},1\le p<\infty ,$$

$${\Vert f\Vert }_{L^{\infty }\left(\Omega \right)}=\text{ess}\underset{x\in \Omega }{\sup }{\Vert f\left(x\right)\Vert }_E.$$

$${\Vert u\Vert }_{W^{m,p}\left(\Omega \right)}={\Vert u\Vert }_{L^p\left(\Omega \right)}+\sum _{k=1}^n{\Vert \frac{{\partial }^{m}u}{\partial x_k^m}\Vert }_{L^p\left(\Omega \right)}<\infty .$$

$$H^{s,p}=H^{s,p}\left(R^n\right)={\left(I-\Delta \right)}^{-\frac{s}{2}}{L}^p\left(R^n\right)$$

$${\Vert u\Vert }_{H^{s,p}}={\Vert {\left(I-\Delta \right)}^{\frac{s}{2}}u\Vert }_{L^p\left(R^n\right)}<\infty .$$

$$B:L_p(R^n)\to L_q(R^n).$$

$${\Vert u\Vert }_{L_q^{\ast }\left(E\right)}={\left(\underset{0}{\overset{\infty }{\int }}{\Vert u\left(t\right)\Vert }_E^q\frac{dt}{t}\right)}^{\frac{1}{q}}<\infty ,1\le q<\infty ,{\Vert u\Vert }_{L_{\infty }^{\ast }\left(E\right)}=\underset{t\in \left(0,\infty \right)}{\sup }{\Vert u\left(t\right)\Vert }_E.$$

$$B_{p,q}^s\left(R^n\right)=\{u\in S^{{}^{\prime }}\left(R^n\right),$$

$${\Vert u\Vert }_{B_{p,q}^s\left(R^n\right)}={\Vert F^{-1}{t}^{\varkappa -s}\left(1+{\left|\xi \right|}^{\frac{\varkappa }{2}}\right)e^{-t{|\xi |}^2}Fu\Vert }_{L_q^{\ast }\left(L^p\left(R^n\right)\right)}\text{,}$$

$${\left|\xi \right|}^2=\sum _{k=1}^n{\xi }_k^2\text{, }\xi =\left({\xi }_1,{\xi }_2,...,{\xi }_n\right),p\in \left(1,\infty \right)\text{, }q\in \left[1,\infty \right]\text{, }\varkappa >s\}.$$

$$u\le C\upsilon .$$

$$u_{tt}-a\Delta u+b\ast u=g\left(x,t\right),x\in R^n,t\in \left(0,T\right),T\in (0,\infty ],$$

$$u\left(x,0\right)=\phi \left(x\right)+\sum _{k=1}^m{\alpha }_{k}u\left(x,{\lambda }_k\right),\text{ for a.e. }x\in R^n,$$

$$u_t\left(x,0\right)=\psi \left(x\right)+\sum _{k=1}^m{\beta }_k{u}_t\left(x,{\lambda }_k\right),\text{ for a.e. }x\in R^n,$$

$$X_p=L^p\left(R^n\right)\text{, }1\le p\le \infty ,Y^{s,p}=H^{s,p}\left(R^n\right),Y_1^{s,p}=$$

$$H^{s,p}\left(R^n\right)\cap L^1\left(R^n\right)\text{, }{Y}_{\infty }^{s,p}=H^{s,p}\left(R^n\right)\cap L^{\infty }\left(R^n\right),Y_p^{s,p}=H^{s,p}\left(R^n\right)\cap L^p\left(R^n\right),$$

$${\Vert u\Vert }_{Y_p^{s,2}}={\Vert u\Vert }_{H^s\left(R^n\right)}+{\Vert u\Vert }_{L^p\left(R^n\right)}<\infty ,1\le p\le \infty .$$

$$\eta =\eta \left(\xi \right)={\left[a{\left|\xi \right|}^2+\hat{b}\left(\xi \right)\right]}^{\frac{1}{2}}.$$

$${\varkappa }_k={\varkappa }_k\left(\xi \right)={\lambda }_k\eta \left(\xi \right),k=1,2,...,m.$$

$$D_0\left(\xi \right)=1-\sum _{k=1}^m\left({\alpha }_k+{\beta }_k\right)\cos {\varkappa }_k+\sum _{i,j=1}^m{\alpha }_i{\beta }_j\cos \left({\varkappa }_i-{\varkappa }_j\right)\ne 0$$

$${\hat{u}}_{tt}\left(\xi ,t\right)+{\eta }^2\left(\xi \right)\hat{u}\left(\xi ,t\right)=\hat{g}\left(\xi ,t\right),$$

$$\hat{u}\left(\xi ,0\right)=\hat{\phi }\left(\xi \right)+\sum _{k=1}^m{\alpha }_k\hat{u}\left(\xi ,{\lambda }_k,\right),{\hat{u}}_t\left(\xi ,0\right)=\hat{\psi }\left(\xi \right)+\sum _{k=1}^m{\beta }_k\hat{u}\left(\xi ,{\lambda }_k\right),$$

$$\hat{u}\left(\xi ,t\right)=g_1\left(\xi \right)\cos \eta t+g_2\left(\xi \right)\cos \eta t+\frac{1}{2\eta }\underset{0}{\overset{t}{\int }}\sin \eta \left(t-\tau \right)\hat{g}\left(\xi ,\tau \right)d\tau ,$$

$$g_1-\sum _{k=1}^m{\alpha }_k{\gamma }_k+g_1\cos {\varkappa }_k+g_2\sin {\varkappa }_k=\hat{\phi }\left(\xi \right),$$

$$\eta g_2-\sum _{k=1}^m{\beta }_k\left[{\eta }_k+\eta \left(-g_1\sin {\varkappa }_k+g_2\cos {\varkappa }_k\right)\right]=\hat{\psi }\left(\xi \right),$$

Full text

Existence local and global solution of multipoint Cauchy problem for nonlocal nonlinear equations

Veli B. Shakhmurov

Department of Mechanical Engineering, Okan University, Akfirat, Tuzla 34959 Istanbul, E-mail: veli.sahmurov@okan.edu.tr;

Rishad Shahmurov

shahmurov@hotmail.com

University of Alabama Tuscaloosa USA, AL 35487

Abstract

In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equatıons are studied.The equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data together with some growth conditions on the nonlinear term

Key Word:  Boussinesq equations, Hyperbolic equations, differential operators, Fourier multipliers

AMS: 35Lxx, 35Qxx, 47D

1. Introduction

The aim in this paper is to study the existence and uniqueness of solution of the multipoint initial value problem (IVP) for nonlocal nonlinear wave equatıon

[TABLE]

[TABLE]

[TABLE]

where $m$ is an integer, $\lambda_{k}\in\left(0,\infty\right),$ $\alpha_{k},$ $\beta_{k}$ are complex numbers $g\left(x\right)$, $b\left(x\right)$ are measurable functions on $\left(0,\infty\right)$; $\ a\geq 0,$ $\Delta$ denotes the Laplace operator in $R^{n},$ $f(u)$ is the given nonlinear function, $\varphi\left(x\right)$ and $\psi\left(x\right)$ are the given initial value functions. Note that for $\alpha_{k}=$ $\beta_{k}=0$ we obtain classical Cauchy problem for nonlocal equation

[TABLE]

[TABLE]

The predictions of classical (local) elasticity theory become inaccurate when the characteristic length of an elasticity problem is comparable to the atomic length scale. To solution this situation, a nonlocal theory of elasticity was introduced (see $[1-3]$ and the references cited therein) and the main feature of the new theory is the fact that its predictions were more down to earth than those of the classical theory. For other generalizations of elasticity we refer the reader to $[4-6]$.  The global existence of the classical Cauchy problem for Boussinesq type nonlocal equations has been studied by many authors (see $\left[7-11\right]$ ). Note that, the existence of solutions and regularity properties for different type Boussinesq equations are considered e.g. in $\left[\text{8-15}\right]$. Boussinesq type equations occur in a wide variety of physical systems, such as in the propagation of longitudinal deformation waves in an elastic rod, hydro-dynamical process in plasma, in materials science which describe spinodal decomposition and in the absence of mechanical stresses (see $\left[16-19\right]$).

The $L^{p}-$well-posedness of the classıcal Cauchy problem $(1.3)$ depends crucially on the presence of a suitable kernel. Then the question that naturally arises is which of the possible forms of the kernel functions are relevant for the global well-posedness of the multipoint initial-value problem (IVP) $\left(1.1\right)-\left(1.2\right)$. In this study, as a partial answer to this question, we consider multipoint IVP $(1.1)-\left(1.2\right)$ with a general class of kernel functions and provide local, global existence and blow-up results for the solutions of the problem $(1.1)-(1.2)$ in fram of $L^{p}$ spaces. The kernel functions most frequently used in the literature are particular cases of this general class of kernel functions. Note that nonlocal Cauchy problem for wave equations were studied e.g. in $\left[\text{20, 21}\right].$

The strategy is to express the equation  $\left(1.1\right)$ as an integral equation. To treat the nonlinearity as a small perturbation of the linear part of the equation, the contraction mapping theorem is used. Also, a priori estimates on $L^{p}$ norms of solutions of the linearized version are utilized. The key step is the derivation of the uniform estimate of the solutions of the linearized Boussinesq equation. The methods of harmonic analysis, operator theory, interpolation of Banach Spaces and embedding theorems in Sobolev spaces are the main tools implemented to carry out the analysis.

In order to state our results precisely, we introduce some notations and some function spaces.

Definitions and Background

Let $E$ be a Banach space. $L^{p}\left(\Omega;E\right)$ denotes the space of strongly measurable $E$-valued functions that are defined on the measurable subset $\Omega\subset R^{n}$ with the norm

[TABLE]

[TABLE]

Let $\mathbb{C}$ denote the set of complex numbers. For $E=\mathbb{C}$ the $L^{p}\left(\Omega;E\right)$ denotes by $L^{p}\left(\Omega\right).$

Let $E_{1}$ and $E_{2}$ be two Banach spaces. $\left(E_{1},E_{2}\right)_{\theta,p}$ for $\theta\in\left(0,1\right),$ $p\in\left[1,\infty\right]$ denotes the interpolation spaces defined by $K$-method $\left[\text{22, \S 1.3.2}\right]$.

Let $m$ be a positive integer. $W^{m,p}\left(\Omega\right)$ denotes the Sobolev space, i.e. space of all functions $u\in L^{p}\left(\Omega\right)$ that have the generalized derivatives $\frac{\partial^{m}u}{\partial x_{k}^{m}}\in L^{p}\left(\Omega\right),$ $1\leq p\leq\infty$ with the norm

[TABLE]

Let $H^{s,p}\left(R^{n}\right)$, $\infty<s<\infty$ denotes fractionanal Sobolev space of order $s$ which is defined as:

[TABLE]

with the norm

[TABLE]

It clear that $H^{0,p}\left(R^{n}\right)=L^{p}\left(R^{n}\right).$ It is known that $H^{m,p}\left(R^{n}\right)=W^{m,p}\left(R^{n}\right)$ for the positive integer $m$ (see e.g. $\left[\text{23, \S\ 15}\right]$). For $p=2,$ the space $H^{s,p}\left(R^{n}\right)$ will be denoted by $H^{s}\left(R^{n}\right).$  Let $S\left(R^{n}\right)$ denote Schwartz class, i.e., the space of rapidly decreasing smooth functions on $R^{n},$ equipped with its usual topology generated by seminorms. Let $S^{{}^{\prime}}\left(R^{n}\right)$ denote the space of all continuous linear operators $L:S\left(R^{n}\right)\rightarrow\mathbb{C},$ equipped with the bounded convergence topology. Recall $S\left(R^{n}\right)$ is norm dense in $L^{p}\left(R^{n}\right)$ when $1\leq p<\infty.$

Let $1\leq p\leq q<\infty.$  A function $\Psi\in L_{\infty}(R^{n})$ is called a Fourier multiplier from $L_{p}(R^{n})$ to $L_{q}(R^{n})$ if the map $B:$ $u\rightarrow F^{-1}\Psi(\xi)Fu$ for $u\in S(R^{n})$ is well defined and extends to a bounded linear operator

[TABLE]

Let $L_{q}^{\ast}\left(E\right)$ denote the space of all $E-$valued function space such that

[TABLE]

Here, $F$ denote the Fourier transform. Fourier-analytic representation of Besov spaces on $R^{n}$ is defined as:

[TABLE]

[TABLE]

[TABLE]

It should be note that, the norm of Besov space does not depends on $\varkappa$ (see e.g. ( $\left[\text{22, \S\ 2.3}\right]$). For $p=q$ the space $B_{p,q}^{s}\left(R^{n}\right)$ will be denoted by $B_{p}^{s}\left(R^{n}\right).$

Sometimes we use one and the same symbol $C$ without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say $\alpha$, we write $C_{\alpha}$. Moreover, for $u$, $\upsilon>0$ the relation $u\lesssim\upsilon$ means that there exists a constant $C>0$  independent on $u$ and $\upsilon$ such that

[TABLE]

The paper is organized as follows: In Section 1, some definitions and background are given. In Section 2, we obtain the existence of unique solution and a priory estimates for solution of the linearized problem $(1.1)-\left(1.2\right).$ In Section 3, we show the existence and uniqueness of local strong solution of the problem $(1.1)-\left(1.2\right)$. In the Section 4 we show the same applications of the problem $(1.1)-\left(1.2\right).$

Sometimes we use one and the same symbol $C$ without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say $h$, we write $C_{h}$.

2. Estimates for linearized equation

In this section, we make the necessary estimates for solutions of the Cauchy problem

[TABLE]

[TABLE]

[TABLE]

Here,

[TABLE]

[TABLE]

[TABLE]

Let $\hat{b}\left(\xi\right)$ is the Fourier transformation of $b\left(x\right),$ i.e. $\hat{b}\left(\xi\right)=Fb$ and let

[TABLE]

Here,

[TABLE]

**Condition 2.1. **Assume $a\geq 0,$ $b$ is an integrable function whose $\hat{b}\left(\xi\right)\geq 0$ and $a+\hat{b}\left(\xi\right)>0$ for all $\ \xi\in R^{n}.$ Moreover, let ****

[TABLE]

for all $\xi\in R^{n}$ with $\xi\neq 0.$

First we need the following lemmas:

**Lemma 2.1. **Let the Condition 2.1. holds. Then, the problem $\left(2.1\right)-\left(2.2\right)$ has a unique solution.

**Proof. **By using of the Fourier transform, we get from $(2.1)-\left(2.2\right)$:

[TABLE]

[TABLE]

where $\hat{u}\left(\xi,t\right)$ is a Fourier transform of $u\left(x,t\right)$ with respect to $x$ and $\hat{\varphi}\left(\xi\right),$ $\hat{\psi}\left(\xi\right)$ are Fourier transform of $\varphi,$ $\psi,$ respectively.

By using the variation of constants it is easy to see that the general solution of $\left(2.3\right)$ is represented as

[TABLE]

where $g_{1}$, $g_{2}$ are general continuous differentiable functions. By taking the multipoint condition $\left(2.4\right)$ from $\left(2.5\right)$ we get that $(2.3)-\left(2.4\right)$ has a solution for $\xi\in R^{n},$ when $g_{1}$, $g_{2}$ are solution of the following system

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

By Condition 2.1 we get

[TABLE]

for all $\xi\neq 0.$ Solving the system $\left(2.6\right),$ we obtain

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

here,

[TABLE]

[TABLE]

Hense, problem $\left(2.3\right)-\left(2.4\right)$ has a unique solution expressed as $\left(2.5\right),$ where $g_{1}$ and $g_{2}$ are defined by $\left(2.7\right),$ i.e. problem $(2.1)-\left(2.2\right)$ has a unique solution

[TABLE]

**Theorem 2.1. **Let   the Condition 2.1 holds and $s>\frac{n}{2}$. Then for $\varphi,$ $\psi,$ $g\left(x,t\right)\in Y_{1}^{s,p}$ the solution $\left(2.1\right)-\left(2.2\right)$ satisfies the following uniformly in $t\in\left[0,T\right]$ estimate

[TABLE]

[TABLE]

where the positive constant $C$ depends only on initial data.

**Proof. **From $\left(2.7\right)$ we deduced that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then, from $\left(2.5\right)$, $\left(2.8\right)$ and $\left(2.10\right)$ we obtain that the solution $\left(2.1\right)-\left(2.2\right)$ can be expressed as

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

here,

[TABLE]

[TABLE]

By Condition 2.1,

[TABLE]

are uniformly bounded.  From $\left(2.11\right)$ and $\left(2.8\right)$ we obtain

[TABLE]

[TABLE]

where

[TABLE]

Let $N\in\mathbb{N}$ and

[TABLE]

From $\left(2.8\right)$ we dedused that

[TABLE]

[TABLE]

[TABLE]

From $\left(2.10\right)-$ $\left(2.13\right)$ due to uniform boundedness of $D_{0}^{-1}\left(\xi\right)$ and $C\left(\xi,t\right)$, $S\left(\xi,t\right)$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In view of $\left(2.12\right),$ by using the Minkowski’s inequality for integrals from above we get

[TABLE]

[TABLE]

Moreover, by $\left(2.11\right)$ and $\left(2.12\right)$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By using $\left(2.5\right)$, $\left(2.7\right)$ and $\left(2.12\right)$ we get

[TABLE]

[TABLE]

for $s>\frac{n}{p},$ $\alpha=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$, $\alpha_{k}\in\left\{0,1\right\}$, $\xi\in R^{n}$ and uniformly in $t\in\left[0,T\right].$ By multiplier theorems (see e.g. $\left[24\right]$) from $\left(2.16\right)$ we get that the functions $\left(1+\left|\xi\right|^{2}\right)^{-\frac{s}{2}}C\left(\xi,t\right),$ $\left(1+\left|\xi\right|^{2}\right)^{-\frac{s}{2}}S\left(\xi,t\right)$ are $L^{p}\left(R^{n}\right)\rightarrow L^{\infty}\left(R^{n}\right)$ Fourier multipliers. Then by Minkowski’s inequality for integrals, from $\left(2.11\right)$ and $\left(2.14\right)-\left(2.16\right)$ we obtain

[TABLE]

[TABLE]

By reasoning as the above we have

[TABLE]

Thus, from $\left(2.8\right)$ and $\left(2.15\right)$ we obtain

[TABLE]

[TABLE]

By using $\left(2.5\right),$ $\left(2.7\right)$ and in view of $\left(2.17\right)$ in similar way, we deduced the estimate of type $\left(2.19\right)$ for $u_{t}$, i.e. we obtain the assertion.

**Theorem 2.2. **Let the Condition 2.1 holds and $s>\frac{n}{2}$. Then for $\varphi,$ $\psi,$ $g\left(x,t\right)\in Y^{s,p}$ the solution of $\left(2.1\right)-\left(2.2\right)$ satisfies the following uniform estimate

[TABLE]

**Proof. **From $\left(2.7\right)$ and $\left(2.12\right)$ we have the following uniform estimate

[TABLE]

[TABLE]

[TABLE]

By Condition 2.1 and by virtue of Fourier multiplier theorems (see $\left[\text{24, \S\ 2.2\ }\right]$) we get that $C\left(\xi,t\right)$, $S\left(\xi,t\right)$ and $\Phi\left(\xi\right)$ are Fourier multipliers in $L^{p}\left(R^{n}\right)$ uniformly with respect to $t\in\left[0,T\right].$ So, the estimate $\left(2.21\right)$ by using the Minkowski’s inequality for integrals implies $\left(2.20\right).$

3. Local well posedness of IVP for nonlinear nonlocal equation

In this section, we will show the local existence and uniqueness of solution for the Cauchy problem $(1.1)-(1.2).$ For the study of the nonlinear problem $\left(1.1\right)-\left(1.2\right)$ we need the following lemmas

Lemma 3.1 (Nirenberg’s inequality) $\left[25\right]$. Assume that $u\in L^{p}\left(\Omega\right)$, $D^{m}u$ $\in L^{q}\left(\Omega\right)$, $p,q\in\left(1,\infty\right)$. Then for $i$ with $0\leq i\leq m,$ $m>\frac{n}{q}$ we have

[TABLE]

where

[TABLE]

**Lemma 3.2 $\left[26\right].$ **Assume that $u\in$ $W^{m,p}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)$ and $f\left(u\right)$ possesses continuous derivatives up to order $m\geq 1$. Then $f\left(u\right)-f\left(0\right)\in W^{m,p}\left(\Omega\right)$ and

[TABLE]

[TABLE]

where $C_{0}$ $\geq 1$ is a constant.

Let

[TABLE]

**Remark 3.1. **By using J.Lions-I. Petree result (see e.g $\left[\text{21, \S\ 1.8.}\right]$) we obtain that the map $u\rightarrow u\left(t_{0}\right)$, $t_{0}\in\left[0,T\right]$ is continuous and surjective from $W^{2,p}\left(0,T\right)$ onto $E_{0}$ and there is a constant $C_{1}$ such that

[TABLE]

First all of, we define the space $Y\left(T\right)=C\left(\left[0,T\right];Y_{\infty}^{2,p}\right)$ equipped with the norm defined by

[TABLE]

It is easy to see that $Y\left(T\right)$ is a Banach space. For $\varphi$, $\psi\in Y^{2,p}$, let

[TABLE]

**Definition 3.1. **For any $T>0$ if $\upsilon,$ $\psi\in Y_{\infty}^{2,p}$ and $u$ $\in C\left(\left[0,T\right];Y_{\infty}^{2,p}\right)$ satisfies the equation $(1.1)-(1.2)$ then $u\left(x,t\right)$ is called the continuous solution or the strong solution of the problem $(1.1)-(1.2).$ If $T<\infty$, then $u\left(x,t\right)$ is called the local strong solution of the problem $(1.1)-(1.2).$ If $T=\infty$, then $u\left(x,t\right)$ is called the global strong solution of the problem $(1.1)-(1.2)$.

**Condition 3.1. **Assume:

(1) Assume that the kernel $g$ is an integrable function whose Fourier transform satisfies

[TABLE]

(2) The Condition 2.1 holds, $\varphi,$ $\psi$ $\in Y_{\infty}^{2,p}$ for $1<p<\infty$   and $\frac{n}{p}<2$;

(3) the function $u\rightarrow$ $f\left(x,t,u\right)$: $R^{n}\times\left[0,T\right]\times E_{0}\rightarrow E$ is a measurable in $\left(x,t\right)\in R^{n}\times\left[0,T\right]$ for $u\in E_{0};$ $f\left(x,t,u\right)$. Moreover, $F\left(x,t,u\right)$ is continuous in $u\in E_{0}$ and $f\left(x,t,u\right)\in C^{\left(3\right)}\left(E_{0};E\right)$ uniformly with respect to $x\in R^{n},$ $t\in\left[0,T\right].$

Main aim of this section is to prove the following result:

**Theorem 3.1. **Let the Condition 3.1. holds. Then problem $\left(1.1\right)-\left(1.2\right)$ has a unique local strange solution $u\in C^{\left(2\right)}\left(\left[0,\right.\left.T_{0}\right);Y_{\infty}^{2,p}\right)$, where $T_{0}$ is a maximal time interval that is appropriately small relative to $M$. Moreover, if

[TABLE]

then $T_{0}=\infty.$

**Proof. **First, we are going to prove the existence and the uniqueness of the local continuous solution of the problem $(1.1)-\left(1.2\right)$ by contraction mapping principle. Consider a map $G$ on $Y\left(T\right)$ such that $G(u)$ is the solution of the Cauchy problem

[TABLE]

[TABLE]

[TABLE]

From Lemma 3.2 we know that $F(u)\in$ $L^{p}\left(0,T;Y_{\infty}^{2,p}\right)$ for any $T>0$. Thus, by Lemma 2.1, problem $\left(3.4\right)$ has a solution which can be written as

[TABLE]

[TABLE]

where $S_{1}\left(x,t\right)$, $S_{2}\left(x,t\right),$ $\Phi$ are operator functions defined by $\left(2.10\right)$ and $\left(2.11\right)$,   where $g$ replaced by $g\ast f\left(G\left(u\right)\right).$ From Lemma 3.2 it is easy to see that the map $G$ is well defined for $f\in C^{\left(2\right)}\left(X_{0};\mathbb{C}\right)$. We put

[TABLE]

First, by reasoning as in $\left[9\right]$ let us prove that the map $G$ has a unique fixed point in $Q\left(M;T\right).$ For this aim, it is sufficient to show that the operator $G$ maps $Q\left(M;T\right)$ into $Q\left(M;T\right)$ and $G:$ $Q\left(M;T\right)$ $\rightarrow$ $Q\left(M;T\right)$ is strictly contractive if $T$ is appropriately small relative to $M.$ Consider the function  $\bar{f}\left(\xi\right)$: $\left[0,\right.$ $\left.\infty\right)\rightarrow\left[0,\right.$ $\left.\infty\right)$ defined by

[TABLE]

It is clear to see that the function $\bar{f}\left(\xi\right)$ is continuous and nondecreasing on $\left[0,\right.$ $\left.\infty\right).$ From Lemma 3.2 we have

[TABLE]

[TABLE]

[TABLE]

In view of the assumptıon (1) and by using Minkowski’s inequality for integralsö we obtain from $\left(3.5\right)$:

[TABLE]

[TABLE]

Thus, from $\left(3.6\right)-\left(3.8\right)$ and Lemma 3.2 we get

[TABLE]

If $T$ satisfies

[TABLE]

then

[TABLE]

Therefore, if $\left(3.9\right)$ holds, then $G$ maps $Q\left(M;T\right)$ into $Q\left(M;T\right).$ Now, we are going to prove that the map $G$ is strictly contractive. Assume $T>0$ and $u_{1},$ $u_{2}\in$ $Q\left(M;T\right)$ given. We get

[TABLE]

[TABLE]

By using the assumption (3) and the mean value theorem, we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $0<\eta_{i}<1,$ $i=1,2,3,4.$ Thus, using Hölder’s and Nirenberg’s inequality, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $C$ is the constant in Lemma $3.1$. From $\left(3.10\right)-\left(3.11\right)$, using Minkowski’s inequality for integrals, Fourier multiplier theorems in $X_{p}$ spaces and Young’s inequality, we obtain

[TABLE]

[TABLE]

[TABLE]

where $C_{1}$ is a constant. If $T$ satisfies $\left(3.9\right)$ and the following inequality holds

[TABLE]

then

[TABLE]

That is, $G$ is a contructive map. By contraction mapping principle we know that $G(u)$ has a fixed point $u(x,t)\in$ $Q\left(M;T\right)$ that is a solution of $(1.1)-(1.2)$. From $\left(2.9\right)-\left(2.11\right)$ we get that $u$ is a solution of the following integral equation

[TABLE]

[TABLE]

Let us show that this solution is a unique in $Y\left(T\right)$. Let $u_{1}$, $u_{2}\in Y\left(T\right)$ are two solution of the problem $(1.1)-(1.2)$. Then

[TABLE]

By the definition of the space $Y\left(T\right)$, we can assume that

[TABLE]

Hence, by Minkowski’s inequality for integrals and Theorem 2.2 we obtain from $\left(3.15\right)$

[TABLE]

From $(3.16)$ and Gronwall’s inequality, we have $\left\|u_{1}-u_{2}\right\|_{Y^{2,p}}=0$, i.e. problem $(1.1)-(1.2)$ has a unique solution which belongs to $Y\left(T\right).$ That is, we obtain the first part of the assertion.

Now, let $\left[0\right.,\left.T_{0}\right)$ be the maximal time interval of existence for $u\in Y\left(T_{0}\right)$. It remains only to show that if $(3.3)$ is satisfied, then $T_{0}=\infty$. Assume contrary that, $\left(3.3\right)$ holds and $T_{0}<\infty.$ For $T\in\left[0\right.,\left.T_{0}\right),$ we consider the following integral equation

[TABLE]

[TABLE]

By virtue of $(3.3)$, for $T^{\prime}>T$ we have

[TABLE]

By reasoning as a first part of theorem and by contraction mapping principle, there is a $T^{\ast}\in\left(0,T_{0}\right)$ such that for each $T\in\left[0\right.,\left.T_{0}\right),$ the equation $\left(3.17\right)$ has a unique solution $\upsilon\in Y\left(T^{\ast}\right).$ The estimates $\left(3.9\right)$ and $\left(3.14\right)$ imply that $T^{\ast}$ can be selected independently of $T\in\left[0\right.,\left.T_{0}\right).$ Set $T=T_{0}-\frac{T^{\ast}}{2}$ and define

[TABLE]

By construction $\tilde{u}\left(x,t\right)$ is a solution of the problem $(1.1)-(1.2)$ on $\left[T,T_{0}+\frac{T^{\ast}}{2}\right]$ and in view of local uniqueness, $\tilde{u}\left(x,t\right)$ extends $u.$ This is against to the maximality of $\left[0\right.,\left.T_{0}\right)$, i.e we obtain $T_{0}=\infty.$

Consider the problem $\left(1.1\right)-\left(1.2\right),$ when $\varphi,$ $\psi\in H^{s}.$

We first need two lemmas concerning the behaviour of the nonlinear term $[$8, 13, 27$]$.

Lemma 3.3. Let $s\geq 0,$ $f\in C^{\left[s\right]+1}\left(R\right)$ with $f(0)=0$. Then for any $u\in H^{s}\cap L^{\infty}$, we have $f(u)\in H^{s}\cap L^{\infty}.$ Moreover there is some constant $A(M)$ depending on $M$ such that for all $u\in H^{s}\cap L^{\infty}$ with $\left\|u\right\|_{L^{m}}\leq M,$

[TABLE]

**Lemma 3.4. ** Let $s\geq 0,$ $f\in C^{\left[s\right]+1}\left(R\right)$. Then for for any $M$ there is some constant $B(M)$ depending on $M$ such that for all $u$, $\upsilon\in H^{s}\cap L^{\infty}$ with $\left\|u\right\|_{L^{m}}\leq M,$ $\left\|\upsilon\right\|_{L^{\infty}}\leq M,$ $\left\|u\right\|_{H^{8}}\leq M,$ $\left\|\upsilon\right\|_{H^{s}}\leq M,$

[TABLE]

By reasoning as in $\left[\text{13, Lemma 3.4}\right]$ we have

Corollary 3.1. Let $s>\frac{n}{2},$ $f\in C^{\left[s\right]+1}\left(R\right)$. Then for for any $B$ there is some constant $B(M)$ depending on $M$ such that for all $u$, $\upsilon\in H^{s}$ with $\left\|u\right\|_{H^{s}}\leq M,$ $\left\|\upsilon\right\|_{H^{s}}\leq M,$

[TABLE]

**Lemma 3.5. ** If $s>0$, then $Y_{\infty}^{s,2}$ is an algebra. Moreover, for  $f,$ $g\in Y_{\infty}^{s,2},$

[TABLE]

Lemma 3.6 $\left[\text{28, Lemma X 4}\right]$. Let $s\geq 0,$ $f\in C^{\left[s\right]+1}\left(R\right)$ and $f\left(u\right)=O\left(\left|u\right|^{\alpha+1}\right)$  for $u\rightarrow 0$, $\alpha\geq 1$ be a positive integer. If $u\in Y_{\infty}^{s,2}$ and $\left\|u\right\|_{\infty}\leq M$,  then

[TABLE]

[TABLE]

Lemma 3.7 $\left[\text{13, Lemma 3.4}\right]$. Let $s\geq 0,$ $f\in C^{\left[s\right]+1}\left(R\right)$ and $f\left(u\right)=O\left(\left|u\right|^{\alpha+1}\right)$  for $u\rightarrow 0$, $\alpha\geq 0$ be a positive integer. If $u,$ $\upsilon\in Y_{\infty}^{s,2},$ $\left\|u\right\|_{H^{s}}\leq M$,  $\left\|\upsilon\right\|_{H^{s}}\leq M$ and $\left\|u\right\|_{\infty}\leq M$,  $\left\|\upsilon\right\|_{\infty}\leq M,$ then

[TABLE]

[TABLE]

[TABLE]

By reasoning as in Theorem 3.1 and $\left[\text{13, Theorem 1.1}\right]$ we have:

**Theorem 3.2. **Let the Condition 3.1 hold. Assume $f\in C^{k}\left(R\right),$ with $k$ an integer $k\geq s>\frac{n}{2},$ satisfies $f\left(u\right)=O\left(\left|u\right|^{\alpha+1}\right)$  for $u\rightarrow 0.$ Then there exists a constant $\delta>0,$ such that for any $\varphi,$ $\psi\in Y_{1}^{s,2}$ satisfying

[TABLE]

problem $\left(1.1\right)-\left(1.2\right)$ has a unique local strange solution $u\in C^{\left(2\right)}\left(\left[0,\right.\left.\infty\right);Y^{s,2}\right)$. Moreover,

[TABLE]

where the constant $C$ only depends on $f$  and initial data.

**Proof.  **Consider a metric space  defined by

[TABLE]

equipped with the norm

[TABLE]

where $\delta>0$  satisfies $\left(3.19\right)$ and $C_{0}$  is a constant in Theorem 2.1.  It is easy to prove that $W^{s}$ is a complete metric space. From Sobolev imbedding theorem we know that $\left\|u\right\|_{\infty}\leq 1$ if we take that $\delta$ is enough small. Consider the problem $\left(3.4\right)$. From Lemma 3.6 we get that $f\left(u\right)\in L^{2}\left(0,T;Y_{1}^{s,2}\right)$ for any $T>0$. Thus the problem $\left(3.4\right)$ has a unique solution which can be written as $\left(3.5\right).$ We should prove that the operator $G\left(u\right)$ defined by $\left(3.5\right)$ is strictly contractive if $\delta$ is suitable small.  In fact, by $(2.9)$ in Theorem 2.1 and Lemma 3.6 we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

On the other hand, by $(2.20)$ in Theorem 2.2 and Lemma 3.6 we have

[TABLE]

[TABLE]

[TABLE]

Therefore, combining $(3.21)$ with $(3.22)$ yields

[TABLE]

Taking that $\delta$ is enough small such that $C\left(3C_{0}\delta\right)^{\alpha}$ $<1/3$, from $\left(3.23\right)$ and from Theorems 2.1, 2.2 we dedused that $G$ maps $W^{s}$ into $W^{s}$. Then, by reasoning as the remaining part of $\left[\text{13, Theorem 1.1}\right]$ we obtain that $G$ :$W^{s}\rightarrow W^{s}$ is strictly contractive. Using the contraction mapping principle, we know that $G(u)$ has a unique fixed point $u(x,t)\in$ $C^{2}([0,\infty),H^{s})$ and $u(x,t)$ is the solution of the problem $(1.1)-(1.2)$.

We claim that the solution $u(x,t)$ of the problem $(1.1)-(1.2)$ is also unique in $C^{2}([0,\infty),H^{s})$. In fact, let $u_{1}$ and $u_{2}$ be two solutions of the problem $(1.1)-(1.2)$ and $u_{1}$, $u_{2}\in C^{2}([0,\infty),H^{s})$. Let $u=u_{1}-u_{2}$; then

[TABLE]

This fact is derived in a similar way as in Theorem 3.2, by using Theorems 2.1, 2.2 and Gronwall’s inequality.

**Condition 3.2. **Let the Condition 2.1 holds. Assume $f\in C^{\left[s\right]+1}\left(R\right)$ with $f(0)=0$ for some $s\geq 0$;

(2) Assume that the kernel $g$ is an integrable function whose Fourier transform satisfies

[TABLE]

**Theorem 3.3. **Let the Condition 3.2 hold. Moreover, $s\geq 0$ and $r\geq 2$. Then there is some $T>0$ such that the multipoint IVB $(1.1)-(1.2)$ is well posed with solution in $C^{2}\left(\left[0,T\right];H^{s}\right)$ for initial data $\varphi,$ $\psi\in H^{s}.$

**Proof. **Consider the convolution operator $u\rightarrow Ku=\Delta\left[g\ast f\left(u\right)\right].$ In view of assumptios we have

[TABLE]

i.e. $\Delta g\ast\upsilon$ is a bounded linear operator on $H^{s}.$ Then by Corollary 3.1, $K\left(u\right)$ is locally Lipschitz on $H^{s}$. Then by reasoning as in Theorem 3.2 and $\left[\text{13, Theorem 1.1}\right]$ we obtain that $G$: $H^{s}\rightarrow H^{s}$ is strictly contractive. Using the contraction mapping principle, we get that the operator $G(u)$ defined by $\left(3.5\right)$ has a unique fixed point $u(x,t)\in$ $C^{2}([0,\infty),H^{s})$  and $u(x,t)$ is the solution of the problem $(1.1)-(1.2)$. Moreover, we show that the solution $u(x,t)$ of $(1.1)-(1.2)$ is also unique in $C^{2}([0,\infty),H^{s})$. In fact, let $u_{1}$ and $u_{2}$ be two solutions of the problem $(1.1)-(1.2)$ and $u_{1}$, $u_{2}\in C^{2}([0,\infty),H^{s})$. Let $u=u_{1}-u_{2}$; then

[TABLE]

This fact is derived in a similar way as in Theorem 3.2, by using Theorems 2.1, 2.2 and Gronwall’s inequality.

**Theorem 3.4. **Let the Condition 3.2 hold and $r>2+\frac{n}{2}$. Then there is some $T>0$ such that the multipoint IVB $(1.1)-(1.2)$ is well posed with solution in $C^{2}\left(\left[0,T\right];Y_{\infty}^{s,2}\right)$ for initial data $\varphi,$ $\psi\in Y_{\infty}^{s,2}.$

**Proof. ** All we need here, is to show that $K\ast f(u)$ is Lipschitz on $Y_{\infty}^{s,2}$. Indeed, by reasoning as in Theorem 3.3 we have

[TABLE]

Then $\Delta g\ast\upsilon$ is a bounded linear map from $H^{s}$ into $H^{s+r-2}$. Since $s\geq 0$ and $r$ $>2+\frac{n}{2}$  we get $s+r-2>\frac{n}{2}.$ The Sobolev embedding theorem implies that $\Delta g\ast\upsilon$ is a bounded linear map from $Y_{\infty}^{s,2}$ into $Y_{\infty}^{s,2}$. Lemma 3.4 implies the Lipschitz condition on $Y_{\infty}^{s,2}$. Then, by reasoning as in Theorem 3.3 we obtain the assertion.

The solution in theorems 3.2-3.4 can be extended to a maximal interval $[0,T_{\max}),$ where finite $T_{\max}$ is characterized by the blow-up condition

[TABLE]

Lemma 3.8. Suppose the conditions of theorems 3.4, 3.5 hold and $u$ is the solution of multipoint IVP $(1.1)-(1.2).$ Then there is a global solution if for any $T<\infty$ we have

[TABLE]

**Proof. **Indeed, by reasoning as in the second part of the proof of Theorem 3.1, by using a continuation of local solution of $(1.1)-(1.2)$ and assuming contrary that, $\left(3.25\right)$ holds and $T_{0}<\infty$  we obtain contradiction, i.e. we get $T_{0}=T_{\max}=\infty.$

**4. Conservation of energy and global existence. **

In this section, we prove the existence and the uniqueness of the global strong solution and the global classical solution for the problem $(1.1)-(1.2).$  For this purpose, we are going to make a priori estimates of the local strong solution for the problem $(1.1)-(1.2).$

**Condition 4.1. Let the Condition 2.1 holds. **Assume that the kernel $g$ is an integrable function whose Fourier transform satisfies

[TABLE]

Let $F^{-1}$ denote the inverse Fourie rtransform. We consider the operator $B$ defined by

[TABLE]

Then it is clear to see that

[TABLE]

First, we show the following

**Lemma 4.1. **Suppose the conditions of theorems 3.4, 3.5 hold with $\hat{g}\left(\xi\right)>0$ and the solution of multipoint IVP $(1.1)-(1.2)$ exists in $C^{2}\left(\left[0,T\right];Y_{\infty}^{s,2}\right)$ for some $s\geq 0.$ If $B\varphi\in L^{2}$ and $B\psi\in L^{2},$ then $Bu,$ $Bu_{t}\in C^{1}\left(\left[0,\right.\left.T\right);L^{2}\right).$

Proof. By Lemma 2.1, problem $\left(1.1\right)-\left(1.2\right)$ is equıvalent to following integral equation ,

[TABLE]

[TABLE]

where $S_{1}\left(x,t\right)$, $S_{2}\left(x,t\right),$ $\Phi$ are are operator functions defined by $\left(2.10\right)$ and $\left(2.11\right)$,   where $g$ replaced by $g\ast f\left(u\right).$

From $\left(4.2\right)$ we get that

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since

[TABLE]

are uniformly bounded for fixet $t$  by $\left(4.1\right)$, $\left(4.4\right)$ $\left(4.5\right)$ we have

[TABLE]

[TABLE]

For fixed $t$, we have $f(u)\in H^{s}.$ Since  $D_{0}^{-1}\left(\xi\right),$ $\cos\left(\eta t\right),$ $\sin\left(\eta t\right)$ are uniformly bounded, from $\left(4.1\right),$ $\left(2.11\right)$ and $\left(4.6\right)$ we get

[TABLE]

From $\left(4.8\right)$ we have

[TABLE]

[TABLE]

Then from $\left(4.2\right),$ $\left(4.7\right),$ $\left(4.8\right)$ and $\left(4.9\right)$ we obtain the assertion.

Remark 4.1. Due to nonlocality of initial conditions the additional conditions appears in Theorem 4.1. For classical Cauchy problem this extra conditions are not required

Lemma 4.2. Assume the conditions of theorems 3.4, 3.5 hold with $a=0,$ $\hat{g}\left(\xi\right)>0$ and

[TABLE]

Suppose the solution of $(1.1)-(1.2)$ exists in $C^{2}\left(\left[0,\right.\left.T\right);Y_{\infty}^{s,2}\right)$ for some $s\geq 0.$ If $B\psi\in L^{2},$ $Bu_{t}\left(x,\lambda_{k}\right)\in L^{2},k=1,2,...,m,$ then $Bu_{t}\in C^{2}\left(\left[0,\right.\left.T\right);L^{2}\right).$ Moreover, if $B\varphi\in L^{2},$ $Bu\left(x,\lambda_{k}\right)\in L^{2}$ then $Bu\in C^{1}\left(\left[0,\right.\left.T\right);L^{2}\right).$

**Proof. ** Integrating the equation $\left(1.1\right)$ for $a=0,$ twice and calculating the resulting double integral as an iterated integral, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From $\left(4.1\right)$ and $\left(4.11\right)$ for fixed $t$ and $\tau$ we get

[TABLE]

[TABLE]

By assumption on $b$, $g$ and by $\left(4.1\right)$ for fixed $\tau$ we have

[TABLE]

Moreover, by Lemma 3.3 for all $t$ we have $f\left(u\right)\in H^{s}.$ Also

[TABLE]

Then from $\left(4.12\right)-\left(4.14\right)$ we obtain $Bu_{t}\in C^{2}\left(\left[0,\right.\left.T\right);L^{2}\right).$ The second statement follows similarly from $\left(4.10\right).$

From Lemma 4.2 we obtain the following result.

Result 4.1. Assume the conditions of theorems 3.4, 3.5 hold with $a=0,$ $\hat{g}\left(\xi\right)>0,$ $\alpha_{k}=\beta_{k}=0$ and

[TABLE]

Suppose the solution of $(1.1)-(1.2)$ exists in $C^{2}\left(\left[0,T\right];Y_{\infty}^{s,2}\right)$ for some $s\geq 0.$ If $B\psi\in L^{2}$ then $Bu_{t}\in C^{2}\left(\left[0,\right.\left.T\right);L^{2}\right).$ Moreover, if $B\varphi\in L^{2},$ then $Bu\in C^{1}\left(\left[0,\right.\left.T\right);L^{2}\right).$

Here,

[TABLE]

**Lemma 4.3. **Assume the all conditions of Lemma 4.2 are satisfied. Let $B\psi\in L^{2},$ $Bu_{t}\left(x,\lambda_{k}\right)\in L^{2},$ $k=1,2,...,m$ and $G\left(\varphi\right)\in L^{1}$. Then for any $t\in\left[0,\right.\left.T\right)$ the energy

[TABLE]

is constant $\left[0,\right.\left.T\right).$

**Proof. ** By use of  equation $\left(1.1\right)$, it follows from straightforward calculation that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\left(u,\upsilon\right)$ denotes the inner product of $L^{2}$ space.  Integrating the above equality with respect to $t$, we have $\left(4.15\right)$.

By using the above lemmas we obtain the following results

**Theorem 4.1. **Let the Condition 3.2 hold $a=0,$ $\hat{g}\left(\xi\right)>0$ and

[TABLE]

Moreover, let $B\psi\in L^{2},$ $Bu_{t}\left(x,\lambda_{k}\right)\in L^{2},$ $k=1,2,...,m$ and $G\left(\varphi\right)\in L^{1}$, $s\geq 0$ , $r>3$ and there is some $k>0$ so that  $G\left(r\right)\geq-kr^{2}$ for $r\in\mathbb{R}$. Then there is some $T>0$ such that the multipoint IVB $(1.1)-(1.2)$ has a global solution $u\in C^{2}\left(\left[0,\right.\left.\infty\right);Y_{\infty}^{s,2}\right).$

References

A. C. Eringen, Nonlocal Continuum Field Theories (2002), New York, Springer. 2. 2.

Z. Huang, Formulations of nonlocal continuum mechanics based on a new definition of stress tensor Acta Mech. (2006)187, 11–27. 3. 3.

C. Polizzotto, Nonlocal elasticity and related variational principles Int. J. Solids Struct. ( 2001) 38 7359–80. 4. 4.

C. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces J. Mech. Phys. Solids (2000)48 175-209. 5. 5.

M. Arndt and M. Griebel, Derivation of higher order gradient continuum models from atomistic models for crystalline solids Multiscale Modeling Simul. (2005)4, 531–62.. 6. 6.

X. Blanc, C. LeBris, P. L. Lions, Atomistic to continuum limits for computational materials science, ESAIM— Math. Modelling Numer. Anal. (2007)41, 391–426. 7. 7.

A. De Godefroy, Blow up of solutions of a generalized Boussinesq equation IMA J. Appl. Math.(1998) 60 123–38. 8. 8.

A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Diff.Integral Eqns. (2002)15, 1061–72. 9. 9.

G. Chen and S. Wang, Existence and nonexistence of global solutions for the generalized IMBq equation Nonlinear Anal.—Theory Methods Appl. (1999)36, 961–80. 10. 10.

M. Lazar, G. A. Maugin and E. C. Aifantis, On a theory of nonlocal elasticity of bi-Helmholtz type and some applications Int. J. Solids and Struct. (2006)43, 1404–21. 11. 11.

N. Duruk, H.A. Erbay and A. Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, (2010)23, 107–118. 12. 12.

S. Wang, G. Chen, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl. 274 (2002) 846–866. 13. 13.

Wang S and Chen G, Cauchy problem of the generalized double dispersion equation Nonlinear Anal.— Theory Methods Appl. (2006 )64 159–73. 14. 14.

J.L. Bona, R.L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988), 15–29. 15. 15.

F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations 106 (1993), 257–293. 16. 16.

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal. 26 (1995), 1527–1546. 17. 17.

V.G. Makhankov, Dynamics of classical solutions (in non-integrable systems), Phys. Lett. C 35(1978), 1–128. 18. 18.

G.B. Whitham, Linear and Nonlinear Waves, Wiley–Interscience, New York, 1975. 19. 19.

N.J. Zabusky, Nonlinear Partial Differential Equations, Academic Press, New York, 1967. 20. 20.

A. Ashyralyev, N. Aggez, Nonlocal boundary value hyperbolic problems involving Integral conditions, Bound.Value Probl., 2014 V (2014):214. 21. 21.

L. S. Pulkina, A non local problem with integral conditions for hyperbolice quations, Electron.J.Differ.Equ.(1999)45, 1-6. 22. 22.

Girardi, M., Lutz, W., Operator-valued Fourier multiplier theorems on $L_{p}(X)$ and geometry of Banach spaces, J. Funct. Anal., ( 2003)204(2), 320–354. 23. 23.

H. Triebel, Interpolation theory, Function spaces, Differential operators, North-Holland, Amsterdam, 1978. 24. 24.

H. Triebel, Fractals and spectra, Birkhauser Verlag, Related to Fourier analysis and function spaces, Basel, 1997. 25. 25.

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (1959)13 , 115–162. 26. 26.

S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math.(1980)33 , 43–101. 27. 27.

R. Coifman and Y. Meyer, Wavelets. Calder´on-Zygmund and Multilinear Operators, Cambridge University Press, 1997. 28. 28.

T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. (1988)41, 891–907.