Cauchy Problem of the Singularly Perturbed Sixth Order Boussinesq Type Equation

Hong Li, Changming Song, Li Chen

College of Science, Zhongyuan University of Technology, Zhengzhou, China.

DOI: 10.4236/jamp.2017.59137                

Abstract

In this paper, the existence and uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of the singularly perturbed sixth order Boussinesq type equation are proved.

Keywords

Boussinesq Equation, Cauchy Problem, Global Generalized Solution, Global Classical Solution

1. Introduction

In this paper, we consider the following Cauchy problem

$u_{tt}=u_{xx}+\sigma {(u)}_{xx}+\alpha u_{x^4}+\beta u_{x^6},x\in R,t>0,$ (1.1)

$u(x,0)=u_0(x),u_t(x,0)=u_1(x),x\in R,$ (1.2)

where $u(x,t)$ is the unknown function, subscript $x$ and $t$ indicate partial derivatives, $\sigma (s)$ is the given function, $\alpha >0$ and $\beta >0$ are real numbers, $u_0(x)$ and $u_1(x)$ are given functions defined on $R$ .

There are also several equations which are closely related to Equation (1.1). In the numerical study of the ill-posed Boussinesq equation

$u_{tt}=u_{xx}+{(u^2)}_{xx}+u_{xxxx},$ (1.3)

In [1], Darapi and Hua proposed the singularly perturbed Boussinesq equation

$u_{tt}=u_{xx}+{(u^2)}_{xx}+u_{xxxx}+\delta u_{xxxxxx}$ (1.4)

as a dispersive regularization of the ill-posed classical Boussinesq Equation (1.3), where $\delta >0$ is small parameter. The authors use both filtering and regularization techniques to control growth of errors and to provide better approximate solutions of this equation.

Dash and Daripi presented a formal derivation of (1.4) from two-dimensional potential flow equations for water waves through an asymptotic series expansion for small amplitude and long wave length in [2] [3]. The physical relevance of equation (1.4) describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3.

In [4], Feng investigated the generalized Boussinesq equation including the singularly perturbed Boussinesq equation

$u_{tt}={[Q(u)]}_{xx}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}}{b}_i{u}_{(2i+2)x},$ (1.5)

where $Q(u)=u+b_0{u}^r,r$ and $b_i(i=1,\cdots ,n)$ are all real constants. By the means of two proper ansatzs, the author obtained explicit traveling solitary wave solutions.

In [5], Song et al. studied the existence and uniqueness of the global generalized solution and the global classical for the initial boundary value problem of Equation (1.1). In [6], Song et al. also studied the nonexistence of the global solutions for the initial boundary value problem of Equation (1.1).

The aim of the present article is to prove that, by virtue of the Galerkin method and prior estimates, the problem (1.1), (1.2) has a unique global generalized solution and a unique global classical solution.

In order to prove that the Cauchy problem (1.1), (1.2) has a unique global solution, we shall consider the following auxiliary problem

$v_{tt}=v_{xx}+\sigma {(v_x)}_x+\alpha v_{x^4}+\beta v_{x^6},x\in R,t>0,$ (1.6)

$v(x,0)=v_0(x),v_t(x,0)=v_1(x),x\in R.$ (1.7)

First of all, we shall prove that the periodic boundary value problem of Equation (1.6) has a unique global solution by the Galerkin method. Next, we prove that the Cauchy problem (1.6), (1.7) has a unique global solution by constructing a sequence of periodic boundary value problem of Equation (1.6). Then, we can obtain a unique global solution of the Cauchy problem (1.1), (1.2) from (1.6), (1.7) by setting $v_x(x,t)=u(x,t)$ , $v_{0x}(x)=u_0(x)$ and $v_{1x}(x)=u_1(x)$ .

2. Periodic Boundary Value Problem of (1.6), (1.7)

To obtain the global solution for the Cauchy problem (1.6), (1.7), we first discuss the following periodic boundary value problem on ${\bar{Q}}_T$

$v_{tt}=v_{xx}+\sigma {(v_x)}_x+\alpha v_{x^4}+\beta v_{x^6},$ (2.1)

$v_x(x,t)=v_x(x+2D,t),$ (2.2)

$v(x,0)=v_0(x),v_t(x,0)=v_1(x),$ (2.3)

where ${\bar{Q}}_T=\bar{\Omega }\times [0,T],\bar{\Omega }=[-D,D],D>0,v_0(x)$ and $v_1(x)$ are given functions defined on $\bar{\Omega }$ and satisfy (2.2).

Let $\{y_n(x)\}=\{\frac{1}{\sqrt{2D}},\frac{1}{\sqrt{D}}cos{\delta }_{n}x,\frac{1}{\sqrt{D}}sin{\delta }_{n}x,n=1,2,\cdots \}$ be the orthogonal

base in $L^2(\Omega )$ composed of the eigenfunctions of the eigenvalue problem

$y^{″}+\lambda y=0,x\in \Omega ,\text{}$

$y_x(x)=y_x(x+2D)$

corresponding to eigenvalue ${\lambda }_i={\delta }_i^2={(\frac{i\pi }{D})}^2(i=1,2,\cdots )$ . Let

$v_N(x,t)={\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\alpha }_{Ni}(t)y_i(x)$

be the Galerkin approximate solution of the problem (2.1)-(2.3), where ${\alpha }_{Ni}(t)$ are the undermined functions, $N$ is a natural number.

Substituting $v_N(x,t)$ and the approximations $v_0(x),\text{}{v}_1(x)$ into the problem (2.1)-(2.3), we get

$v_{Ntt}=v_{Nxx}+\sigma {(v_{Nx})}_x+\alpha v_{N{x}^4}+\beta v_{N{x}^6},$ (2.4)

$v_{Nx}(x,t)=v_{Nx}(x+2D,t),$ (2.5)

$v_N(x,0)=v_{0N}(x),v_{Nt}(x,0)=v_{1N}(x).$ (2.6)

Multiplying both sides of (2.4) and (2.6) by $y_s(x)$ , summing up for $s=1,2,\cdots ,N$ and integrating on $\Omega$ , we have

$(v_{Ntt}-v_{Nxx}-\sigma {(v_{Nx})}_x-\alpha v_{N{x}^4}-\beta v_{N{x}^6},y_s)=0,$ (2.7)

$v_{Ns}(0)={\beta }_s,v_{Nst}(0)={\gamma }_s,s=1,2,\cdots ,N.$ (2.8)

Lemma 2.1. (Adams [7]) There exist constants $\epsilon >0$ and $C(\epsilon )>0$ such that for any integers $j$ and $m,0\le j\le m$ , the following inequality holds

${\Vert D_x^{j}u\Vert }^2\le C(\epsilon ){\Vert u\Vert }^2+\epsilon {\Vert D_x^{m}u\Vert }^2.$

Lemma 2.2. Assume that $\sigma \in C^1(R),{\sigma }^{\prime }(s)$ is bounded blow, i.e., there is a constant $C_0$ such that ${\sigma }^{\prime }(s)\ge C_0$ for any $s\in R,v_0\in H^3(\Omega ),v_1\in L^2(\Omega )$ . Then for any $N$ , the problem (2.7), (2.8) has a global classical solution ${\alpha }_{Ns}(t)\in C^2[0,T]$ $(s=1,2,\cdots ,N)$ . Moreover, we have the following estimate

${\Vert v_N(\cdot ,t)\Vert }_{H^3(\Omega )}^2+{\Vert v_{Nt}(\cdot ,t)\Vert }^2\le C_1(T),t\in [0,T],$ (2.9)

where and in the sequel $C_i(T)(i=1,2,\cdots )$ are constants which depend on $T$ , but not on $N$ and $\Omega$ .

Proof: Let ${\sigma }_1(s)=\sigma (s)-k_{0}s-\sigma (0),k_0=min\{C_0,0\}\le 0$ , then ${\sigma }_1(0)=0$ , ${{\sigma }^{\prime }}_1(s)={\sigma }^{\prime }(s)-k_0\ge 0$ and ${\sigma }_1(s)$ is a monotonically increasing function, and thus

${\displaystyle {\int }_0^s}{\sigma }_1(\tau )d\tau \ge 0,\forall s\in R$ .

Obviously, system (2.7) is equivalent to the following system

$(v_{Ntt}-(1+k_0)v_{Nxx}-\alpha v_{N{x}^4}-\beta v_{N{x}^6}-{\sigma }_1{(v_{Nx})}_x,y_s)=0,s=1,2,\cdots ,N.$ (2.10)

Multiplying both sides of (2.10) by $2{\stackrel{\dot{}}{\alpha }}_{Ns}(t)$ , summing up for $s=1,2,\cdots ,N$ , and integrating by parts, we obtain

$\frac{d}{dt}({\Vert v_{Nt}\Vert }^2+(1+k_0){\Vert v_{Nx}\Vert }^2-\alpha {\Vert v_{Nxx}\Vert }^2+\beta {\Vert v_{N{x}^3}\Vert }^2+2{\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_0^{v_{Nx}}}{\sigma }_1(s)dsdx)=0$ (2.11)

When $k_0<-1$ , by virtue of Lemma 2.1, there is constants $C_1>0$ and $C_2>0$ such that

${\Vert v_{Nx}(\cdot ,t)\Vert }^2\le C_1{\Vert v_N(\cdot ,t)\Vert }^2-\frac{\beta }{4(1+k_0)}{\Vert v_{N{x}^3}(\cdot ,t)\Vert }^2,t>0,$ (2.12)

${\Vert v_{N{x}^2}(\cdot ,t)\Vert }^2\le C_2{\Vert v_N(\cdot ,t)\Vert }^2+\frac{\beta }{4\alpha }{\Vert v_{N{x}^3}(\cdot ,t)\Vert }^2,t>0.$ (2.13)

Adding $2[1-C_1(1+k_0)+C_2\alpha ](v_N,v_{Nt})$ to the both sides of (2.11), integrating in $[0,t]$ , making use of (2.12), (2.13), we get

$\begin{array}{l}{\Vert v_N(\cdot ,t)\Vert }^2+{\Vert v_{Nt}(\cdot ,t)\Vert }^2+\frac{\beta }{2}{\Vert v_{N{x}^3}(\cdot ,t)\Vert }^2+2{\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_o^{v_{Nx}}}{\sigma }_1(s)dsdx\\ \le {\Vert v_0\Vert }^2+{\Vert v_1\Vert }^2+\beta {\Vert {v^{‴}}_0\Vert }^2+2{\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_o^{v_0^{\text{<msup> <mo> ′ </mo> </msup>}}}}{\sigma }_1(s)dsdx\\ +[1-C_1(1+k_0)+C_2\alpha ]{\displaystyle {\int }_0^t}({\Vert v_N(\cdot ,\tau )\Vert }^2+{\Vert v_{Nt}(\cdot ,\tau )\Vert }^2)d\tau ,t\in [0,T].\end{array}$ (2.14)

Applying the Gronwall inequality to (2.14), we can obtain (2.9). When $-1\le k<0$ , adding $2(1+C_2\alpha )(v_N,v_{Nt})$ to the both sides of (2.11), integrating the product over $[0,t]$ , making use of (2.13) and the Cauchy inequality and Gronwall inequality, we get (2.9) immediately.

Using (2.9) and the Leray-Schauder fixed point theorem [8], we can prove that the problem (2.7), (2.8) has a solution ${\alpha }_{Ns}\in C^2[0,T](s=1,2,\cdots ,N)$ . Lemma 2.2 is proved.

Lemma 2.3. (Zhou and Fu [9]) Assume that $G(z)$ is a k-times $(k\ge 1)$ continuously differentiable function with respect to variables $z$ and $z\in L^{\infty }([0,T];H^k(\Omega ))$ . Then

${\Vert \frac{{\partial }^k}{\partial x^k}G(z)\Vert }^2\le C(m,k){\Vert z(\cdot ,t)\Vert }_{H^k(\Omega )}^2,$

where $m=\underset{(x,t)\in [0,T]\times \bar{\Omega }}{max}|z(x,t)|,C(m,k)$ is a positive constant depending only on

$m$ and $k$ .

Lemma 2.4. Assume that the assumption of Lemma 2.2 hold, $\sigma \in C^{k+1}(R)$ , $v_0\in H^{k+3}(\Omega ),v_1\in H^k(\Omega )$ , then there exist the estimates

${\Vert v_N(\cdot ,t)\Vert }_{H^{k+3}(\Omega )}^2+{\Vert v_{Nt}(\cdot ,t)\Vert }_{H^k(\Omega )}^2\le C_2(T),k\ge 0,$ (2.15)

${\Vert v_{Ntt}(\cdot ,t)\Vert }_{H^{p_1}(\Omega )}^2\le C_3(T),k=3+p_1,p_1\ge 0.$ (2.16)

Proof: We apply the mathematical induction to prove the estimate (2.15). The estimate (2.9) is the estimate (2.15) when $k=0$ . Suppose that when $k=n$ , the estimate (2.15) holds. We shall prove that, when $k=n+1$ , the estimate (2.15) holds too.

Multiply both sides of (2.7) by $2{(-1)}^{n+1}{\lambda }_s^{n+1}{\alpha }_{Nst}(t)$ , summing up for $s=1,2,\cdots ,N$ , integrating by parts, we obtain

$\begin{array}{l}\frac{d}{dt}({\Vert v_{N{x}^{n+1}t}(\cdot ,t)\Vert }^2+{\Vert v_{N{x}^{n+2}}(\cdot ,t)\Vert }^2-\alpha {\Vert v_{N{x}^{n+3}}(\cdot ,t)\Vert }^2+\beta {\Vert v_{N{x}^{n+4}}(\cdot ,t)\Vert }^2)\\ =2(\sigma {(}_{v_{Nx}},v_{N{x}^{n+1}t}).\end{array}$ (2.17)

By lemma 2.1, there is a constant $C_3>0$ , such that

${\Vert v_{N{x}^{n+3}}(\cdot ,t)\Vert }^2\le C_3{\Vert v_N(\cdot ,t)\Vert }^2+\frac{\beta }{2\alpha }{\Vert v_{N{x}^{n+4}}(\cdot ,t)\Vert }^2,t>0.$ (2.18)

Adding $2(1+C_3\alpha )(v_N,v_{Nt})$ to the both sides of (2.17), integrating the product over $[0,t]$ , Cauchy inequality, Lemma 2.3, (2.9) and (2.18), we have

$\begin{array}{l}{\Vert v_N(\cdot ,t)\Vert }^2+{\Vert v_{N{x}^{n+1}t}(\cdot ,t)\Vert }^2+{\Vert v_{N{x}^{n+2}}(\cdot ,t)\Vert }^2+\frac{\beta }{2}{\Vert v_{N{x}^{n+4}}(\cdot ,t)\Vert }^2\\ \le {\Vert v_0\Vert }^2+{\Vert v_{1x^{n+1}}\Vert }^2+{\Vert v_{0x^{n+2}}\Vert }^2+\beta {\Vert v_{0x^{n+4}}\Vert }^2+{\displaystyle {\int }_0^t}[C(m,n+2){\Vert v_{Nx}(\cdot ,\tau )\Vert }_{H^{n+2}(\Omega )}^2\\ +{\Vert v_{N{x}^{n+1}\tau }(\cdot ,\tau )\Vert }^2+(1+C_3\alpha )({\Vert v_N(\cdot ,\tau )\Vert }^2+{\Vert v_{N\tau }(\cdot ,\tau )\Vert }^2)]d\tau ,\end{array}$ (2.19)

where $m=\underset{(x,t)\in [0,T]\times \bar{\Omega }}{max}|v_{Nx}(x,t)|$ . It follows from (2.9), (2.19) and the Gronwall

inequality, we get

${\Vert v_{N{x}^{p_1}tt}\Vert }^2\le (\Vert v_{N{x}^{p_1+2}}\Vert +\Vert \sigma {(v_{Nx})}_{x^{p_1+1}}\Vert +\alpha \Vert v_{N{x}^{p_1+4}}\Vert +\beta \Vert v_{N{x}^{p_1+6}}\Vert )\Vert v_{N{x}^{p_1}tt}\Vert$ (2.20)

Multiply both sides of (2.7) by ${(-1)}^{p_1}{\lambda }_s^{p_1}{\alpha }_{Nstt}(t)$ , summing up for $s=1,2,\cdots ,N$ , integrating by parts, Holder inequality, (2.15), we get

$\Vert v_{N{x}^{p_1}tt}\Vert \le C_5(T),t\in [0,T].$ (2.21)

Theorem 2.1. Under the assumptions of Lemma 2.4, if $k\ge 3$ , then the problem (2.1) - (2.3) has a unique generalized global solution $v(x,t)$ , which has continuous derivatives $v_{x^s}(x,t)(0\le s\le 2)$ and generalized derivatives $v_{x^s}(x,t)(0\le s\le 6)$ , $v_{x^{s}t}(x,t)(0\le s\le 3)$ and $v_{tt}(x,t)$ .

Proof: First we give the definition of the generalized solution, which $v(x,t)$ satisfies the identity

${\displaystyle {\int }_0^T}{\displaystyle {\int }_{\Omega }}[v_{tt}-v_{xx}-\sigma {(v_x)}_x-\alpha v_{x^4}-\beta v_{x^6}]g(x,t)dxdt=0,\forall g(x,t)\in L^2(Q_T)$ (2.22)

and the periodic boundary value conditions (2.2), (2.3) in the classical sense.

By Lemma 2.4, we have

${\Vert v_N(\cdot ,t)\Vert }_{H^{k+3}(\Omega )}^2+{\Vert v_{Nt}(\cdot ,t)\Vert }_{H^k(\Omega )}^2\le C_2(T),t\in [0,T],$

${\Vert v_{Ntt}(\cdot ,t)\Vert }_{H^{p_1}(\Omega )}^2\le C_3(T),t\in [0,T],$

It follows from Sobolev embedding theorem, when $k=3$ , we know

$v_{N{x}^s}\in C([0,T]\times \bar{\Omega })（0\le s\le 5）,v_{N{x}^{s}t}\in C([0,T]\times \bar{\Omega })（0\le s\le 2）.$

We can select a subsequence of $\{v_N(x,t)\}$ and a function $v(x,t)$ and $N\to \infty$ , the subsequence uniformly converges to the limiting function $v(x,t)$ in ${\bar{Q}}_T$ . In fact, $\{v_N(x,t)\}$ is the uniformly bound in ${\bar{Q}}_T$ . Meanwhile

$\begin{array}{l}|v_N(x+\Delta x,t+\Delta t)-v_N(x,t)|\\ \le |v_N(x+\Delta x,t+\Delta t)-v_N(x,t+\Delta t)|+|v_N(x,t+\Delta t)-v_N(x,t)|\\ =|v_{Nx}(x+{\theta }_1\Delta x,t+\Delta t)|\left|\Delta x\right|+|v_{Nt}(x,t+{\theta }_2\Delta t)|\left|\Delta t\right|,\end{array}$ (2.23)

where $0<{\theta }_1,{\theta }_2<1,\Delta x$ and $\Delta t$ are the change vectors. Therefore, $\{v_N(x,t)\}$ is equicontinuous in ${\bar{Q}}_T$ .

According to Ascoli-Arzela, we can select a subsequence of $\{v_N(x,t)\}$ , still denoted by $\{v_N(x,t)\}$ , such that there exists a function $v(x,t)$ and $N\to \infty$ , the subsequence $v(x,t)$ uniformly converges to the limiting function $v(x,t)$ in ${\bar{Q}}_T$ . The corresponding subsequences $\{v_{N{x}^s}(x,t)\}(s=1,2)$ also uniformly converges to $v_{x^s}(x,t)(s=1,2)$ in ${\bar{Q}}_T$ , respectively.

Making use of the weakly compact theorem of the space $L^2(Q_T)$ , we know that the subsequences $\{v_{N{x}^s}(x,t)\}(0\le s\le 6)$ , $\{v_{N{x}^{s}t}(x,t)\}(0\le s\le 3)$ and $\{v_{Ntt}(x,t)\}$ weakly converge to $v_{x^s}(x,t)(0\le s\le 6)$ , $v_{x^{s}t}(x,t)(0\le s\le 3)$ and $v_{tt}(x,t)$ in $L^2(Q_T)$ , respectively.

In fact,

$\begin{array}{l}|{\displaystyle {\int }_0^T}{\displaystyle {\int }_{\Omega }}[\sigma {(}_{v_{Nx}}-\sigma {(}_{v_x}]g(x,t)dxdt|\\ =|{\displaystyle {\int }_0^T}{\displaystyle {\int }_{\Omega }}[{\sigma }^{\prime }(v_{Nx})(v_{N{x}^2}-v_{x^2})+{\sigma }^{″}(v_x+{\theta }_3(v_{Nx}-v_x))v_{x^2}]g(x,t)dxdt|\\ \le \underset{{{\displaystyle \bar{Q}}}_T}{max}|{\sigma }^{\prime }(v_{Nx})||{\displaystyle {\int }_0^T}{\displaystyle {\int }_{\Omega }}(v_{N{x}^2}-v_{x^2})g(x,t)dxdt|\\ +\underset{{{\displaystyle \bar{Q}}}_T}{max}|{\sigma }^{″}(v_x+{\theta }_3(v_{Nx}-v_x))v_{x^2}||{\displaystyle {\int }_0^T}{\displaystyle {\int }_{\Omega }}(v_{Nx}-v_x)g(x,t)dxdt|,\end{array}$ (2.24)

where $0<{\theta }_3<1$ . So the identity (2.22) holds. Obviously $v(x,t)$ satisfies the periodic boundary value conditions (2.2), (2.3) in the classical sense. Therefore when $k\ge 3$ , there exists a generalized global solution $v(x,t)$ of the problem (2.1) - (2.3).

It is easy that we can get the uniqueness of the solution of the periodic boundary value problem (2.1) - (2.3). The Theorem 2.1 is proved.

Theorem 2.2. Under the assumptions of Lemma 2.4, if $k\ge 7$ , then the periodic boundary value problem (2.1) - (2.3) has a unique global classical solution $v(x,t)$ .

Proof: Differentiating (2.7) with respect to $t$ , we have

$(v_{N{t}^3}-v_{Nxxt}-\alpha v_{N{x}^{4}t}-\beta v_{N{x}^{6}t}-\sigma {(v_{Nx})}_{xt},y_s)=0,s=1,2,\cdots ,N.$ (2.25)

Multiplying both sides of (2.25) by ${(-1)}^{p_2}{\lambda }_s^{p_2}{\alpha }_{Ns{t}^3}(t)$ , summing up for $s=1,2,\cdots ,N$ , integrating by parts and using the Holder inequality, combining (2.15), we conclude

$\Vert v_{N{x}^{p_2}{t}^3}\Vert \le C_7(T),k=p_2+6,p_2\ge 0.$ (2.26)

Combining (2.26) and Sobolev embedding theorem, we know that

$v_{N{x}^s{t}^3}\in C([0,T]\times \bar{\Omega }),0\le s\le k-7.$

Using the method of Theorem 2.1, when $k\ge 7$ , the periodic boundary value problem (2.1) - (2.3) has a global classical solution $v(x,t)$ . It is easy to prove the uniqueness of solution for the problem (2.1) - (2.3).

3. Cauchy Problem (1.6), (1.7)

Theorem 3.1. Suppose that $\sigma \in C^{k+1}(R),{\sigma }^{\prime }(s)$ is bounded blow, $v_0\in H^{k+3}(R)$ , $v_1\in H^k(R)$ . If $k\ge 3(k=3+p_1,p_1\ge 0)$ , then the Cauchy problem (1.6), (1.7) has a unique global generalized solution $v(x,t)$ .

Proof: Let us take a real sequence $\left\{D_s\right\}(D_s>1)$ such that $D_s\to \infty$ as $s\to \infty$ . For every $s$ , let us construct periodic functions $v_{0s}(x)$ and $v_{1s}(x)$ with period $2D_s$ such that

i) $v_{0s}\in H^{k+3}[-D_s,D_s],v_{1s}\in H^k[-D_s,D_s]$ ;

ii) $v_{0s}(x)=v_0(x),v_{1s}(x)=v_1(x)$ for $x\in [-(D_s-1),D_s-1]$ and

${\Vert v_{0s{x}^i}\Vert }_{L^2[-D_s,D_s]}\le {\Vert v_{0x^i}\Vert }_{L^2(R)},i=0,1,\cdots ,k+3,$

${\Vert v_{1s{x}^i}\Vert }_{L^2[-D_s,D_s]}\le {\Vert v_{1x^i}\Vert }_{L^2(R)},i=0,1,\cdots ,k+1.$

we consider the following periodic boundary value problem

$v_{tt}=v_{xx}+\sigma {(v_x)}_x+\alpha v_{x^4}+\beta v_{x^6},$ (3.1)

$v_x(x,t)=v_x(x+2D_s,t),$ (3.2)

$v(x,0)=v_{0s}(x),v_t(x,0)=v_{1s}(x).$ (3.3)

Let $\{y_n(x)\}=\{\frac{1}{\sqrt{2D}},\frac{1}{\sqrt{D}}cos{\delta }_{n}x,\frac{1}{\sqrt{D}}sin{\delta }_{n}x,n=1,2,\cdots \}$ be the orthogonal

base in $L^2({\Omega }_s)$ composed of eigenfunctions of eigenvalue problem

$\begin{array}{l}{y}^{″}+\lambda y=0,x\in {\Omega }_s,\\ y_x(x)=y_x(x+2D_s)\end{array}$

corresponding to eigenvalue ${\lambda }_i={\delta }_i^2={(\frac{i\pi }{D_s})}^2(i=1,2,\cdots )$ , where ${\Omega }_s=(-D_s,D_s)$ .

Suppose that the Galerkin approximate solution of (3.1)-(3.3) is

$v_{N_s}(x,t)={\displaystyle \underset{i=1}{\overset{N_s}{\sum }}}{\alpha }_{N_{s}i}(t)y_i(x)$

where ${\alpha }_{N_{s}i}(t)$ are the undermined functions.

Let $v_{N_s}(x,t)$ satisfy the following equation and conditions

$v_{N_{s}tt}=v_{N_{s}xx}+\sigma {(v_{N_{s}x})}_x+\alpha v_{N_s{x}^4}+\beta v_{N_s{x}^6},$ (3.4)

$v_{N_{s}x}(x,t)=v_{N_{s}x}(x+2D_s,t),$ (3.5)

$v_{N_s}(x,0)=v_{0s}(x),v_{N_{s}t}(x,0)=v_{1s}(x).$ (3.6)

By the same method as in the estimates (2.15), (2.16), we have

${\Vert v_{N_s}\Vert }_{H^{k+3}({\Omega }_s)}+{\Vert v_{N_{s}t}\Vert }_{H^k({\Omega }_s)}+{\Vert v_{N_s{t}^2}\Vert }_{H^{p_1}({\Omega }_s)}\le C_8(T),t\in [0,T],$ (3.7)

where and in the sequel $C_i(T)(i=8,9,\cdots )$ are constants independent of $N_s$ and $D_s$ . By the Sobolev imbedding theorem when $k=4$ , we get

${\Vert v_{N_s}\Vert }_{H^6({\Omega }_s)}+{\Vert v_{N_{s}t}\Vert }_{H^3({\Omega }_s)}+{\Vert v_{N_s{t}^2}\Vert }_{L^2({\Omega }_s)}\le C_9(T),t\in [0,T],$ (3.8)

${\Vert v_{N_s}\Vert }_{C^5({{\displaystyle \bar{\Omega }}}_s)}+{\Vert v_{N_{s}t}\Vert }_{C^2({{\displaystyle \bar{\Omega }}}_s)}\le C_{10}(T),t\in [0,T].$ (3.9)

By virtue of (3.9) and Ascoli-Arzela theorem, we can select from $\{v_{N_s}(x,t)\}$ a subsequence, still denoted by $\{v_{N_s}(x,t)\}$ , such that when $N_s\to \infty$ ,

$\{v_{N_s{x}^k}(x,t)\}(0\le k\le 2)$ uniformly converge to limiting functions $v_{s{x}^k}(x,t)(0\le k\le 2)$ in ${{\displaystyle \bar{\Omega }}}_s\times [0,T]$ , respectively.

The estimates (3.8) still holds for the above subsequence $\{v_{N_s}(x,t)\}$ . Hence, we can select from $\{v_{N_s}(x,t)\}$ a subsequence, still denoted by $\{v_{N_s}(x,t)\}$ , such

that when $N_s\to \infty$ , the subsequences $\{v_{N_s{x}^k}(x,t)\}(0\le k\le 6)$ , $\{v_{N_s{x}^{k}t}(x,t)\}(0\le k\le 3)$ and $\{v_{N_{s}tt}(x,t)\}$ weakly converge to limiting functions $v_{s{x}^k}(x,t)(0\le k\le 6)$ , $v_{s{x}^{k}t}(x,t)(0\le k\le 3)$ and $v_{stt}(x,t)$ in $L^2((0,T);L^2({\Omega }_s))$ ,

respectively.

From the corollary of the resonance theorem [10], it follows that the estimates (3.8), (3.9) still hold for $v_s(x,t)$ , which is the generalized solution of the problem (3.1)-(3.3). Using Ascoli-Arzela theorem, we can select from $\{v_s(x,t)\}$ a subsequence still denoted by $\{v_s(x,t)\}$ , such that when $s\to \infty$ , the subsequences $\{v_{s{x}^k}(x,t)\}(0\le k\le 2)$ uniformly converge to limiting functions $v_{x^k}(x,t)(0\le k\le 2)$ in any domain $\{-L\le x\le L,0\le t\le T\}$ , respectively.

It follows from (3.8) that we can select from $v_s(x,t)$ a subsequence, still denoted by $v_s(x,t)$ , such that when $s\to \infty$ , in $L^2((0,T);L^2(-L,L))$ , the subsequences $\{v_{s{x}^k}(x,t)\}(0\le k\le 6)$ , $\{v_{s{x}^k}(x,t)\}(0\le k\le 6)$ and $\{v_{stt}(x,t)$ weakly converge to limiting functions $\{v_{x^k}(x,t)\}(0\le k\le 6)$ , $\{v_{x^{k}t}(x,t)\}(0\le k\le 3)$ and $\{v_{tt}(x,t)\}$ , respectively. The obtained limiting function $v(x,t)$ is just the global generalized solution of the auxiliary problem (1.6), (1.7).

Clearly, the generalized solution of the auxiliary problem (1.6), (1.7) is also unique. Therefore when $k\ge 3$ , the Cauchy problem (1.6), (1.7) has a unique global generalized solution.

Theorem 3.2. Assume that the assumptions of Theorem 3.1 hold, If $k\ge 7(k=6+p_2,p_2\ge 0)$ , then the Cauchy problem (1.6), (1.7) has a unique global classical solution $v(x,t)$ .

Proof: Combining the estimates (2.15), (2.16) with (2.35), we obtain

${\Vert v_{N_s}\Vert }_{H^{k+3}({\Omega }_s)}+{\Vert v_{N_{s}t}\Vert }_{H^k({\Omega }_s)}+{\Vert v_{N_s{t}^2}\Vert }_{H^{p_1}({\Omega }_s)}+{\Vert v_{N_s{t}^3}\Vert }_{H^{p_2}({\Omega }_s)}\le C_{11}(T),t\in [0,T]$ (3.10)

By the Sobolev imbedding theorem when $k=7$ , we get

${\Vert v_{N_s}\Vert }_{H^{10}({\Omega }_s)}+{\Vert v_{N_{s}t}\Vert }_{H^7({\Omega }_s)}+{\Vert v_{N_s{t}^2}\Vert }_{H^4({\Omega }_s)}+{\Vert v_{N_s{t}^3}\Vert }_{H^1({\Omega }_s)}\le C_{12}(T),$

${\Vert v_{N_s}\Vert }_{C^9({{\displaystyle \bar{\Omega }}}_s)}+{\Vert v_{N_{s}t}\Vert }_{C^6({{\displaystyle \bar{\Omega }}}_s)}+{\Vert v_{N_{s}tt}\Vert }_{C^3({{\displaystyle \bar{\Omega }}}_s)}+{\Vert v_{N_s{t}^3}\Vert }_{C({{\displaystyle \bar{\Omega }}}_s)}\le C_{13}(T).$

Using the method of Theorem 3.1, when $k\ge 7$ , the Cauchy problem (1.6), (1.7) has a global classical solution. It is easy to prove the uniqueness of solution for the problem (1.6), (1.7).

4. Cauchy Problem (1.1), (1.2)

Lemma 4.1. [11] Suppose that $s=m+\frac{1}{2}+\lambda ,\lambda \in (0,1),m\in Z_{+}$ , then $H^s(R)$

may be embedded into $C^{m+\lambda }(R)$ , and for any $u\in H^s(R)$ , we have

$|\frac{d^k}{d{x}^k}u(x)|\to 0(|x|\to \infty ),\forall k\in Z_{+},0\le k\le m,$

where $Z_{+}$ is a set of nonnegative integers.

Theorem 4.1. Suppose that $\sigma \in C^{k+2}(R)$ , ${\sigma }^{\prime }(s)$ is bounded blow, $u_0\in H^{k+2}(R)$ , $u_1\in H^{k-1}(R)$ . If $k-1\ge 3(k=3+p_1,p_1\ge 0)$ , then the Cauchy problem (1.1), (1.2) has a unique global generalized solution $u(x,t)$ .

Proof: Differentiating (3.4) and (3.6) with respect to $x$ , we have

$v_{N_{s}xtt}=v_{N_s{x}^3}+\sigma {(v_{N_{s}x})}_{xx}+\alpha v_{N_s{x}^5}+\beta v_{N_s{x}^7},$ (4.1)

$v_{N_{s}x}(x,0)=v_{0sx}(x),v_{N_{s}xt}(x,0)=v_{1sx}(x).$ (4.2)

Let

$v_{N_{s}x}(x,t)=u_{N_s}(x,t).$ (4.3)

Substituting (4.3) into (4.1), (3.5) and (4.2), we obtain

$u_{N_{s}tt}=u_{N_{s}xx}+\sigma {(u_{N_s})}_{xx}+\alpha u_{N_s{x}^4}+\beta u_{N_s{x}^6},$ (4.4)

$u_{N_s}(x,t)=u_{N_s}(x+2D_s,t),$ (4.5)

$u_{N_s}(x,0)=u_{0N_s}(x),u_{N_{s}t}(x,0)=u_{1N_s}(x).$ (4.6)

By using the change (4.3), it follows from (3.7) that

${\Vert u_{N_s}\Vert }_{H^{k+2}({\Omega }_s)}+{\Vert u_{N_{s}t}\Vert }_{H^{k-1}({\Omega }_s)}+{\Vert u_{N_s{t}^2}\Vert }_{H^{p_1-1}({\Omega }_s)}\le C_{14}(T),t\in [0,T].$ (4.7)

From (4.7) and the Sobolev imbedding theorem, we know that

${\Vert u_{N_s}\Vert }_{C^{k+1}({{\displaystyle \bar{\Omega }}}_s)}+{\Vert u_{N_{s}t}\Vert }_{C^{k-2}({{\displaystyle \bar{\Omega }}}_s)}+{\Vert u_{N_{s}tt}\Vert }_{C^{p_1-2}({{\displaystyle \bar{\Omega }}}_s)}\le C_{15}(T),t\in [0,T].$ (4.8)

By using the same method as in Section 3, it follows from (4.7) and (4.8) that, when $k\ge 4$ , the Cauchy problem (1.1), (1.2) has a generalized global solution $u(x,t)$ .

It is easy that, we prove the uniqueness of solution for the problem (1.1), (1.2). Hence, Theorem 4.1 is proved.

Theorem 4.2. Assume that the assumptions of Theorem 4.1 hold, If $k-1\ge 7$ Math_277#, then the Cauchy problem (1.1), (1.2) has a unique global classical solution $u(x,t)$ .

Proof: By virtue of Theorem 3.2, when $k\ge 8$ , the problem (1.6), (1.7) has a unique global classical solution $v(x,t)$ . Differentiating Equation (1.6), (1.7) with $x$ and substituting $v_x(x,t)=u(x,t)$ into this equation and $v_{0x}(x)=u_0(x),v_{1x}(x)=u_1(x)$ into the obtained initial value condition, we get $u(x,t)$ is the classical global solution of (1.1), (1.2). The proof is completed.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11671367 and Natural Science Foundation of Henan Province, China under Grant No. 152300410227.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Darapi, P. and Hua, W. (1999) A Numerical Method for Solving an Ill-Posed Bous-sinesq Equation Arising in Water Waves and Nonlinear Lattices. Applied Mathematics and Computation, 101, 159-207. https://doi.org/10.1016/S0096-3003(98)10070-X
[2]	Dash, R.K. and Daripa, P. (2002) Analytical and Numerical Studies of a Singularly Perturbed Boussinesq Equa-tion. Applied Mathematics and Computation, 126, 1-30. https://doi.org/10.1016/S0096-3003(01)00166-7
[3]	Darapi, P. and Hua, W. (2001) Weakly Non-Local Solitary Wave Solutions of a Singularly Perturbed Boussinesq Equation. Mathematics and Computers in Simulation, 55, 393-405. https://doi.org/10.1016/S0378-4754(00)00288-3
[4]	Feng, Z.S. (2003) Traveling Solitary Wave Solutions to the Generalized Boussinesq Equation. Wave Motion, 37, 17-23. https://doi.org/10.1016/S0165-2125(02)00019-7
[5]	Song, C., Li, H. and Li, J. (2013) Initial Boundary Value Problem for the Singularly Perturbed Bous-sineaq-Type Equation. Discrete and Continuous Dynamical Systems, 709-717.
[6]	Song, C., Li, J. and Gao, R. (2014) Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth Order Bous-sinesq Equation. Hindawi Publishing Corporation, Journal of Applied Mathemat-ics.
[7]	Admas, R.A. (1975) Sobolev Space. Academic Press, New York.
[8]	Friedman, A. (1964) Partial Differential Equation of Parabolic Type. Pren-tice Hall, Eagliweed Cliffs.
[9]	Zhou, Y.L. and Fu, H.Y. (1983) Nonlinear Hyper-bolic Systems of Higher Order Generalized Sine-Gordon Type. Acta Mathematica Sinica, 26, 234-249.
[10]	Yosida, K. (1980) Functional Analysis. 6th edition, Springer Verlag, Berlin.
[11]	Wang, Y.D. (1989) L2 Theory of Partial Differential Equations. Peking University Press, Beijing.