﻿ 五次长短波共振方程初值问题解的存在性
 上海理工大学学报  2019, Vol. 41 Issue (4): 313-320 PDF

Existence of Solutions for the Initial Value Problems of a Five-Time Long-Short Wave Resonance Equation
WANG Beibei, ZHANG Weiguo
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The existence of the solutions to the initial value problem of a five-time long-short wave resonance equation was focused. First, the existence of the local solution was proved by using the compression mapping theorem. Then, the local solution with energy conservation properties was established and the prior estimation method was used to, It was also proved that the local solution of high-order nonlinear terms in the five-time long-short wave resonance equation can be extended to the whole domain. Finally, the existence of the global solution was testified.
Key words: compression mapping theorem     a prior estimate     energy conservation
1 问题的提出

Djordjevic等[1] 在研究二维的毛细管重力波时，发现了描述长波和短波之间相互作用的演化方程

 $\left\{ {\begin{array}{*{20}{l}} {i{u_t} + {u_{xx}} = \gamma uv} \\ {{v_t} + \beta {{\left( {|u{|^2}} \right)}_x} = 0} \end{array}} \right.$ (1)

 $\left\{ {\begin{array}{*{20}{l}} {i{u_t} + {u_{xx}} = uv + \gamma |u{|^2}u} \\ {{v_t} = {{\left( {|u{|^2}} \right)}_x}} \end{array}} \right.$ (2)

1988年，Oikawa等[8]研究了双层流体中长波和短波在彼此分界面角度上的传播和共振作用，导出了（2+1）维长短波方程组，并将长短波方程拓展到高维空间，文献[914]分别研究了高维长短波方程及其推广形式解的存在唯一性。

 $\left\{ {\begin{array}{*{20}{l}} {i{u_t} + {u_{xx}} - uv + \gamma q(|u{|^2})u = 0} \\ {{v_t} + {{\left( {|u{|^2}} \right)}_x} = \delta u + \gamma f(|u{|^2})} \end{array}} \right.$ (3)

2010年，Shang[16]研究了广义长短波方程

 $\left\{ {\begin{array}{*{20}{l}} {i{u_t} + {u_{xx}} = uv + \gamma (|u{|^2})u + \delta |u{|^4}u} \\ {{v_t} = {{\left( {|u{|^2}} \right)}_x}} \end{array}} \right.$ (4)

 $\left\{\begin{split} & X_3^T = \{ f:[0,T] \times {\mathbb R} \to {\mathbf{C}}|f \in \left( {[0,T];{H^{5/2}}({\mathbb R})} \right),\\ &{\rm D}_x^kf \in {L^\infty }\left( {{\mathbb R};{L^2}[0,T]} \right)(k = 1,2,3),\\ &{\rm D}_x^{p - 2q}{\rm D}_t^qf\!\! \in\!\! {L^\infty }\left( {{\mathbb R};{L^2}[0,\!T]} \right)\left( {2 \!\!\leqslant\!\! p \!\!\leqslant \!\!3,q \!=\! [p/2]} \right)\} \end{split}\right.$ (5)

 $\begin{split} &\quad\parallel f{\parallel _{X_3^T}} = \mathop {\sup }\limits_{0 \leqslant t \leqslant T} \parallel f(t){\parallel _{{H^{5/2}}}} +\\ &\qquad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\sum\limits_{k = 1}^3 {\int_0^T | } ({\rm D}_x^kf)(x,t){|^2}{\rm d}t} \right)^{1/2}} + \\ &\qquad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\sum\limits_{2 \leqslant p \leqslant 3,q = [p/2]} {\int_0^T | } ({\rm D}_x^{p - 2q}{\rm D}_t^q)f(x,t){|^2}{\rm d}t} \right)^{1/2}}\qquad \end{split}$

 $\begin{gathered} u(0,x) = {u_0}(x),\quad x \in {\mathbb {R}} \\ v(0,x) = {v_0}(x),\quad x \in {\mathbb {R}} \\ \end{gathered}$

$s$ 是实数, 则Sobolev空间 ${H^s}({{\mathbb {R}}^n})$

 ${H^s}({{\mathbb {R}}^n}) = \{ f \in {L^2}({{\mathbb {R}}^n})\mid {\left( {1 + |\xi {|^2}} \right)^{s/2}}\hat f(\xi ) \in {L^2}({{\mathbb {R}}^2})\}$

$f \in {H^s}({{\mathbb{R}}^n})$

 $\parallel f{\parallel _{{H^s}}} = {(2\pi )^{ - n/2}}{\left( {\int_{{{\mathbb{R}}^n}} {{{\left( {1 + |\xi {|^2}} \right)}^s}} |\hat f(\xi ){|^2}{\rm d}\xi } \right)^{1/2}}$

$s$ 是整数时，

 $\begin{split} &\quad\|f\|_{{{H}^{s}}}^{2}={{(2\pi )}^{-n}}\int_{{{{\mathbb{R}}}^{n}}}{\sum\limits_{|\alpha |\leqslant s}{{{C}_{\alpha }}}}|{{\xi }^{\alpha }}\hat{f}(\xi ){{|}^{2}}{\rm d}\xi=\qquad\qquad\qquad\qquad \\ &\qquad\int_{{{{\mathbb{R}}}^{n}}}{\sum\limits_{|\alpha |\leqslant s}{{{C}_{\alpha }}}}|{{{\rm D}}^{\alpha }}f{{|}^{2}}{\rm d}x \end{split}$

 $U(t) = {{\rm e}^{{{\rm{i}}t}({\partial ^2}/\partial {x^2})}} = {{\mathcal {F}}^{ - 1}}{{\rm e}^{ - {\rm i}t{\xi ^2}}}{\mathcal {F}}, \quad t \in {\mathbb {R}}$

2 局部解的存在性

 $\mathop {\sup }\limits_{ - \infty < x < \infty } \left( {\int_{ - \infty }^\infty | {\rm D}_x^{1/2}U(t){u_0}{|^2}{\rm d}t} \right) \leqslant C{\left( {\int_{ - \infty }^\infty | {u_0}(x){|^2}{\rm d}x} \right)^{1/2}}$

 $\begin{split}&\quad\mathop {\sup }\limits_{0 \leqslant t \leqslant T} {\left( {\int_{ - \infty }^\infty | {\rm D}_x^{1/2}\int_0^t U (t - s)f(s,x){\rm d}s{ |^2}{\rm d}x} \right)^{1/2}} \leqslant\qquad\qquad\\ & \qquad C\int_{ - \infty }^\infty {{{\left( {\int_0^T | f(t,x){|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x\end{split}$

 $\begin{split} &\quad\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T \left| {{\rm D}_x}\int_0^t U (t - s)f(s, \cdot ){\rm d}s\right|{^2}{\rm d}t} \right)^{1/2}} \leqslant\qquad\qquad \\ &\qquad C\int_{ - \infty }^\infty {{{\left( {\int_0^T | f(t,x){|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x\end{split}$

 $\begin{split}\int_0^T {{{\left\| {\int_0^t f (s){g_x}(s){\rm d}sh(t)} \right\|}_{{L^2}}}} {\rm d}t \leqslant C{T^{3/2}}\left\| f\right\|_{X_3^T}\left\|g\right\|_{X_3^T}\left\|h\right\|_{X_3^T}\end{split}$

 $\begin{split} &\quad \int_{-\infty }^{\infty }{{{\left[ \int_{0}^{T}{{{\left| {{\rm D}_{x}}\left( \int_{0}^{t}{f}(s){{g}_{x}}(s){\rm d}sh(t) \right) \right|}^{2}}}{\rm d}t \right]}^{1/2}}}{\rm d}x \leqslant \qquad\qquad\\ & \qquad T\underset{-\infty 引理 6 对任意的$f,g,h\in X_{3}^{T}$, 有 $\begin{split}&\quad \int_{-\infty }^{\infty }{{{\left[ \int_{0}^{T}{{{\left| {{\rm D}_{xx}}\left( \int_{0}^{t}{f}(s,x){{g}_{x}}(s,x){\rm d}sh(t,x) \right) \right|}^{2}}} {\rm d}t\right]}^{1/2}}}{\rm d}x\leqslant\qquad\\ &\qquad C(T)\|f{{\|}_{X_{3}^{T}}}\|g{{\|}_{X_{3}^{T}}}\|h{{\|}_{X_{3}^{T}}}\end{split}$其中，$T \to 0$时，$C(T) \to 0$. 证明 运用Cauchy不等式和Holder不等式，可得 $ \begin{split}&\quad\int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {{{\rm D}_{xx}}\left( {\int_0^t f (s,x){g_x}(s,x){\rm d}sh(t,x)} \right)} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x \leqslant \qquad\qquad\\ &\qquad \int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\int_0^t {(2{f_x}{g_{xx}})} {\rm d}sh} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x + \\ &\qquad\int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\int_0^t {(f{g_{xxx}})} {\rm d}sh} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x +\\ & \qquad \int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\int_0^t {({f_{xx}}{g_x})} {\rm d}sh} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x + \\ &\qquad 2\int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\int_0^t {(f{g_{xx}})} {\rm d}s{h_x}} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x+ \\ &\qquad 2\int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\int_0^t {({f_x}{g_x})} {\rm d}s{h_x}} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x +\\ &\qquad\int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\int_0^t f {g_x}{\rm d}s{h_{xx}}} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x\\[-20pt] \end{split}$(6) 估计方程（6）不等式右边的第1项为 $\begin{split}& \quad\int_{ - \infty }^\infty {{{\left[ {\int_0^T {{{\left| {\left( {\int_0^t 2 {f_x}(s,x){g_{xx}}(s,x)} \right){\rm d}sh(t,x)} \right|}^2}} {\rm d}t} \right]}^{1/2}}} {\rm d}x \leqslant\qquad\qquad\\ &\qquad\int_{ - \infty }^\infty {{{\left[ {{{\left( {\int_0^T 2 {f_x}(s,x){g_{xx}}(s,x){\rm d}s} \right)}^2}\int_0^T | h{|^2}{\rm d}t} \right]}^{1/2}}} {\rm d}x\leqslant\\ &\qquad 2\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {g_{xx}}{|^2}{\rm d}t} \right)^{1/2}}{\left( {\int_{ - \infty }^\infty {\int_0^T | } {f_x}{|^2}{\rm d}t{\rm d}x} \right)^{1/2}}{\text{·}}\\ &\qquad {\left( {\int_{ - \infty }^\infty {\int_0^T | } h{|^2}{\rm d}t{\rm d}x} \right)^{1/2}}\leqslant\\ &\qquad 2T\underset{-\infty

 $u(t) = U(t){u_0} - {\rm i}\int_0^t U (t - s){F}(u(s)){\rm d}s$ (7)

 ${F}\left( {u(t)} \right) = \left[ {\int_0^t {(|u(} x,s){|^2}{)_x}{\rm d}s} \right]u + {v_0}u + \gamma |u{|^2}u + \delta |u{|^4}u\!\!\!\!\!\!$ (8)

 $B_{r}^{3}=\left\{ u\in X_{3}^{T}\mid \|u{{\|}_{X_{3}^{T}}}\leqslant r \right\},\quad r>0$

${u_0} \in {H^{5/2}}({\mathbb R}),{v_0} \in {H^2}({\mathbb R})$ , 定义算子 $\varPhi$ ,

 $\left( {\varPhi (\omega )} \right)(t) = U(t){u_0} - {\rm i}\int_0^t U (t - s){ F}(\omega (s)){\rm d}s$ (9)

 $\begin{split} &\quad{F}\left( {\omega (t)} \right) = \left[ {\int_0^t {(|\omega (} x,s){|^2}{)_x}{\rm d}s} \right]\omega + \qquad\qquad\qquad\qquad\qquad\\ &\qquad {v_0}\omega + \gamma |\omega {|^2}\omega + \delta |\omega {|^4}\omega\\[-10pt] \end{split}$ (10)

 $\begin{split} &\quad\|\varPhi (\omega )(t){{\|}_{X_{3}^{T}}}=\underset{0\leqslant t\leqslant T}{\mathop{\sup }}\,\|\varPhi (\omega )(t){{\|}_{{{H}^{5/2}}}}+ \\ & \qquad\mathop {\sup }\limits_{ - \infty < x < \infty } \left( \int_0^T {{{\left| {{{\rm D}_x}\varPhi (\omega )(x,t)} \right|}^2}} {\rm d}t +\right.\\ &\left. \qquad\int_0^T {{{\left| {{\rm D}_x^2\varPhi (\omega )(x,t)} \right|}^2}} {\rm d}t + \int_0^T {{{\left| {{\rm D}_x^3\varPhi (\omega )(x,t)} \right|}^2}} {\rm d}t \right)^{1/2}+\qquad\qquad\\ & \qquad\mathop {\sup }\limits_{ - \infty < x < \infty } \left( \int_0^T | {{\rm D}_t}\varPhi (\omega )(x,t){|^2}{\rm d}t+ \right.\\ &\left. \qquad \int_0^T | {{\rm D}_x}{{\rm D}_t}\varPhi (\omega )(x,t){|^2}{\rm d}t \right)^{1/2}\\[-15pt]\end{split}$ (11)

 $\|\varPhi (\omega )(t){{\|}_{{{L}^{2}}}}\leqslant C\|{{u}_{0}}{{\|}_{{{L}^{2}}}}+\int_{0}^{t}{\|{ F}\left( \omega (s) \right){{\|}_{{{L}^{2}}}}}{\rm d}s$

 $\begin{split}& \quad\int_{0}^{T}{\|{F}\left( \omega (s) \right){{\|}_{{{L}^{2}}}}}{\rm d}t\leqslant C\left( {{T}^{3/2}}+T \right)\|\omega \|_{X_{3}^{T}}^{3}+\qquad\qquad\qquad\\ &\qquad CT\|{{v}_{0}}{{\|}_{{{L}^{\infty }}}}\|\omega {{\|}_{X_{3}^{T}}}+CT\|\omega \|_{X_{3}^{T}}^{5}\\[-15pt]\end{split}$ (12)

 $\begin{split}&\quad\|\varPhi (\omega )(t){{\|}_{{{L}^{2}}}}\leqslant C\|{{u}_{0}}{{\|}_{{{L}^{2}}}}+C\left( {{T}^{3/2}}+T \right)\|\omega \|_{X_{3}^{T}}^{3}+\qquad\qquad\\ &\qquad CT\|{{v}_{0}}{{\|}_{{{L}^{\infty }}}}\|\omega {{\|}_{X_{3}^{T}}}+CT\|\omega \|_{X_{3}^{T}}^{5}\\[-15pt]\end{split}$ (13)

 $\begin{split}&\quad |\omega (t){|^2} \leqslant |\omega (s){|^2} + \left| {\int_s^t {{{\rm D}_\tau }} |\omega (\tau ){|^2}{\rm d}\tau } \right|\leqslant\qquad\qquad\qquad\qquad\\ & \qquad|\omega (s){|^2} + 2{\left( {\int_0^T | {\omega _t}{|^2}{\rm d}t} \right)^{1/2}}{\left( {\int_0^T | \omega {|^2}{\rm d}t} \right)^{1/2}}\\[-15pt] \end{split}$ (14)

 $\begin{split}&\quad\int_{ - \infty }^\infty {{{\left( {\int_0^T | \omega {|^6}{\rm d}t} \right)}^{1/2}}} {\rm d}x\leqslant\\ &\qquad C{{\left( \int_{-\infty }^{\infty }{|}\omega (s){{|}^{4}}{\rm d}x \right)}^{1/2}}{{\left( \int_{-\infty }^{\infty }{\int_{0}^{T}{|}}\omega {{|}^{2}}{\rm d}t{\rm d}x \right)}^{1/2}}+\\ &\qquad CT{{\left( \int_{-\infty }^{\infty }{\int_{0}^{T}{|}}{{\omega }_{t}}|{\rm d}t{\rm d}x \right)}^{1/2}}\|\omega \|_{L_{T}^{\infty }({{L}^{2}})}^{2}\leqslant\\ &\qquad C{{T}^{1/2}}\underset{-\infty $\begin{split}&\quad\int_{ - \infty }^\infty {{{\left( {\int_0^T | \omega {|^{10}}{\rm d}t} \right)}^{1/2}}} {\rm d}x\leqslant \\ &\qquad C{{T}^{1/2}}|\omega (s){{|}^{3}}\|\omega (s){{\|}_{{{L}^{2}}}}\|\omega {{\|}_{X_{3}^{T}}}+\\ &\qquad CT\underset{-\infty \!<\!x\!<\!\infty }{\mathop{\sup }}\,\int_{0}^{T}{|}{{\omega }_{t}}{{|}^{2}}{\rm d}t\int_{-\infty }^{\infty }{\underset{0\!\leqslant\! t\!\leqslant\! T}{\mathop{\sup }}\,}|\omega {{|}^{2}}{{\left( \int_{0}^{T}{|}\omega {{|}^{2}}{\rm d}t \right)}^{1/2}}{\rm d}x \!\leqslant \\ &\qquad C{{T}^{1/2}}|\omega (s){{|}^{3}}\|\omega (s){{\|}_{{{L}^{2}}}}\|\omega {{\|}_{X_{3}^{T}}}+\\ &\qquad CT\underset{-\infty

 $\begin{split}&\quad\int_{ - \infty }^\infty {{{\left( {\int_0^T | {F}(\omega (t,x)){|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x\leqslant \\ &\qquad\int_{ - \infty }^\infty \left( \int_0^T | \left[ {\int_0^t {(|\omega (} s,x){|^2}{)_x}{\rm d}s} \right]\omega +{v_0}\omega + \right.\\ &\left.\qquad\gamma |\omega {|^2}\omega + \delta |\omega {|^4}\omega |^2 {\rm d}t \right)^{1/2} {\rm d}x \leqslant \\ &\qquad CT\underset{-\infty 结合引理1, 引理2 和式（17），可得 $\begin{split}&\|{\rm D}_{x}^{1/2}\varPhi (\omega )(t,x){{\|}_{{{L}^{2}}}}\leqslant C\|{{u}_{0}}{{\|}_{{{H}^{1/2}}}}+\\ &\qquad C(T)\|\omega \|_{X_{3}^{T}}^{3}+C(T)\|\omega \|_{X_{3}^{T}}^{5}+C(T)\|\omega {{\|}_{X_{3}^{T}}}\end{split}$(18) 由引理3和式（17），可以估计式（11）的第2项为 $\begin{split}&\quad\underset{-\infty

 $\begin{split}&\quad\int_{-\infty }^{\infty }{{{\left[ \int_{0}^{T}{{{\left| {{\rm D}_{x}}\left( |\omega (t,x){{|}^{2}}\omega (t,x) \right) \right|}^{2}}}{\rm d}t \right]}^{1/2}}}{\rm d}x\leqslant\qquad\qquad\qquad\\ &\qquad C{{T}^{1/2}}\|\omega {{\|}_{X_{3}^{T}}}+CT\|\omega \|_{X_{3}^{T}}^{3}\\[-15pt]\end{split}$ (20)
 $\begin{split}&\quad\int_{-\infty }^{\infty }{{{\left[ \int_{0}^{T}{{{\left| {{\rm D}_{x}}\left( |\omega (t,x){{|}^{4}}\omega (t,x) \right) \right|}^{2}}}{\rm d}t \right]}^{1/2}}}{\rm d}x\leqslant \qquad\qquad\qquad\\ &\qquad C{{T}^{1/2}}\|\omega {{\|}_{X_{3}^{T}}}+C{{T}^{3/2}}\|\omega \|_{X_{3}^{T}}^{3}+C{{T}^{2}}\|\omega \|_{X_{3}^{T}}^{5}\\[-15pt]\end{split}$ (21)

 $\begin{split}&\quad\int_{ - \infty }^\infty {{{\left( {\int_0^T | {{\rm D}_x}F(\omega (t,x)){|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x \leqslant \\ &\qquad C\left( T+{{T}^{3/2}} \right)\|\omega {{\|}_{X_{3}^{T}}}+C{{T}^{1/2}}\|{{v}_{0}}{{\|}_{{{H}^{1}}}}\|\omega {{\|}_{X_{3}^{T}}}+\\ &\qquad C{{T}^{1/2}}\|\omega {{\|}_{X_{3}^{T}}}+C(T+{{T}^{3/2}})\|\omega \|_{X_{3}^{T}}^{3}+CT\|\omega \|_{X_{3}^{T}}^{5}\leqslant\qquad\qquad\qquad\\ & \qquad C(T)\left( \|\omega {{\|}_{X_{3}^{T}}}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega \|_{X_{3}^{T}}^{5} \right)\\[-13pt] \end{split}$ (22)

 $\begin{split}&\quad\|{\rm D}_{x}^{1/2}{{\rm D}_{x}}\varPhi (\omega )(t,x){{\|}_{{{L}^{2}}}}\leqslant\\ & \qquad C\|{{\rm D}_{x}}{\rm D}_{x}^{1/2}{{u}_{0}}{{\|}_{{{L}^{2}}}}+{{\left\| {\rm D}_{x}^{1/2}\int_{0}^{t}{U}(t-s){{\rm D}_{x}}{F}(\omega (s)){\rm d}s \right\|}_{{{L}^{2}}}}\leqslant\\ & \qquad C\|{{u}_{0}}{{\|}_{{{H}^{3/2}}}}+C\int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{{{\left| {{\rm D}_{x}}{F}(\omega (t,x)) \right|}^{2}}}{\rm d}t \right)}^{1/2}}}{\rm d}x\leqslant\\ & \qquad C\|{{u}_{0}}{{\|}_{{{H}^{3/2}}}}+C(T)\left( \|\omega {{\|}_{X_{3}^{T}}}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega \|_{X_{3}^{T}}^{5} \right)\\[-13pt] \end{split}$ (23)

 $\begin{split} & \quad\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T {{{\left| {{\rm D}_x^2\varPhi (\omega )(x,t)} \right|}^2}} {\rm d}t} \right)^{1/2}}\leqslant\\ & \qquad C\|{{u}_{0}}{{\|}_{{{H}^{3/2}}}}+C\int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{{{\left| {{\rm D}_{x}}F(\omega (t,x)) \right|}^{2}}}{\rm d}t \right)}^{1/2}}}{\rm d}x\leqslant\qquad\qquad\qquad\\ & \qquad C\|{{u}_{0}}{{\|}_{{{H}^{3/2}}}}+C(T)\left( \|\omega {{\|}_{X_{3}^{T}}}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega \|_{X_{3}^{T}}^{5} \right)\\[-15pt] \end{split}$ (24)

 ${\rm i}\;{{\rm D}_t}U(t)f + {{\rm D}_{xx}}U(t)f = 0$

 ${{\rm D}_t}U(t)f = {\rm i}\;{{\rm D}_{xx}}U(t)f$

 $\begin{split}&\quad{{\rm D}_t}\varPhi (\omega )(t) = {\rm {i\;D}}_x^2U(t){u_0} +\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\ &\qquad\int_0^t {{\rm D}_x^2} U(t \!-\! s){F}\left( {\omega (s)} \right){\rm d}s \!-\! {\rm {i\;}}F\left( {\omega (t)} \right)\\[-15pt]\end{split}$ (25)

 $\begin{split} &\quad\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {{\rm D}_t}\varPhi (\omega )(x,t){|^2}{\rm d}t} \right)^{1/2}}\leqslant\\[-2pt] & \qquad\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T {{{\left| {{\rm D}_x^{1/2}U(t){\rm D}_x^{3/2}{u_0}(x)} \right|}^2}} {\rm d}t} \right)^{1/2}} +\\[-2pt] &\qquad C\int_{ - \infty }^\infty {{{\left( {\int_0^T | {{\rm D}_x}{F}(\omega (t,x)){|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x+\\[-2pt] &\qquad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {F}(\omega (t)){|^2}{\rm d}t} \right)^{1/2}}\leqslant\\[-2pt] &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{3/2}}}}+C(T)\left( \|\omega {{\|}_{X_{3}^{T}}}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega \|_{X_{3}^{T}}^{5} \right)+\qquad\qquad\qquad\qquad\\[-2pt] &\qquad\underset{-\infty 对式（26）右边最后一项的估计为 $\begin{split} &\!\!\!\!\!\! \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {F}(\omega (t)){|^2}{\rm d}t} \right)^{1/2}}\leqslant\\[-3pt] &\!\!\!\!\!\!\quad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T {{{\left| {\int_0^t | \omega (s,x)||{\omega _x}(s,x)|{\rm d}s\omega (t,x)} \right|}^2}} {\rm d}t} \right)^{1/2}} +\\[-3pt] &\!\!\!\!\!\!\quad \mathop {\sup }\limits_{ - \infty \!< \!x \!< \!\infty } {\left( {\int_0^T | \omega (t,x){|^6}{\rm d}t} \right)^{1/2}}\!\!\!\!+\!\!\!\!\mathop {\sup }\limits_{ - \infty \! <\! x \!<\! \infty } {\left( {\int_0^T | \omega (t,x){|^{10}}{\rm d}t} \right)^{1/2}} \!\!+\\[-3pt] &\!\!\!\!\!\!\quad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {v_0}(x)\omega (t,x){|^2}{\rm d}t} \right)^{1/2}} \leqslant \\[-3pt] &\!\!\!\!\!\!\quad C{{\left[ \underset{-\infty

 $\begin{split}& \quad\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {{\rm D}_t}\varPhi (\omega )(x,t){|^2}{\rm d}t} \right)^{1/2}}\leqslant\\ &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{3/2}}}}+C(T)\left( \|\omega \|_{X_{3}^{T}}^{5}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega {{\|}_{X_{3}^{T}}} \right)\qquad\qquad \\[-15pt]\end{split}$ (28)

 $\begin{split} &\quad\int_{ - \infty }^\infty {{{\left( {\int_0^T | {\rm D}_x^2F(\omega (t,x)){|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x\leqslant\\[-3pt] & \qquad C(T)\|\omega \|_{X_{3}^{T}}^{3}+C{{T}^{1/2}}\|{{v}_{0}}{{\|}_{{{H}^{2}}}}\|\omega {{\|}_{X_{3}^{T}}}+\qquad\qquad\qquad\qquad\\[-3pt] &\qquad C\int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{{{\left| {\rm D}_{x}^{2}\left( |\omega {{|}^{2}}\omega \right) \right|}^{2}}}{\rm d}t \right)}^{1/2}}}{\rm d}x+\\[-3pt] & \qquad C\int_{ - \infty }^\infty {{{\left( {\int_0^T {{{\left| {{\rm D}_x^2\left( {|\omega {|^4}\omega } \right)} \right|}^2}} {\rm d}t} \right)}^{1/2}}} {\rm d}x\\[-15pt] \end{split}$ (29)

 $\begin{split} & \quad\int_{ - \infty }^\infty {{{\left( {\int_0^T {{{\left| {{\rm D}_x^2\left( {|\omega {|^2}\omega } \right)} \right|}^2}} {\rm d}t} \right)}^{1/2}}} {\rm d}x\leqslant\\ & \qquad \int_{ - \infty }^\infty {{{\left( {\int_0^T {{{\left| {6|\omega ||{\omega _x}{|^2} + 3|\omega {|^2}|{\omega _{xx}}|} \right|}^2}} {\rm d}t} \right)}^{1/2}}} {\rm d}x\leqslant\qquad\qquad\qquad\\ & \qquad C\int_{ - \infty }^\infty {\mathop {\sup }\limits_{0 \leqslant t \leqslant T} } |{\omega _x}{|^2}{\left( {\int_0^T | \omega {|^2}{\rm d}t} \right)^{1/2}}{\rm d}x + \\ &\qquad C\int_{ - \infty }^\infty {\mathop {\sup }\limits_{0 \leqslant t \leqslant T} } |\omega {|^2}{\left( {\int_0^T | {\omega _{xx}}{|^2}{\rm d}t} \right)^{1/2}}{\rm d}x \leqslant\\ &\qquad C{{T}^{1/2}}\|\omega \|_{X_{3}^{T}}^{3}\\[-13pt] \end{split}$ (30)

 $\begin{split} &\quad\int_{ - \infty }^\infty {{{\left( {\int_0^T {{{\left| {{\rm D}_x^2\left( {|\omega {|^4}\omega } \right)} \right|}^2}} {\rm d}t} \right)}^{1/2}}} {\rm d}x \leqslant\\ & \qquad C\int_{ - \infty }^\infty \!\! {{{\left( {\int_0^T | \omega {|^6}|{\omega _x}{|^4}{\rm d}t} \right)}^{1/2}}} \!{\rm d}x +\\ &\qquad C\int_{ - \infty }^\infty {{{\left( {\int_0^T | \omega {|^8}|{\omega _{xx}}{|^2}{\rm d}t} \right)}^{1/2}}} {\rm d}x\! \leqslant\\ & \qquad C{{T}^{1/2}}\int_{-\infty }^{\infty }{\underset{0\leqslant t\leqslant T}{\mathop{\sup }}\,}|\omega {{|}^{2}}{{\left( \int_{0}^{T}{|}\omega {{|}^{2}}{\rm d}t \right)}^{1/2}}{\rm d}x\|\omega \|_{X_{3}^{T}}^{2}+\\ &\qquad \int_{-\infty }^{\infty }{\underset{0\leqslant t\leqslant T}{\mathop{\sup }}\,}|\omega {{|}^{4}}{{\left( \int_{0}^{T}{|}{{\omega }_{xx}}{{|}^{2}}{\rm d}t \right)}^{1/2}}{\rm d}x \leqslant C{{T}^{1/2}}\|\omega {{\|}_{X_{3}^{T}}}+\qquad\qquad\\ &\qquad C(T+{{T}^{3/2}})\|\omega \|_{X_{3}^{T}}^{3}+ C({{T}^{3/2}}+{{T}^{2}})\|\omega \|_{X_{3}^{T}}^{5}\\[-13pt]\end{split}$

 $\begin{split}&\quad \int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{|}{\rm D}_{x}^{2}F(\omega (t,x)){{|}^{2}}{\rm d}t \right)}^{1/2}}}{\rm d}x\leqslant \qquad\qquad\qquad\qquad\qquad\\[-2pt] &\qquad C(T)\left( \|\omega {{\|}_{X_{3}^{T}}}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega \|_{X_{3}^{T}}^{5} \right)\\[-15pt]\end{split}$ (32)

 $\begin{split} &\quad\|{\rm D}_{x}^{1/2}{\rm D}_{x}^{2}\varPhi (\omega )(t,x){{\|}_{{{L}^{2}}}}\leqslant \\[-2pt] &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+C\int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{{{\left|{\rm D}_{x}^{2}{F}(\omega (t,x)) \right|}^{2}}}{\rm d}t \right)}^{1/2}}}{\rm d}x\leqslant \qquad\quad\\[-2pt] &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+C(T)\left( \|\omega {{\|}_{X_{3}^{T}}}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega \|_{X_{3}^{T}}^{5} \right)\\[-15pt]\end{split}$ (33)

 $\begin{split} &\quad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {\rm D}_x^3\varPhi (\omega )(x,t){|^2}{\rm d}t} \right)^2}\leqslant \\[-2pt] &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+C\int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{|}{\rm D}_{x}^{2}{F}(\omega (t,x)){{|}^{2}}{\rm d}t \right)}^{1/2}}}{\rm d}x\leqslant\qquad\quad\\[-2pt] & \qquad C\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+C(T)\left( \|\omega \|_{X_{3}^{T}}^{5}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega {{\|}_{X_{3}^{T}}} \right)\\[-15pt] \end{split}$ (34)

 $\begin{split} & \quad \mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {{\rm D}_x}{{\rm D}_t}\varPhi (\omega )(x,t){|^2}{\rm d}t} \right)^{1/2}}\leqslant\\[-2pt] &\qquad \mathop {\sup }\limits_{ - \infty < x < \infty } \left( \int_0^T \left| {{\rm D}_x}[ {\rm {iD}}_x^2U(t){u_0} + \int_0^t {{\rm D}_x^2} U(t - s){\text{·}}\right.\right.\qquad\qquad\qquad\\[-2pt] &\qquad \left.\left. F(\omega (s)){\rm d}s -{\rm {i}}F(\omega (t)) ] \right|^2 {\rm d}t \right)^{1/2} \leqslant\\[-2pt] &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+C\int_{-\infty }^{\infty }{{{\left( \int_{0}^{T}{|}{\rm D}_{x}^{2}{F}(\omega ){{|}^{2}}{\rm d}t \right)}^{1/2}}}{\rm d}x+\\[-2pt] &\qquad\underset{-\infty 由方程（27）, 可得对式（35）的第3项的估计为 $\begin{split} &\quad\underset{-\infty

 $\begin{split}&\quad\mathop {\sup }\limits_{ - \infty < x < \infty } {\left( {\int_0^T | {{\rm D}_x}{{\rm D}_t}\varPhi (\omega )(x,t){|^2}{\rm d}t} \right)^{1/2}}\leqslant\\ &\qquad C\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+C(T)\left( \|\omega \|_{X_{3}^{T}}^{5}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega {{\|}_{X_{3}^{T}}} \right)\qquad\qquad\qquad\\[-15pt]\end{split}$ (36)

 $\begin{split} &\quad\|\varPhi (\omega ){{\|}_{X_{3}^{T}}}\leqslant {{C}_{7}}\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+\\ &\qquad{{C}_{8}}(T)\left( \|\omega \|_{X_{3}^{T}}^{5}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega {{\|}_{X_{3}^{T}}} \right)\qquad\qquad\qquad\qquad\\[-15pt]\end{split}$ (37)

 ${{C}_{7}}\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}\leqslant \frac{r}{2}$

 ${{C}_{8}}(T)\left( \|\omega \|_{X_{3}^{T}}^{5}+\|\omega \|_{X_{3}^{T}}^{3}+\|\omega {{\|}_{X_{3}^{T}}} \right)<\frac{r}{2}$

 $\varPhi :B_r^3 \to B_r^3$

$\varPhi$ 是在 $B_r^3$ 上的一个压缩。采用与估计不等式（37）同样的方法，可得

 $\begin{split} &\quad\|\varPhi ({{\omega }_{1}})-\varPhi ({{\omega }_{2}}){{\|}_{X_{3}^{T}}}\leqslant{{C}_{9}}(T)\left( \|{{\omega }_{1}}\|_{X_{3}^{T}}^{4}+\|{{\omega }_{2}}\|_{X_{3}^{T}}^{4}+\right.\qquad\qquad\qquad\quad\\ &\qquad\left.\|{{\omega }_{1}}\|_{X_{3}^{T}}^{2}+\|{{\omega }_{2}}\|_{X_{3}^{T}}^{2}+1 \right)\|{{\omega }_{1}}-{{\omega }_{2}}{{\|}_{X_{3}^{T}}}\end{split}$

$T$ 足够小，满足

 ${C_9}(T)(2{r^4} + 2{r^2} + 1) < 1$

3 全局解的存在性

 $\qquad\qquad\quad\int_{ - \infty }^\infty | u(t,x){|^2}{\rm d}x = \int_{ - \infty }^\infty | {u_0}(x){|^2}{\rm d}x$ (38)

 $\begin{split}&\quad{\rm i}\langle {u_t},u\rangle + \langle {u_{xx}},u\rangle = \langle vu,u\rangle + \langle \gamma |u{|^2}u,u\rangle +\qquad\qquad\qquad\qquad\qquad\\ &\qquad\langle \delta |u{|^4}u,u\rangle\\[-10pt]\end{split}$ (39)

 $\langle {{u}_{t}},u\rangle =\frac{1}{2}\frac{{\rm d}}{{\rm d}t}\|u\|_{{{L}^{2}}}^{2}=0$ (40)

 $\int_{ - \infty }^\infty | u(t,x){|^2}{\rm d}x = \int_{ - \infty }^\infty | {u_0}(x){|^2}{\rm d}x$

 $\begin{split}&\quad\int_{ - \infty }^\infty {\left( {v(t,x)|u(t,x){|^2} + |{u_x}(t,x){|^2} + \dfrac{\gamma }{2}|u(t,x){|^4} + \dfrac{\delta }{3}|u(t,x){|^6}} \right)} {\rm d}x=\\ &\qquad \int_{ - \infty }^\infty {\left( {{v_0}(t,x)|{u_0}(t,x){|^2} +|{u_{{0_x}}}(t,x){|^2} + \dfrac{\gamma }{2}|{u_0}(t,x){|^4} + \dfrac{\delta }{3}|{u_0}(t,x){|^6}} \right)} {\rm d}x\end{split}$ (41)
 $\int_{ - \infty }^\infty {\left( {{v^2}(t,x) + 2{\rm {Im}}\left( {u(t,x)\overline {{u_x}(t,x)} } \right)} \right)} {\rm d}x= \int_{ - \infty }^\infty {\left( {v_0^2(t,x) + 2{\rm{Im}}\left( {{u_0}(t,x)\overline {{u_{{0_x}}}(t,x)} } \right)} \right)} {\rm d}x$ (42)

 $\frac{{{\rm d}\vec S}}{{{\rm d}t}} = JE'(\vec S)$

 $J = \left( {\begin{array}{*{20}{c}} { - {\rm i}}&{} \\ {}&{2\dfrac{\partial }{{\partial x}}} \end{array}} \right)$
 $E'(\vec S) = \left( {\begin{array}{*{20}{c}} { - {u_{xx}} + uv + \gamma |u{|^2}u + \delta |u{|^4}u} \\ {\dfrac{1}{2}|u{|^2}} \end{array}} \right)$

 $\begin{split} &\quad E(\vec S) = \frac{1}{2}\int_{ - \infty }^\infty \left( v(t,x)|u(t,x){|^2} + |{u_x}(t,x){|^2} + \right.\qquad\qquad\qquad\qquad\qquad\qquad\\ &\qquad\left.\frac{\gamma }{2}|u(t,x){|^4} + \frac{\delta }{3}|u(t,x){|^6} \right) {\rm d}x\end{split}$

$u = {u_1} + {\rm i}{u_2}$ ，所以，有

 $\begin{split}&\quad\frac{{{\rm d}E(\vec S)}}{{{\rm d}t}} = {\rm {Re}}\int_{\mathbb{R}} {\frac{1}{2}} |u{|^2}{\left( {|u{|^2}} \right)_x}{\rm d}x=\\ & \qquad\int_{\mathbb{R}} {\left( {u_1^2{u_2}{u_{2x}} + {u_1}{u_{1x}}u_2^2} \right)} {\rm d}x {\rm{ = 0}}\qquad\qquad\qquad\qquad\qquad\\[-15pt]\end{split}$ (43)

 $\|u{{\|}_{L_{T}^{\infty }({{L}^{2}})}}\leqslant \|{{u}_{0}}{{\|}_{{{L}^{2}}}}$ (44)

 $\|u{{\|}_{L_{T}^{\infty }({{H}^{1}})}}\leqslant {{M}_{1}}$ (45)

 $\begin{split} &\quad\int_{ - \infty }^\infty | {u_x}{|^2}{\rm d}x \leqslant {\left( {\int_{ - \infty }^\infty | v{|^2}{\rm d}x} \right)^{1/2}}{\left( {\int_{ - \infty }^\infty | u{|^4}{\rm d}x} \right)^{1/2}} +\\ &\qquad \frac{{|\gamma |}}{2}\int_{ - \infty }^\infty | u{|^4}{\rm d}x + \frac{{|\delta |}}{3}\int_{ - \infty }^\infty | u{|^6}{\rm d}x + C\leqslant\\ & \qquad C{\left( {\int_{ - \infty }^\infty | v{|^2}{\rm d}x} \right)^{1/2}}{\left( {\int_{ - \infty }^\infty | {u_x}{|^2}{\rm d}x} \right)^{1/4}}{\left( {\int_{ - \infty }^\infty | {u_0}{|^2}{\rm d}x} \right)^{3/4}} +\qquad\qquad\\ &\qquad C{\left( {\int_{ - \infty }^\infty | {u_x}{|^2}{\rm d}x} \right)^{1/2}}{\left( {\int_{ - \infty }^\infty | {u_0}{|^2}{\rm d}x} \right)^{3/2}} +\\ & \qquad C\frac{{|\delta |}}{3}\left( {\int_{ - \infty }^\infty | {u_x}{|^2}{\rm d}x} \right){\left( {\int_{ - \infty }^\infty | {u_0}{|^2}{\rm d}x} \right)^2} +C \\ & \quad \int_{ - \infty }^\infty | v{|^2}{\rm d}x \leqslant2{\left( {\int_{ - \infty }^\infty | {u_0}{|^2}{\rm d}x} \right)^{1/2}}{\left( {\int_{ - \infty }^\infty | {u_x}{|^2}{\rm d}x} \right)^{1/2}} + C\qquad \end{split}$

 $\begin{split} &\quad \int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x\leqslant C{{\left( \int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x \right)}^{1/2}}+\\ & \qquad C{{\left( \int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x \right)}^{1/4}}\!+\!C\frac{|\delta |}{3}\|{{u}_{0}}\|_{{{L}^{2}}}^{4}\int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x\!+\!C\qquad\qquad\\[-15pt]\end{split}$ (46)

 $\begin{split} &\quad\left(1-C\frac{|\delta |}{3}\|{{u}_{0}}\|_{{{L}^{2}}}^{4}\right)\int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x\leqslant\\ & \qquad C{{\left( \int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x \right)}^{1/2}}+C{{\left( \int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x \right)}^{1/4}}+C\leqslant \qquad\qquad\qquad\qquad\qquad\\ &\qquad \frac{1-C\dfrac{|\delta |}{3}\|{{u}_{0}}\|_{{{L}^{2}}}^{4}}{2}\int_{-\infty }^{\infty }{|}{{u}_{x}}{{|}^{2}}{\rm d}x+C\end{split}$

 $\|{{u}_{t}}{{\|}_{L_{T}^{\infty }({{L}^{2}})}}\leqslant {{M}_{2}}$ (47)

 $\|u{{\|}_{L_{T}^{\infty }({{H}^{2}})}}\leqslant {{M}_{2}}$ (48)

 $\begin{split} &\quad{\rm i}{\psi _t} + {\psi _{xx}} = {(|u(x,s){|^2})_x}u + \left[ {\int_0^t {(|u(} x,s){|^2}{)_x}{\rm d}s} \right]\psi +\qquad\qquad\\ & \qquad{v_0}(x)\psi + 2\gamma |u{|^2}\psi +\gamma {u^2}\bar \psi + 2\delta |u{|^2}{u^2}\bar \psi + 3\delta |u{|^4}\psi\quad\\[-10pt] \end{split}$ (49)

 $\begin{split} &\quad{\rm i}\langle {\psi _t},\psi \rangle + \langle {\psi _{xx}},\psi \rangle = \langle {(|u(x,s){|^2})_x}u,\psi \rangle +\\ &\qquad\left\langle {\left[ {\int_0^t {(|u(} x,s){|^2}{)_x}{\rm d}s} \right]\psi ,\psi } \right\rangle \!+\! \langle {v_0}(x)\psi ,\psi \rangle \!+\! \langle 2\gamma |u{|^2}\psi ,\psi \rangle +\quad\\ &\qquad \langle \gamma {u^2}\bar \psi ,\psi \rangle + \langle 2\delta |u{|^2}{u^2}\bar \psi ,\psi \rangle + \langle 3\delta |u{|^4}\psi ,\psi \rangle\\[-10pt] \end{split}$ (50)

 $\begin{split} &\quad 2\langle {{\psi }_{t}},\psi \rangle =\frac{{\rm d}}{{\rm d}t}\|\psi \|_{{{L}^{2}}}^{2}= 2{\rm {Im}}\langle {(|u(x,s){|^2})_x}u,\psi \rangle +\\ & \qquad 2{\rm {Im}}\langle \gamma {u^2}\bar \psi ,\psi \rangle + 2{\rm {Im}}\langle 2\delta |u{|^2}u\bar \psi ,\psi \rangle \leqslant\\ &\qquad C\|u\|_{L_{T}^{\infty }({{H}^{1}})}^{3}\|\psi {{\|}_{{{L}^{2}}}}\!+\!C\|u\|_{L_{T}^{\infty }({{L}^{2}})}^{2}\|\psi \|_{{{L}^{2}}}^{2}\!+\!C\|u\|_{L_{T}^{\infty }({{L}^{2}})}^{4}\|\psi \|_{{{L}^{2}}}^{2}\quad \end{split}$

 $\|\psi {{\|}_{L_{T}^{\infty }({{L}^{2}})}}\leqslant {{M}_{2}}$

 $\begin{split}&\quad\left\| \left| u \right| \right\|_{p}^{T}=\underset{-\infty 参照对方程（11）等式右边各项的估计，可以对$\left\| {\left| u \right|} \right\|_p^T(p \leqslant 3)$作出估计，这里只估计$\left\| {\left| u \right|} \right\|_3^T$。用与式（19），（23），（24），（33），（34）相同的估计方法，可得 $\begin{split} &\quad\left\| \left| u \right| \right\|_{3}^{T}=\underset{-\infty

 $\begin{split}&\quad(1-{{C}_{11}}TM_{1}^{2})\||u|\|_{3}^{T}\leqslant {{C}_{10}}\|{{u}_{0}}{{\|}_{{{H}^{5/2}}}}+\\ &\qquad{{C}_{12}}(T)\|{{v}_{0}}{{\|}_{{{H}^{2}}}}{{M}_{2}}+{{C}_{13}}(T)M_{2}^{3}+{{C}_{14}}(T)M_{2}^{5}\qquad\qquad\qquad\qquad\qquad\qquad\end{split}$

$\hat T > 0$ ，使得

 $\hat T = {\rm {min}}\;\left\{ {\frac{1}{{2{C_{11}}M_1^2}},{T_3}} \right\}$

 $\left\| {\left| u \right|} \right\|_3^{\hat T} \leqslant {K_1}$ (52)

 $\begin{split}&\quad {\rm i}u_t^{(1)} + u_{xx}^{(1)} = \left[ {\int_0^T {{{\left( {{{\left| {{u^{(1)}}(x,s)} \right|}^2}} \right)}_x}} {\rm d}s} \right]{u^{(1)}} + {v_0}{u^{(1)}} +\qquad\qquad\qquad\qquad\qquad\qquad\\ &\qquad\gamma {\left| {{u^{(1)}}} \right|^2}{u^{(1)}} + \delta {\left| {{u^{(1)}}} \right|^4}{u^{(1)}}\end{split}$

 $\left\| {\left| u \right|} \right\|_3^{2\hat T} \leqslant K_1^{(1)}$ (53)

 $\quad\begin{split}u(t,x)& =\left\{ {\begin{array}{*{20}{l}} {u(t,x), 0 \leqslant t \leqslant \hat T} \\ {{u^{(1)}}(t,x), \hat T \leqslant t \leqslant 2\hat T} \end{array}} \right.\\ &\left\| {\left| u \right|} \right\|_3^{2\hat T} \leqslant {\rm {max}}\;\{ {K_1},K_1^{(1)}\}\end{split}$

$n \in {N^ + },n\hat T \geqslant {T_1}$ ，采用与估计式（52）和式（53）同样的方法，将区间 $\hat T$ 逐步延拓到 $n\hat T$ ，存在正常数 $K$ ，使得

 $\left\| {\left| u \right|} \right\|_3^{n\hat T} \leqslant K$

 $\left\{ {\begin{array}{*{20}{l}} {{\rm i}{u_t} + {u_{xx}} = uv + \gamma |u{|^2}u + \delta |u{|^{2m}}u} \\ {{v_t} = {{(|u{|^2})}_x}} \end{array}} \right.$ (54)

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