西南大学学报 (自然科学版)  2019, Vol. 41 Issue (2): 60-63.  DOI: 10.13718/j.cnki.xdzk.2019.02.009
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  • 非局部非线性Schrödinger方程组解的渐近行为    [PDF全文]
    魏娟, 朱朝生     
    西南大学 数学与统计学院, 重庆 400715
    摘要:研究了临界带有非局部非线性项的Schrödinger方程组解的渐近行为,通过对方程组解的衰减估计证明其渐近自由解的非存在性.
    关键词Schrödinger方程组    时间衰减估计    渐近自由解    

    本文我们研究如下带非局部项的非线性Schrödinger方程组:

    $ \left\{ \begin{gathered} i{\partial _t}{w_1} + \frac{1}{{2{m_1}}}\Delta {w_1} + {\gamma _1}{w_1}\int_{ - \infty }^x {{{\left| {{w_1}} \right|}^2}{\text{d}}s} = {\alpha _1}{\left| {{w_1}} \right|^2}{w_1} + {\beta _1}\overline {w_1^2} {w_2}\;\;\;\;\;\;\;\;\left( {t,x} \right) \in \mathbb{R} \times \mathbb{R} \hfill \\ i{\partial _t}{w_2} + \frac{1}{{2{m_2}}}\Delta {w_2} + {\gamma _2}{w_2}\int_{ - \infty }^x {{{\left| {{w_2}} \right|}^2}{\text{d}}s} = {\alpha _2}{\left| {{w_2}} \right|^2}{w_2} + {\beta _2}w_1^3\;\;\;\;\;\;\;\;\;\;\left( {t,x} \right) \in \mathbb{R} \times \mathbb{R} \hfill \\ {w_1}\left( {0,x} \right) = {\psi _1}\left( x \right),{w_2}\left( {0,x} \right) = {\psi _2}\left( x \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathbb{R} \hfill \\ \end{gathered} \right. $ (1)

    其中:mj为粒子的质量;βj是耗散参数;αjγjβj$\mathbb{C}$j=1,2;wj为未知复值函数.

    Schrödinger方程在非线性光学、激光、孤波的传播中有重要应用.文献[1]研究了方程组(1)在没有非局部项的情况下其渐近自由解的非存在性.文献[2]证明了Schrödinger方程在能量空间H1($\mathbb{R}$),L2($\mathbb{R}$)中Cauchy问题的适定性.文献[3]证明了一类非线性Schrödinger方程存在正解以及解的聚集性,同时给出了解的衰减性估计.文献[4]研究了带有非线性项|u|pu的高阶非线性Schrödinger方程的Cauchy问题.对于Schrödinger方程的其他相关研究,可以参考文献[5-7].本文的主要目的是在文献[1]的基础上进一步证明方程组(1)的渐近自由解的非存在性.

    首先定义f的Fourier变换如下:

    $ \begin{array}{*{20}{c}} {F\left[ f \right]\left( x \right) = \frac{1}{{\sqrt {2{\rm{ \pi}}} }}\int_\mathbb{R} {{{\text{e}}^{ - i\zeta \cdot x}}f\left( \zeta \right){\text{d}}\zeta } }&{x \in \mathbb{R}} \end{array} $

    ms$\mathbb{R}$,Sobolev空间Hm, s($\mathbb{R}$)满足

    $ {H^{m,s}}\left( \mathbb{R} \right) = \left\{ {f \in {L^2}\left( \mathbb{R} \right):{{\left\| {{{\left( {1 + {{\left| x \right|}^2}} \right)}^{\frac{1}{2}}}{{\left( {1 - \Delta } \right)}^{\frac{m}{2}}}f} \right\|}_{{L^2}}} < \infty } \right\} $

    C表示不同的正常数.由方程组(1)可以得到相应的自由Schrödinger方程组

    $ \left\{ \begin{array}{l} i{\partial _t}{u_1} + \frac{1}{{2{m_1}}}\Delta {u_1} + {\gamma _1}{u_1}\int_{ - \infty }^x {{{\left| {{u_1}} \right|}^2}{\rm{d}}s} = 0\\ i{\partial _t}{u_2} + \frac{1}{{2{m_2}}}\Delta {u_2} + {\gamma _2}{u_2}\int_{ - \infty }^x {{{\left| {{u_2}} \right|}^2}{\rm{d}}s} = 0 \end{array} \right. $ (2)

    其中uj(0,x)=ϕj(x)(j=1,2).若存在方程组(2)的L2-自由解(u1u2),使得

    $ \mathop {\lim }\limits_{t \to \pm \infty } \left( {{{\left\| {{w_1} - {u_1}} \right\|}_{{L^2}}} + {{\left\| {{w_2} - {u_2}} \right\|}_{{L^2}}}} \right) = 0 $

    则称方程组(1)的解(w1w2)是渐近自由的.

    将方程组(1)的各方程两边分别乘以${{\overline{w}}_{1}}$${{\overline{w}}_{2}}$,取虚部,在$\mathbb{R}$上积分可得

    $ \begin{gathered} \frac{{\text{d}}}{{{\text{d}}t}}\left( {\left\| {{w_1}} \right\|_{{L^2}}^2 + \left\| {{w_2}} \right\|_{{L^2}}^2} \right) = 2\operatorname{Im} \left( {{\alpha _1}\left\| {{w_1}} \right\|_{{L^4}}^4 + {\alpha _2}\left\| {{w_2}} \right\|_{{L^4}}^4} \right) + 2\operatorname{Im} \left( {{\beta _1}\int_\mathbb{R} {\overline {w_1^3} {w_2}{\text{d}}x} + {\beta _2}\int_\mathbb{R} {w_1^3\overline {{w_2}} {\text{d}}x} } \right) - \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\operatorname{Im} \left( {\int_\mathbb{R} {{\gamma _1}{{\left| {{w_1}} \right|}^2}} \int_{ - \infty }^x {{{\left| {{w_1}} \right|}^2}{\text{d}}s{\text{d}}x} + \int_\mathbb{R} {{\gamma _2}{{\left| {{w_2}} \right|}^2}} \int_{ - \infty }^x {{{\left| {{w_2}} \right|}^2}{\text{d}}s{\text{d}}x} } \right) \hfill \\ \end{gathered} $ (3)

    假设Im αj≤0,Im γj≥0,j=1,2且β1=${{\overline{\beta }}_{2}}$,则有$\frac{\text{d}}{\text{d}t}({{\left\| {{w}_{1}} \right\|}_{{{L}^{2}}}}^{2}+{{\left\| {{w}_{2}} \right\|}_{{{L}^{2}}}}^{2})\le 0$(见文献[8-9]).应用文献[10]中的方法,容易得到方程组(1)的Cauchy问题解的存在性和解的L-时间衰减估计,即下面的引理1:

    引理1  设Im αj≤0,Im γj≥0,3m1=m2ψj(x)∈H1,1($\mathbb{R}$),j=1,2,β1=${{\overline{\beta }}_{2}}$.假设对某个ε>0,有‖ψ1(x)‖H1,1+‖ψ2(x)‖H1,1ε,则存在方程组(1)的解w=(wj)j=1,2,使得wC([0, ∞];H1,1($\mathbb{R}$)),且${{\left\| w \right\|}_{{{L}^{\infty }}}}\le C{{\left(1+t \right)}^{-\frac{1}{2}}}$.

    引理2[11]  设(u1u2)为方程组(2)的光滑解.若ϕjL1($\mathbb{R}$)∩L2($\mathbb{R}$)(j=1,2),且2≤q≤∞,则:

    (ⅰ)存在正常数Cj,使得${{\left\| {{u}_{j}} \right\|}_{{{L}^{q}}}}\ge {{C}_{j}}{{t}^{-\left(\frac{1}{2}-\frac{1}{q} \right)}}(\forall t\ge {{T}_{0}})$

    (ⅱ)存在常数cj,使得${{\left\| {{u}_{j}} \right\|}_{{{L}^{q}}}}\le {{c}_{j}}{{t}^{-\left(\frac{1}{2}-\frac{1}{q} \right)}}\left(\forall t\ge 0 \right)$.

    本文的主要结果如下:

    定理1  设3m1=m2,Re αj>0,Im αj≤0,Im γj≥0,j=1,2且β1=${{\overline{\beta }}_{2}}$.若方程组(1)的解wC([0,∞);H1,1($\mathbb{R}$))满足衰减估计${{\left\| w \right\|}_{{{L}^{\infty }}}}\le C{{\left(1+t \right)}^{-\frac{1}{2}}}$,则不存在方程组(2)的自由解(u1u2),使得

    $ \begin{array}{*{20}{c}} {\phi = \left( {{\phi _1}\left( x \right),{\phi _2}\left( x \right)} \right) \ne \left( {0,0} \right)}&{\phi \in {H^{1,1}}\left( \mathbb{R} \right)} \end{array} $

    $ \mathop {\lim }\limits_{t \to \infty } \left( {{{\left\| {{w_1}\left( t \right) - {u_1}\left( t \right)} \right\|}_{{L^2}}} + {{\left\| {{w_2}\left( t \right) - {u_2}\left( t \right)} \right\|}_{{L^2}}}} \right) = 0 $

     假设存在方程组(2)的解(u1u2),使得

    $ \mathop {\lim }\limits_{t \to \infty } \left( {{{\left\| {{w_1} - {u_1}} \right\|}_{{L^2}}} + {{\left\| {{w_2} - {u_2}} \right\|}_{{L^2}}}} \right) = 0 $ (4)

    将方程组(1)的各方程两边分别乘以${{\overline{u}}_{1}}$${{\overline{u}}_{2}}$,取实部,在$\mathbb{R}$上积分可得:

    $ \begin{gathered} \operatorname{Im} \int_\mathbb{R} {{\partial _t}{{\overline w }_1} \cdot {u_1}{\text{d}}x} = - \frac{1}{{2{m_1}}}\operatorname{Re} \int_\mathbb{R} {\Delta {{\overline w }_1} \cdot {u_1}{\text{d}}x} - \operatorname{Re} \int_\mathbb{R} {{u_1}{{\bar \gamma }_1}{{\overline w }_1}} \overline {\int_{ - \infty }^x {{{\left| {{w_1}} \right|}^2}{\text{d}}s{\text{d}}x} } + \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\operatorname{Re} \int_\mathbb{R} {{\alpha _1}{{\left| {{w_1}} \right|}^2}{{\overline w }_1} \cdot {u_1}{\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {{{\bar \beta }_1}{{\left| {{w_1}} \right|}^2}{{\overline w }_2} \cdot {u_1}{\text{d}}x} \hfill \\ \end{gathered} $ (5)
    $ \begin{gathered} \operatorname{Im} \int_\mathbb{R} {{\partial _t}{{\overline w }_2} \cdot {u_2}{\text{d}}x} = - \frac{1}{{2{m_2}}}\operatorname{Re} \int_\mathbb{R} {\Delta {{\overline w }_2} \cdot {u_2}{\text{d}}x} - \operatorname{Re} \int_\mathbb{R} {{{\bar \gamma }_2}{u_2}{{\overline w }_2}} \overline {\int_{ - \infty }^x {{{\left| {{w_2}} \right|}^2}{\text{d}}s{\text{d}}x} } + \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\operatorname{Re} \int_\mathbb{R} {{{\bar \alpha }_2}{{\left| {{w_2}} \right|}^2}{{\overline w }_2} \cdot {u_2}{\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {{{\bar \beta }_2}\overline w _1^3 \cdot {u_2}{\text{d}}x} \hfill \\ \end{gathered} $ (6)

    将方程组(2)的各方程两边分别乘以${{\overline{w}}_{1}}$${{\overline{w}}_{2}}$,取实部,在$\mathbb{R}$上积分可得:

    $ \operatorname{Im} \int_\mathbb{R} {{\partial _t}{u_1} \cdot {{\overline w }_1}{\text{d}}x} = \frac{1}{{2{m_1}}}\operatorname{Re} \int_\mathbb{R} {\Delta {{\overline w }_1} \cdot {u_1}{\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {{\gamma _1}{u_1}{{\overline w }_1}} \int_{ - \infty }^x {{{\left| {{u_1}} \right|}^2}{\text{d}}s{\text{d}}x} $ (7)
    $ \operatorname{Im} \int_\mathbb{R} {{\partial _t}{u_2} \cdot {{\overline w }_2}{\text{d}}x} = \frac{1}{{2{m_2}}}\operatorname{Re} \int_\mathbb{R} {\Delta {{\overline w }_2} \cdot {u_2}{\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {{\gamma _2}{u_2}{{\overline w }_2}} \int_{ - \infty }^x {{{\left| {{u_2}} \right|}^2}{\text{d}}s{\text{d}}x} $ (8)

    由(5)-(8)式可得

    $ \begin{gathered} \operatorname{Im} \int_\mathbb{R} {\left( {{\partial _t}{{\overline w }_1} \cdot {u_1} + {\partial _t}{{\overline w }_2} \cdot {u_2} + {\partial _t}{u_1} \cdot {{\overline w }_1} + {\partial _t}{u_2} \cdot {{\overline w }_2}} \right){\text{d}}x} = \hfill \\ \operatorname{Re} \int_\mathbb{R} {\left( {{{\bar \alpha }_1}{{\left| {{w_1}} \right|}^2}{{\overline w }_1} \cdot {u_1} + {{\bar \beta }_1}{{\left| {{w_1}} \right|}^2}{{\overline w }_2} \cdot {u_1} + {{\bar \alpha }_2}{{\left| {{w_2}} \right|}^2}{{\overline w }_2} \cdot {u_2} + {{\bar \beta }_2}\overline w _1^3 \cdot {u_2}} \right){\text{d}}x} - \hfill \\ \operatorname{Re} \int_\mathbb{R} {{u_1}{{\bar \gamma }_1}{{\bar w}_1}} \int_{ - \infty }^x {{{\left| {{w_1}} \right|}^2}{\text{d}}s{\text{d}}x} - \operatorname{Re} \int_\mathbb{R} {{{\bar \gamma }_2}{u_2}{{\bar w}_2}} \int_{ - \infty }^x {{{\left| {{w_2}} \right|}^2}{\text{d}}s{\text{d}}x} + \hfill \\ \operatorname{Re} \int_\mathbb{R} {{\gamma _1}{u_1}{{\bar w}_1}} \int_{ - \infty }^x {{{\left| {{u_1}} \right|}^2}{\text{d}}s{\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {{\gamma _2}{u_2}{{\bar w}_2}} \int_{ - \infty }^x {{{\left| {{u_2}} \right|}^2}{\text{d}}s{\text{d}}x} = \hfill \\ \operatorname{Re} \int_\mathbb{R} {\left( {{{\bar \alpha }_1}{{\left| {{u_1}} \right|}^4} + {{\bar \alpha }_2}{{\left| {{u_2}} \right|}^4}} \right){\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {\left( {{{\bar \alpha }_1}{{\left| {{w_1}} \right|}^2}{{\bar w}_1} \cdot {u_1} - {{\bar \alpha }_1}{{\left| {{u_1}} \right|}^4}} \right){\text{d}}x} + \hfill \\ \operatorname{Re} \int_\mathbb{R} {\left( {{{\bar \alpha }_2}{{\left| {{w_2}} \right|}^2}{{\bar w}_2} \cdot {u_2} - {{\bar \alpha }_2}{{\left| {{u_2}} \right|}^4}} \right){\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {\left( {{{\bar \beta }_1}{{\left| {{w_1}} \right|}^2}{{\bar w}_2} \cdot {u_1} + {{\bar \beta }_2}\bar w_1^3 \cdot {u_2}} \right){\text{d}}x} - \hfill \\ \operatorname{Re} \int_\mathbb{R} {\left( {{u_1}{{\bar \gamma }_1}{{\bar w}_1}\int_{ - \infty }^x {{{\left| {{w_1}} \right|}^2}{\text{d}}s} } \right){\text{d}}x} - \operatorname{Re} \int_\mathbb{R} {\left( {{{\bar \gamma }_2}{u_2}{{\bar w}_2}\int_{ - \infty }^x {{{\left| {{w_2}} \right|}^2}{\text{d}}s} } \right){\text{d}}x} + \hfill \\ \operatorname{Re} \int_\mathbb{R} {\left( {{\gamma _1}{u_1}{{\bar w}_1}\int_{ - \infty }^x {{{\left| {{u_1}} \right|}^2}{\text{d}}s} } \right){\text{d}}x} + \operatorname{Re} \int_\mathbb{R} {\left( {{\gamma _2}{u_2}{{\bar w}_2}\int_{ - \infty }^x {{{\left| {{u_2}} \right|}^2}{\text{d}}s} } \right){\text{d}}x} = \hfill \\ {\mathit{\Phi }_1} + {\mathit{\Phi }_2} + {\mathit{\Phi }_3} + {\mathit{\Phi }_4} + {\mathit{\Phi }_5} + {\mathit{\Phi }_6} + {\mathit{\Phi }_7} + {\mathit{\Phi }_8} \hfill \\ \end{gathered} $ (9)

    下面估计方程(9)右边的各项.首先由Re αj>0(j=1,2)及引理2可知Φ1Ct-1.其次考虑Φ2.因为

    $ \begin{gathered} {\mathit{\Phi }_2} \leqslant \int_\mathbb{R} {\left( {\left| {{\alpha _1}} \right|\left| {{u_1}} \right|{{\left| {{w_1}} \right|}^3} - \left| {{\alpha _1}} \right|{{\left| {{u_1}} \right|}^4}} \right){\text{d}}x} \leqslant \int_\mathbb{R} {\left( {\left| {{\alpha _1}} \right|\left| {{u_1}} \right|{{\left| {{w_1}} \right|}^3}} \right){\text{d}}x} \leqslant \hfill \\ \;\;\;\;\;\;\;\;{\left\| {{u_1}} \right\|_{{L^q}}} \cdot \left| {{\alpha _1}} \right|\left\| {{w_1}} \right\|_{{L^{\frac{{3q}}{{q - 1}}}}}^3 \leqslant \frac{1}{2}\left\| {{u_1}} \right\|_{{L^q}}^2 + \frac{1}{2}{\left| {{\alpha _1}} \right|^2}\left\| {{w_1}} \right\|_{{L^{\frac{{3q}}{{q - 1}}}}}^6\;\;\;\;\;\;\;q \geqslant 2 \hfill \\ \end{gathered} $ (10)

    下面给出(10)式右边两项的估计.由不等式${{\left\| {{u}_{j}} \right\|}_{{{L}^{q}}}}\le {{c}_{j}}{{t}^{-\left(\frac{1}{2}-\frac{1}{q} \right)}}$(2≤q≤∞,j=1,2)可得

    $ \frac{1}{2}\left\| {{u_1}} \right\|_{{L^q}}^2 \le c_1^2{t^{ - \frac{{q - 2}}{q}}} $

    由于

    $ \mathop {\lim }\limits_{t \to 0} \frac{{{c_1}{t^{ - \frac{{q - 2}}{q}}}}}{{{t^{ - 1}}}} = 0 $

    因此

    $ \mathop {\lim }\limits_{t \to 0} \frac{{\frac{1}{2}\left\| {{u_1}} \right\|_{{L^q}}^2}}{{{t^{ - 1}}}} = 0 $ (11)

    又由Sobolev嵌入${{L}^{\infty }}\circlearrowleft {{L}^{\frac{3q}{q-1}}}$及引理1可知

    $ \begin{array}{*{20}{c}} {\left\| {{w_1}} \right\|_{{L^{\frac{{3q}}{{q - 1}}}}}^6 \le C\left\| {{w_1}} \right\|_{{L^\infty }}^6 \le C{{\left( {1 + t} \right)}^{ - 3}}}&{q \ge 2} \end{array} $

    由于

    $ \mathop {\lim }\limits_{t \to 0} \frac{{C{{\left( {1 + t} \right)}^{ - 3}}}}{{{t^{ - 1}}}} = \mathop {\lim }\limits_{t \to 0} \frac{t}{{C{{\left( {1 + t} \right)}^3}}} = 0 $

    因此

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{t \to 0} \frac{{\frac{1}{2}{{\left| {{\alpha _1}} \right|}^3}\left\| {{w_1}} \right\|_{{L^{\frac{{3q}}{{q - 1}}}}}^6}}{{{t^{ - 1}}}} = 0}&{q \ge 2} \end{array} $ (12)

    由(11)-(12)式可知,当t→0时有Φ2=o(t-1).同理可知,方程(9)右边第3项、第4项,当t→0时有Φ3=o(t-1),Φ4=o(t-1).

    下面考虑方程(9)右边第5项:

    $ {\mathit{\Phi }_5} \leqslant \left| {{\gamma _1}} \right|\int_\mathbb{R} {{{\left| {{w_1}} \right|}^2}{\text{d}}x} \cdot \int_\mathbb{R} {\left| {{u_1}} \right|\left| {{w_1}} \right|{\text{d}}x} \leqslant \left| {{\gamma _1}} \right|\left\| {{w_1}} \right\|_{{L^2}}^3{\left\| {{u_1}} \right\|_{{L^2}}} $

    重复(10)-(12)式的过程可知,Φ5=o(t-1).同理有Φ6=o(t-1),Φ7=o(t-1),Φ8=o(t-1).综上所述,当tT(假设T≥1)时有

    $ \frac{{\text{d}}}{{{\text{d}}t}}\varphi \left( t \right) = \operatorname{Im} \int_\mathbb{R} {\left( {{\partial _t}{{\bar w}_1} \cdot {u_1} + {\partial _t}{{\bar w}_2} \cdot {u_2} + {\partial _t}{u_1} \cdot {{\bar w}_1} + {\partial _t}{u_2} \cdot {{\bar w}_2}} \right){\text{d}}x} \geqslant C{t^{ - 1}} $ (13)

    其中

    $ \varphi \left( t \right) = \operatorname{Im} \int_\mathbb{R} {\left( {{u_1}{{\bar w}_1} + {u_2}{{\bar w}_2}} \right){\text{d}}x} $

    将(13)式两边同时关于t积分,可得

    $ \varphi \left( {10T} \right) - \varphi \left( T \right) \ge C\int_T^{10T} {{t^{ - 1}}{\rm{d}}t} = C\ln 10 > 0 $ (14)

    此外,由φ(t)的定义可知

    $ \varphi \left( t \right) = \operatorname{Im} \int_\mathbb{R} {\left( {{u_1}\left( {{{\bar w}_1} - {{\bar u}_1}} \right) + {u_2}\left( {{{\bar w}_2} - {{\bar u}_2}} \right)} \right){\text{d}}x} $

    利用Schwartz不等式,有

    $ \left| {\varphi \left( t \right)} \right| \le {\left\| {{u_1}} \right\|_{{L^2}}}{\left\| {{w_1} - {u_1}} \right\|_{{L^2}}} + {\left\| {{u_2}} \right\|_{{L^2}}}{\left\| {{w_2} - {u_2}} \right\|_{{L^2}}} $

    由假设可得$\underset{t\to \infty }{\mathop{\text{lim}}}\, \left| \varphi \left(t \right) \right|=0$,这与(14)式矛盾,故定理1得证.

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    Asymptotic Behavior of Solutions for Nonlocal Nonlinear Schr dinger Equations
    WEI Juan, ZHU Chao-sheng     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: This paper studies the asymptotic behavior of the solutions of the critical Schrödinger equations with nonlocal nonlinear terms, and prove the nonexistence of the asymptotic free solutions for this system by decay estimates of the solutions of the equations.
    Key words: Schrödinger equations    time decay estimate    asymptotic free solution    
    X