The existence of solutions of coupled nonlinear Schrödinger equations will be considered in this section. We apply the Galerkin method to prove the existence of global smooth solution for problem (1.10)–(1.11). Let \(w_{j} =(w_{1j},w_{2j} )^{T}\) be the normalized eigenfunction of the equation \(-\Delta w_{j} +\lambda _{j} w_{j} =0\) with the Dirichlet boundary condition corresponding to eigenvalue \(\lambda _{j}\), and \(\{w_{j} (x)\}_{j=1}^{\infty }\in V\) forms a normalized orthogonal system of eigenfunctions.
For every \(m\in N\), we denote the approximate solution \(U_{m} (x,t)\) of (1.10)–(1.11) by the following form:
$$ U_{m} (x,t)=\sum_{j=1}^{m} {\beta _{jm} (t)w_{j} } (x),\quad t\in [0,T], $$
(7.1)
where \(\beta _{jm} (t)(j=1,2,\ldots,m)\) are coefficient functions of variable \(t\in (0,L)\). According to the Galerkin method, the coefficient \(\beta _{jm} (t)\) is assumed to satisfy the following system of nonlinear ordinary equations of the first order:
$$ \bigl(i{U}'_{m} +\alpha (-\Delta )^{s} U_{m} +g \bigl(\vert U_{m} \vert ^{2} \bigr)U_{m} +iQU_{m} +\beta U_{m} -F, {w}_{j} \bigr)=0, $$
(7.2)
where \(\vert U_{m} \vert ^{2}=\vert u_{1m} \vert ^{2}+\vert u_{2m} \vert ^{2}\), \(j=1,2,\ldots,m\), with the initial condition
$$ \bigl(U_{m} (x,0), {w}_{j} (x) \bigr)= \bigl(U_{0} (x), {w}_{j} (x) \bigr). $$
(7.3)
It is obvious that
$$\begin{aligned}& \bigl({U}'_{m} (x,t), {w}_{j} (x) \bigr)={ \beta }'_{jm} (t), \\& \bigl(U_{m} (x,0), {w}_{j} (x) \bigr)=\beta _{jm} (0), \end{aligned}$$
and \(U_{0j} (x)=(U_{0} (x), {w}_{j} (x))\) (\(j=1,2,\ldots,m\)) are coefficients in the approximate expansion \(\sum_{j=1}^{m} {U_{0j} w_{j} (x)} \) of function \(U_{0} (x)\).
Let us prove that (7.2) has solution about the unknown function \(\beta _{jm} (t)\). By using the characteristic of normalized eigenfunction
$$\begin{aligned}& (w_{1j}, {w}_{1i} )=(w_{2j}, {w}_{2i} )=1,\quad i=j; \\& (w_{1j}, {w}_{1i} )=(w_{2j}, {w}_{2i} )_{0} =0,\quad i\ne j, \end{aligned}$$
from (3.2) we get
$$\begin{aligned} 0&= \Biggl(i\sum_{j=1}^{m} {{\beta }'_{jm} w_{1j} }, {w}_{1j} \Biggr)- \alpha \Biggl(\sum_{j=1}^{m} {\beta _{jm} (-\Delta )^{s} w_{1j} }, {w}_{1j} \Biggr) \\ &\quad {}+ \Biggl(g \Biggl( \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{1j} } \Biggr\vert ^{2}+ \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{2j} } \Biggr\vert ^{2} \Biggr)\sum _{j=1}^{m} {\beta _{jm} w_{1j} }, {w}_{1j} \Biggr) \\ &\quad {}+ \Biggl(ir\sum_{j=1}^{m} { \beta _{jm} w_{1j} }, {w}_{1i} \Biggr)+ \Biggl(\beta \sum_{j=1}^{m} {\beta _{jm} w_{1j} }, {w}_{1j} \Biggr)- \bigl(f(x),\overline{w}_{1j} \bigr) \\ &=i \int _{\Omega }{{\beta }'_{jm}\,dx} -\alpha \int _{\Omega }{\sum_{j=1}^{m}{ \beta _{jm}} (-\Delta )^{s} w_{1j} \overline{w}_{1j}\,dx} \\ &\quad {} + \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m} {\beta _{jm} w_{1j} } \Biggr\vert ^{2}+ \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{2j} } \Biggr\vert ^{2} \Biggr) \beta _{jm} }\,dx \\ &\quad {} +(ir+\beta ) \int _{\Omega }{\beta _{jm}\,dx} - \int _{\Omega }{f(x)\overline{w}_{1j}\,dx} \\ & =i\vert \Omega \vert {\beta }'_{jm} + \alpha \sum _{j=1}^{m} {\beta _{jm} } \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1j} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} \\ &\quad {} +\beta _{jm} \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m} {\beta _{jm} w_{1j} } \Biggr\vert ^{2}+ \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{2j} } \Biggr\vert ^{2} \Biggr)} \,dx \\ &\quad {} +(ir+\beta )\vert \Omega \vert \beta _{jm} - \int _{\Omega }{f_{1} (x)\overline{w}_{1j} \,dx}. \end{aligned}$$
(7.4)
That is,
$$\begin{aligned} 0&=i\vert \Omega \vert {\beta }'_{jm} \\ &\quad {}+\alpha \sum_{j=1}^{m} {\beta _{jm} } \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1j} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} \\ &\quad {}+\beta _{jm} \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m} {\beta _{jm} w_{1j} } \Biggr\vert ^{2}+ \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{2j} } \Biggr\vert ^{2} \Biggr)} \,dx \\ &\quad {}+(ir+\beta )\vert \Omega \vert \beta _{jm} - \int _{\Omega }{f_{1} (x)\overline{w}_{1j} \,dx} . \end{aligned}$$
(7.5)
And as to \(w_{2j} \), we have a similar conclusion
$$\begin{aligned} 0&=i\vert \Omega \vert {\beta }'_{jm} +\alpha \sum _{j=1}^{m} {\beta _{jm} } \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{2j} (-\Delta )^{\frac{s}{2}} \overline{w}_{2i}\,dx} \\ &\quad {}+\beta _{jm} \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m} {\beta _{jm} w_{1j} } \Biggr\vert ^{2}+ \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{2j} } \Biggr\vert ^{2} \Biggr)} \,dx \\ &\quad {}+(ir+\beta )\vert \Omega \vert \beta _{jm} - \int _{\Omega }{f_{2} (x)\overline{w}_{2i}\,dx} . \end{aligned}$$
(7.6)
We know that (7.7) and (7.6) are the first order ordinary equations of unknown functions \(\beta _{jm} \), \(j=1,2,\ldots,m\). If (7.7) and (7.6) have common solution, it must satisfy that
$$\begin{aligned} &h(\beta _{1m},\beta _{2m},\ldots,\beta _{mm} ) \\ &\quad =\alpha \sum_{j=1}^{m} {\beta _{jm} } \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1j} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} \\ &\quad \quad {} +\beta _{jm} \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m} {\beta _{jm} w_{1j} } \Biggr\vert ^{2}+ \Biggl\vert \sum_{j=1}^{m} {\beta _{jm} w_{2j} } \Biggr\vert ^{2} \Biggr)} \,dx \\ &\quad \quad {}+(ir+\beta )\vert \Omega \vert \beta _{jm} - \int _{\Omega }{f_{1} (x)\overline{w}_{1j} \,dx}. \end{aligned}$$
(7.7)
It is locally Lipschitz continuous in H.
We set \(\theta (t)=(\beta _{1m} (t),\beta _{2m} (t),\ldots,\beta _{mm} (t))\), \(\tilde{\theta }(t)=(\tilde{\beta }_{1m} (t),\tilde{\beta }_{2m} (t),\ldots,\tilde{\beta }_{mm} (t))\), and Lipschitz continuous functions \(h(\theta (t))\) are considered to satisfy
$$ \bigl\vert h\bigl(\theta (t)\bigr)-h \bigl(\tilde{\theta }(t) \bigr) \bigr\vert \le C \bigl\vert \theta (t)-\tilde{\theta }(t) \bigr\vert . $$
Then
$$\begin{aligned} h \bigl(\theta (t) \bigr) & =\alpha \sum_{j=1}^{m} {\beta _{jm} } (t) \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} \\ &\quad {}+\beta _{jm} \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m}\beta _{jm}w_{1j} \Biggr\vert ^{2} + \Biggl\vert \sum_{j=1}^{m} \beta _{jm}w_{2j} \Biggr\vert ^{2} \Biggr)\,dx} \\ &\quad {}+(ir+\beta )\vert \Omega \vert \beta _{jm} (t) - \int _{\Omega }{f_{1} (x)\overline{w}_{1i} \,dx}, \\ h \bigl(\tilde{\theta }(t) \bigr)& =\alpha \sum_{j=1}^{m} {\tilde{\beta }_{jm} } (t) \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} \\ &\quad {}+ \tilde{\beta }_{jm} (t) \int _{\Omega }{g \Biggl( \Biggl\vert \sum _{j=1}^{m} \tilde{\beta }_{jm} (t)w_{1j} \Biggr\vert ^{2}+ \bigl\vert \tilde{\beta }_{jm} (t) w_{2j} \bigr\vert ^{2} \Biggr)\,dx} \\ &\quad {} +(ir+\beta )\vert \Omega \vert \tilde{\beta }_{jm} (t) - \int _{\Omega }{f_{1} (x)\overline{w}_{1i} \,dx}, \end{aligned}$$
and
$$\begin{aligned}& h\bigl(\theta (t)\bigr)-h \bigl(\tilde{\theta }(t) \bigr) \\& \quad =\alpha \Biggl( {\sum_{j=1}^{m} { \beta _{jm} } (t)-\sum_{j=1}^{m} { \tilde{\beta }_{jm} } (t)} \Biggr) \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} \\& \quad \quad {}+ \bigl( {\beta _{jm} (t)-\tilde{\beta }_{jm} (t)} \bigr) \int _{\Omega }{g \bigl(\vert U_{m} \vert ^{2} \bigr)\,dx} \\& \quad \quad {}+(ir+\beta )\vert \Omega \vert \bigl( { \beta _{jm} (t)-\tilde{ \beta }_{jm} (t)} \bigr). \end{aligned}$$
Because of
$$ \bigl\vert \theta (t)-\tilde{\theta }(t) \bigr\vert =\sum _{j=1}^{m} {\vert {\beta _{jm} -\tilde{ \beta }_{jm} } \vert }, $$
and
$$\begin{aligned}& \bigl\vert \theta (t) \bigr\vert =\sqrt{\beta _{1m}^{2} +\beta _{2m}^{2} +\cdots +\beta _{mm}^{2} }, \\& \bigl\vert \tilde{\theta }(t) \bigr\vert =\sqrt{\tilde{ \beta }_{1m}^{2} +\tilde{\beta }_{2m}^{2} + \cdots +\tilde{\beta }_{mm}^{2} }, \\& \bigl\vert {h \bigl(\theta (t)-h \bigl(\tilde{\theta }(t) \bigr) \bigr)} \bigr\vert \\& \quad \le \alpha \Biggl\vert {\sum_{j=1}^{m} {\beta _{jm} } (t)-\sum_{j=1}^{m} {\tilde{\beta }_{jm} } (t)} \Biggr\vert \biggl\vert { \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} } \biggr\vert \\& \quad \quad {}+ \bigl\vert \beta _{jm} (t)-\tilde{\beta }_{jm} (t) \bigr\vert { \int _{\Omega }{g \bigl(\vert U_{m} \vert ^{2} \bigr)}\,dx}+(r+\beta )\vert \Omega \vert \bigl\vert \beta _{jm}(t)-\tilde{\beta }_{jm} (t) \bigr\vert \\& \quad \le \alpha \sum_{j=1}^{m} {\vert { \beta _{jm} -\tilde{\beta }_{jm} } \vert } \biggl\vert { \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} } \biggr\vert \\& \quad \quad {}+\sum_{j=1}^{m} {\vert { \beta _{jm} -\tilde{\beta }_{jm} } \vert } { \int _{\Omega }{g \bigl(\vert U_{m} \vert ^{2} \bigr)}\,dx} +(r+\beta )\vert \Omega \vert \sum _{j=1}^{m} {\vert {\beta _{jm} -\tilde{ \beta }_{jm} } \vert } \\& \quad = \vert \theta -\tilde{\theta } \vert \biggl( {\alpha \biggl\vert { \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} (-\Delta )^{\frac{s}{2}} \overline{w}_{1i}\,dx} } \biggr\vert + { \int _{\Omega }{g \bigl(\vert U_{m} \vert ^{2} \bigr)\,dx}} +(r+\beta )\vert \Omega \vert } \biggr). \end{aligned}$$
If \(\int _{\Omega }{g(\vert U_{m} \vert ^{2})\,dx}\le M\), \(\{w_{j} (x)\}_{j=1}^{\infty }\in V\), because of \(w_{1j},w_{1i},w_{2j},w_{2i} \in V\), and \(V\to H\) is compact, \(\Vert w_{ij} \Vert _{V}^{2} \) is bounded and \(\Vert w_{ij} \Vert _{H}^{2} \) also is bounded, then
$$ \alpha \Biggl\vert { \int _{\Omega }{(-\Delta )^{\frac{s}{2}} w_{1m} \sum _{i=1}^{m} {(-\Delta )^{\frac{s}{2}} \overline{w}_{1i} }\,dx} } \Biggr\vert \le N_{1} $$
and
$$ \frac{3}{2}\sum_{j=1}^{m} { \int _{\Omega }{\vert w_{1j} \vert ^{2}} + \vert w_{2j} \vert ^{2}\,dx} \le N_{2} $$
leads to
$$ \bigl\vert {h(\theta )-h(\tilde{\theta })} \bigr\vert \le \bigl(N_{1} +{C}'N_{2} +r\vert \Omega \vert +\beta \vert \Omega \vert \bigr)\vert \theta -\tilde{\theta } \vert . $$
We finally get \(h(\theta (t))\) is a Lipschitz continuous function and know that the ordinary differential equations (7.2) have common solutions for the unknown functions \(\beta _{jm} (t)\), \(j=1,2,\ldots,m\).
Theorem 7.1
For the given functions
F, \(U_{0}\),
$$ F\in H(\Omega ), \qquad U_{0} \in H(\Omega )\cap V(\Omega ). $$
If
\(g(s)\ge 0\), \(G(s)=\int _{0}^{s}g(s)\,ds\le g(s)s\)
and
\(\vert g'(s) \vert \le c_{0}s\) (\(s\ge 0\)), where
\(c>0\), then there exists a unique solution
\(U(x,t)\)
for problem (1.10)–(1.11), and it satisfies the condition
$$ U\in L^{\infty } \bigl(0,T;H(\Omega )\cap V(\Omega ) \bigr). $$
(7.8)
Proof
Under the condition above in this section, we continue to get the existence of the solution of problem (1.10)–(1.11). Firstly, we multiply (7.2) by \({\beta }_{jm} (t)\) and make sum about j,
$$\begin{aligned} &i \bigl({U}'_{m}, {U}_{m} \bigr)+\alpha \bigl((-\Delta )^{s} U_{m}, {U}_{m} \bigr)+ \bigl(g \bigl(\vert U_{m} \vert ^{2} \bigr)U_{m}, {U}_{m} \bigr)+i(QU_{m}, {U}_{m} )+(\beta U_{m}, {U}_{m} ) \\ &\quad =(F, \overline{U}_{m} ). \end{aligned}$$
(7.9)
It is similar to the process of (6.2), (6.4), from (6.5), we obtain
$$ \frac{d}{dt}\Vert U _{m} \Vert _{H}^{2} +\gamma \Vert U _{m} \Vert _{H}^{2} \le \frac{1}{\gamma }\Vert F \Vert _{H}^{2}, $$
because of \(F\in H(\Omega )\), \(\Vert F\Vert _{H}^{2} \le M_{1} \), and \(U_{0m} \in H(\Omega )\cap V(\Omega )\). For condition (7.3), by using Gronwall’s inequality, we have a conclusion similar to (6.1)
$$ \bigl\Vert U _{m} (t) \bigr\Vert _{H}^{2} \le \bigl\Vert U _{m} (x,0) \bigr\Vert _{H}^{2} e^{-\gamma t}+\frac{\Vert F\Vert _{H}^{2} }{\gamma } \bigl(1-e^{-\gamma t} \bigr), $$
then
$$ \mathop{\limsup }_{t\to \infty } \Vert U _{m} \Vert _{H}^{2} \le \frac{M_{1} }{\gamma ^{2}}. $$
Therefore \(U_{m} (t)\) is bounded in H.
Secondly, we choose \({w}'_{j} \) instead of \(w_{j} \) in (7.2), then multiply (7.2) by \(\overline{\beta }_{jm} (t)\), to make sum about j:
$$\begin{aligned}& i \bigl({U}'_{m},{ {U}}'_{m} \bigr)+\alpha \bigl((-\Delta )^{s} U_{m},{ {U}}'_{m} \bigr)+ \bigl(g \bigl(\vert U_{m} \vert ^{2} \bigr)U_{m},{ {U}}'_{m} \bigr)+i \bigl(QU_{m},{ {U}}'_{m} \bigr)+ \beta \bigl(U_{m},{ {U}}'_{m} \bigr) \\& \quad = \bigl(F,{ {U}}'_{m} \bigr), \end{aligned}$$
and the real part of the equation above is
$$\begin{aligned} & \operatorname{Re} \alpha \bigl((-\Delta )^{\frac{s}{2}} U_{m},(- \Delta)^{\frac{s}{2}} {\overline{U}}'_{m} \bigr)+ \operatorname{Re} \int _{\Omega }{g \bigl(\vert U_{m} \vert ^{2} \bigr)U_{m} {\overline{U}}'_{m} \,dx} \\ &\quad \quad {}-\operatorname{Im} \int _{\Omega }{ru_{1m} {\overline{u}}'_{1m} +\sigma u_{2m} {\overline{u}}'_{2m}\,dx} + \operatorname{Re} \beta \bigl(U_{m},{ {U}}'_{m} \bigr) \\ &\quad = \operatorname{Re} \bigl(F,{ {U}}'_{m} \bigr). \end{aligned}$$
(7.10)
Return to see the real part of (7.9),
$$ \operatorname{Im} \bigl(U_{m},{ {U}}'_{m} \bigr)+ \bigl\Vert (-\Delta )^{\frac{s}{2}} U_{m} \bigr\Vert _{H}^{2} + \int _{\Omega }{g \bigl(\vert U_{m} \vert ^{2} \bigr)\vert U_{m} \vert ^{2}\,dx} +\beta \Vert U _{m} \Vert _{H}^{2} = \operatorname{Re}(F, {U}_{m} ), $$
(7.11)
where \(G(\vert U_{m} \vert ^{2})=\int _{0}^{\vert U_{m} \vert ^{2}}g(s)\,ds\). Combining (7.10) with (7.11), we finally get
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \biggl\{ \alpha \bigl\Vert (-\Delta )^{\frac{s}{2}} U_{m} \bigr\Vert _{H}^{2}+ \int _{\Omega }{G \bigl(\vert U_{m} \vert ^{2} \bigr)\,dx} +\beta \Vert U _{m} \Vert _{H}^{2} -2\operatorname{Re}(F, {U}_{m} ) \biggr\} \\& \quad \quad{} +\gamma \biggl\{ \alpha \bigl\Vert (-\Delta )^{\frac{s}{2}} U_{m} \bigr\Vert _{H}^{2} + \int _{\Omega }{G \bigl(\vert U_{m} \vert ^{2} \bigr)\,dx+\beta \Vert U _{m} \Vert _{H}^{2} -2\operatorname{Re}(F, {U}_{m} )} \biggr\} \\& \quad \le \operatorname{Re} \int _{\Omega }{(r-2\gamma )f_{1} \overline{u}_{1m} +(\sigma -2\gamma )f_{2} \overline{u}_{2m}\,dx} . \end{aligned}$$
(7.12)
Introducing a functional equation about (7.12)
$$ \eta (U_{m} )=\beta \Vert U _{m} \Vert _{H}^{2} +\alpha \bigl\Vert (-\Delta )^{\frac{s}{2}} U_{m} \bigr\Vert _{H}^{2} + \int _{\Omega }{G \bigl(\vert U_{m} \vert ^{2} \bigr)\,dx} -2\operatorname{Re}(F, {U}_{m} ) $$
and rewriting (7.12) as
$$ \frac{1}{2}\frac{d}{dt}\eta (U_{m} )+\delta \eta (U_{m} )\le \frac{1}{2}(\delta +2\gamma ) \bigl(\Vert U _{m} \Vert _{H}^{2} +\Vert F \Vert _{H}^{2} \bigr), $$
we have
$$ \frac{d}{dt}\eta (U)+2\gamma \eta (U)\le 2\gamma {C}'_{\infty }. $$
Finally, we can get
$$ \mathop{\limsup }_{t\to \infty } \eta (U_{m} )\le {C}'_{\infty } $$
and
$$ \beta \Vert U _{m} \Vert _{H}^{2} +\alpha \bigl\Vert (-\Delta )^{\frac{s}{2}} U_{m} \bigr\Vert _{H}^{2} + \int _{\Omega }{G \bigl(\vert U_{m} \vert ^{2} \bigr)\,dx} \le {C}'_{\infty }+\frac{1}{2} \biggl(M_{1} +\frac{M_{1} }{\gamma ^{2}} \biggr). $$
Because of \(\beta \Vert U_{m} \Vert _{H}^{2} \ge 0\), \(\int _{\Omega }{G(\vert U_{m} \vert ^{2})\,dx} \ge 0\), we know that \(\Vert (-\Delta )^{\frac{s}{2}} U_{m} \Vert _{H}^{2} =\Vert U_{m} \Vert _{V}^{2} \), and \(U_{m} (t)\) is bounded in V.
Hence from the sequence \(\{U_{m} (x,t)\}\) of approximate solutions, we can select a subsequence \(\{U_{\mu }(x,t)\}\) and have a function \(U(x,t)\in L^{\infty }(0,T;H)\) such that
$$\begin{aligned}& U_{\mu }(x,t)\to U(x,t)\quad \mbox{in }U(x,t)\in L^{\infty }(0,T;H) \mbox{ weakly star, }\mu \to \infty. \\& (-\Delta )^{\frac{s}{2}} U_{\mu }(x,t)\to\!\!(-\Delta )^{\frac{s}{2}} U(x,t)\quad\!\mbox{in }U(x,t)\in L^{\infty }(0,T;V) \mbox{ weakly star and a.e.}, \mu \to \infty. \\& g \bigl(\vert U_{\mu } \vert ^{2} \bigr)\vert U_{\mu } \vert ^{2}\to g \bigl(\vert U \vert ^{2} \bigr)\vert U \vert ^{2} \quad \mbox{in }U(x,t)\in L^{\infty} \bigl(0,T;L^{6}(\Omega ) \bigr) \mbox{ weakly star, } \mu \to \infty. \end{aligned}$$
And
$$\begin{aligned}& U_{\mu }(x,t)\to U(x,t)\quad \mbox{in }U(x,t)\in L^{2}(0,T;H) \quad \mbox{weakly, }\mu \to \infty. \\& (-\Delta )^{\frac{s}{2}} U_{\mu }(x,t)\to (-\Delta )^{\frac{s}{2}} U(x,t)\quad \mbox{in }U(x,t)\in L^{2}(0,T;V)\mbox{ weakly and a.e., } \mu \to \infty. \\& g \bigl(\vert U_{\mu } \vert ^{2} \bigr)\vert U_{\mu } \vert ^{2}\to g \bigl(\vert U \vert ^{2} \bigr)\vert U \vert ^{2} \quad \mbox{in }U(x,t)\in L^{2} \bigl(0,T;L^{6}(\Omega ) \bigr)\mbox{ weakly, }\mu \to \infty. \end{aligned}$$
From
$$ \bigl(i{U}'_{\mu }-\alpha (-\Delta )^{s} U_{\mu }+G_{\mu }U_{\mu }+iQU_{\mu }+ \beta U_{\mu }-F,U_{\mu } \bigr)=0, $$
(7.13)
hence the function \(U(x,t)\) satisfies equation (1.11) everywhere and the boundary initial conditions (1.10). So the existence of solution for problem (1.10)–(1.11) has been proved. □