In the previous section, when we employ the Gronwall’s inequality to deduce the estimates for higher regularity, it holds only with \(C=C(T)\). That is why we just obtain the local existence for the solutions. However, we found in the proof of Lemma 2 that the boundedness of the estimates is independent of the time T. Thus we need to ask, if the initial values \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\), does problem (1)-(5) admit a global unique solution? If it does, what are the conditions? In this section, we will answer these questions.
First, from the results in Lemma 2 and previous section theorem, we know that for any \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega )\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega )\), problem (1)-(5) admits a unique solution
$$ (u, u_{t}, n,V)\in C\bigl([0, T_{\max}); H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}( \Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}( \Omega)\bigr), $$
(38)
where \(t\in[0, T_{\mathrm{max}})\) (\(0< T_{\mathrm{max}}<\infty\)). And from (16), the following conserved energy holds:
$$ E(u,u_{t},n,V)=E(u_{0}, u_{1}, n_{0},V_{0}), $$
(39)
where \(E(u,u_{t},n,V)\) is defined as
$$ \begin{aligned}[b] E(u,u_{t},n,V)&= \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac{1}{2} \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \frac{1}{2} \Vert V \Vert ^{2}+ \int_{\Omega}n \vert u \vert ^{2}\,dx. \end{aligned} $$
(40)
As we have noticed, system (1)-(3) includes derivative nonlinearity and different-degree nonlinearities, thus we need to imply some proper techniques to handle these terms. To deal with the derivative nonlinearity, we first introduce a homogeneous Sobolev space \(\dot{H}_{\mathrm{per}}^{-1}(\Omega)\) defined by
$$ \begin{aligned}[b] \dot{H}_{\mathrm{per}}^{-1}( \Omega)&=\bigl\{ n\vert\exists\nu:\Omega\rightarrow\Omega \mbox{ such that } n=- \nabla\cdot\nu, \nu\in L_{\mathrm{per}}^{2}(\Omega) \\ &\quad\mbox{ and } \Vert n \Vert _{\dot{H}_{\mathrm{per}}^{-1}(\Omega )}= \Vert \nu \Vert _{L_{\mathrm{per}}^{2}(\Omega)}\bigr\} . \end{aligned} $$
(41)
Second, we make the assumption that there exists a real vector-valued function \(g(x,t)\in L_{\mathrm{per}}^{2}(\Omega)\) such that
$$ g_{t}(x,t)=V(x,t). $$
(42)
For the different-degree nonlinearities, we introduce some functionals and manifolds to handle it. That is to say, for any \((\phi,\psi)\in H_{\mathrm{per}}^{1}(\Omega)\times H_{\mathrm{per}}^{1}(\Omega)\), we define
$$\begin{aligned}& F(\phi,\psi):= \Vert \phi \Vert ^{2}+ \Vert \nabla \phi \Vert ^{2}+\frac{1}{2} \Vert \phi \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert \psi \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla\psi \Vert ^{2}+ \int_{\Omega}\psi \vert \phi \vert ^{2}\,dx, \end{aligned}$$
(43)
$$\begin{aligned}& G(\phi,\psi):=2 \Vert \phi \Vert ^{2}+2 \Vert \nabla\phi \Vert ^{2}+2 \Vert \phi \Vert _{L^{4}}^{4}+ \Vert \psi \Vert ^{2}+H^{2} \Vert \nabla\psi \Vert ^{2}+3 \int_{\Omega}\psi \vert \phi \vert ^{2}\,dx, \end{aligned}$$
(44)
$$\begin{aligned}& P(\phi,\psi):=F(\phi,\psi)-\frac{1}{\lambda+1}G(\phi,\psi), \end{aligned}$$
(45)
and
$$\begin{aligned}& \Phi:=\bigl\{ (\phi,\psi)\in H_{\mathrm{per}}^{1}(\Omega) \times H_{\mathrm{per}}^{1}(\Omega)\vert G(\phi,\psi)=0, (\phi, \psi) \neq(0,0)\bigr\} , \end{aligned}$$
(46)
$$\begin{aligned}& \Phi^{-}:=\bigl\{ (\phi,\psi)\in H_{\mathrm{per}}^{1}( \Omega)\times H_{\mathrm{per}}^{1}(\Omega)\vert G(\phi,\psi)\leq0, ( \phi, \psi)\neq(0,0)\bigr\} , \end{aligned}$$
(47)
where \(\lambda>1\) is a constant.
Additionally, we define two constrained variational problems
$$ d_{\Phi}:=\inf_{(\phi,\psi)\in\Phi }F(\phi,\psi),\qquad d_{\Phi^{-}}:=\inf_{(\phi,\psi)\in\Phi^{-}}P(\phi,\psi). $$
(48)
According to the definitions, it is easy to find that the energy functional E can be rewritten as
$$ E(u,u_{t},n,V)= \Vert u_{t} \Vert ^{2}+\frac{1}{2} \Vert V \Vert ^{2}+F(u,n), $$
(49)
or
$$ E(u,u_{t},n,V)= \Vert u_{t} \Vert ^{2}+\frac{1}{2} \Vert V \Vert ^{2}- \frac{1}{2} \Vert u \Vert _{L^{4}}^{4}- \frac{1}{2} \int _{\Omega}n \vert u \vert ^{2}\,dx+ \frac{1}{2}G(u,n). $$
(50)
For the properties of \(F(\phi,\psi)\) and \(d_{\Phi}\), we have the following results.
Proposition 4.1
\(F(\phi,\psi)\)
is bounded below on Φ, \(F(\phi,\psi)>0\)
for all
\((\phi,\psi)\in\Phi \)
and
\(d_{\Phi}>0\).
Proof
First, from (43) and (44), we have
$$ 3F(\phi,\psi)-G(\phi,\psi)= \Vert \phi \Vert ^{2}+ \Vert \nabla\phi \Vert ^{2}-\frac{1}{2} \Vert \phi \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert \psi \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla\psi \Vert ^{2}. $$
(51)
Thus on Φ there holds \(G(\phi,\psi)=0\) and
$$ F(\phi,\psi)=\frac{1}{3} \Vert \phi \Vert ^{2}+\frac{1}{3} \Vert \nabla\phi \Vert ^{2}+ \frac{1}{6} \Vert \psi \Vert ^{2}+\frac{H^{2}}{6} \Vert \nabla\psi \Vert ^{2}-\frac{1}{6} \Vert \phi \Vert _{L^{4}}^{4}. $$
(52)
On the other hand, from (44) we have
$$ \begin{aligned}[b] 2 \Vert \phi \Vert ^{2}+2 \Vert \nabla\phi \Vert ^{2}+2 \Vert \phi \Vert _{L^{4}}^{4}+ \Vert \psi \Vert ^{2}+H^{2} \Vert \nabla\psi \Vert ^{2}&=-3 \int_{\Omega}\psi \vert \phi \vert ^{2}\,dx \\ &\leq\frac{3}{2} \Vert \psi \Vert ^{2}+\frac{3}{2} \Vert \phi \Vert _{L^{4}}^{4}, \end{aligned} $$
(53)
which implies
$$ 0\leq2 \Vert \phi \Vert ^{2}+2 \Vert \nabla\phi \Vert ^{2}+H^{2} \Vert \nabla\psi \Vert ^{2} \leq\frac{1}{2} \Vert \psi \Vert ^{2}-\frac{1}{2} \Vert \phi \Vert _{L^{4}}^{4}. $$
(54)
That is also to say
$$ \Vert \psi \Vert ^{2}> \Vert \phi \Vert _{L^{4}}^{4} \quad \mbox{on } \Phi. $$
(55)
Combining (52) and (55), we can complete the proof of Proposition 4.1. □
For the functional \(P(\phi,\psi)\) and \(d_{\Phi^{-}}\), we have the following results.
Proposition 4.2
If
\(G(\phi,\psi)\leq0\)
and
\((\phi, \psi)\neq(0,0)\), then
\(P(\theta ^{\frac{1}{2}}\phi,\theta\psi)\)
is an increasing function of
\(\theta\in (0,\infty)\). And
$$ d_{\Phi}=\inf_{(\phi,\psi)\in\Phi }F(\phi, \psi)=d_{\Phi^{-}}=\inf_{(\phi,\psi)\in\Phi^{-}}P(\phi,\psi). $$
(56)
Furthermore, if
\(G(\phi,\psi)<0\), then
$$ P(\phi,\psi)>d_{\Phi}. $$
(57)
Proof
The first result that \(P(\theta^{\frac{1}{2}}\phi,\theta\psi)\) is an increasing function of θ can be proved directly by computing the derivation with respect to θ. Here we prove (56).
First, from definition (44), we have
$$ \begin{aligned}[b] G\bigl(\theta^{\frac{1}{2}}\phi, \theta\psi\bigr)&=2\theta \Vert \phi \Vert ^{2}+2\theta \Vert \nabla\phi \Vert ^{2}+2\theta^{2} \Vert \phi \Vert _{L^{4}}^{4}+\theta^{2} \Vert \psi \Vert ^{2} \\ &\quad{}+H^{2}\theta^{2} \Vert \nabla\psi \Vert ^{2}+3\theta^{2} \int _{\Omega}\psi \vert \phi \vert ^{2}\,dx. \end{aligned} $$
(58)
Then, for \(\theta=1\), \(G(\phi,\psi)\leq0\), and for \(\theta>0\) close to zero, \(G(\theta^{\frac{1}{2}}\phi,\theta\psi)>0\). Thus from the continuity there exists \(\theta_{0}\in(0,1)\) such that \(G(\theta_{0}^{\frac {1}{2}}\phi,\theta_{0}\psi)=0\), which also implies \((\theta_{0}^{\frac {1}{2}}\phi,\theta_{0}\psi)\in\Phi\). Noticing \(\Phi\subset\Phi^{-}\), and from (48), there holds
$$ \begin{aligned}[b] d_{\Phi}&\leq\inf _{(\theta_{0}^{\frac{1}{2}}\phi,\theta_{0}\psi)\in\Phi} F\bigl(\theta_{0}^{\frac{1}{2}}\phi, \theta_{0}\psi\bigr) \\ &\leq\inf_{(\theta_{0}^{\frac{1}{2}}\phi,\theta_{0}\psi)\in\Phi^{-}} \biggl(P\bigl(\theta_{0}^{\frac{1}{2}} \phi,\theta_{0}\psi\bigr)+\frac{1}{\lambda+1}G\bigl(\theta _{0}^{\frac{1}{2}}\phi,\theta_{0}\psi\bigr) \biggr) \\ &=\inf_{(\theta_{0}^{\frac{1}{2}}\phi,\theta_{0}\psi)\in\Phi^{-}} P\bigl(\theta _{0}^{\frac{1}{2}}\phi, \theta_{0}\psi\bigr)\leq\inf_{(\phi,\psi)\in\Phi^{-}} P(\phi,\psi) \\ &=d_{\Phi^{-}}. \end{aligned} $$
(59)
Meanwhile, we have
$$ \begin{aligned}[b] d_{\Phi}&= \inf _{(\phi,\psi)\in\Phi} F(\phi,\psi)=\inf_{(\phi,\psi)\in \Phi} \biggl(P(\phi, \psi)+\frac{1}{\lambda+1}G(\phi,\psi) \biggr) \\ &=\inf_{(\phi,\psi)\in\Phi} P(\phi,\psi)\geq\inf_{(\phi,\psi)\in\Phi ^{-}} P( \phi,\psi)=d_{\Phi^{-}}. \end{aligned} $$
(60)
Thus one combines (59) and (60) to get (56).
Finally, since \(G(\phi,\psi)<0\), then there also exists \(\theta\in (0,1)\) such that \(G(\theta^{\frac{1}{2}}\phi,\theta\psi)=0\) and \((\phi, \psi)\neq(0,0)\). Thus there holds
$$ P(\phi,\psi)>P\bigl(\theta^{\frac{1}{2}}\phi,\theta\psi\bigr)=F \bigl(\theta^{\frac {1}{2}}\phi,\theta\psi\bigr)-\frac{1}{\lambda+1}G\bigl( \theta^{\frac{1}{2}}\phi,\theta\psi\bigr)=F\bigl(\theta^{\frac{1}{2}}\phi, \theta\psi\bigr)\geq d_{\Phi}, $$
(61)
which leads to (57). The proof of Proposition 4.2 is completed. □
Since \(d_{\Phi}>0\), then we define a set \(\mathcal{S}\) as
$$\begin{aligned}[b] \mathcal{S}&:=\bigl\{ (\phi_{1}, \phi_{2},\psi_{1},\psi_{2})\in H_{\mathrm{per}}^{1}( \Omega )\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}( \Omega)\times L_{\mathrm{per}}^{2}(\Omega ): \\ &\quad E(\phi_{1},\phi_{2},\psi_{1}, \psi_{2})< d_{\Phi}\bigr\} , \end{aligned} $$
and introduce two invariant sets as
$$\begin{aligned}[b] &\mathcal{S}_{1}:=\bigl\{ ( \phi_{1}, \phi_{2},\psi_{1},\psi_{2})\in \mathcal{S}\vert G(\phi_{1},\psi_{1})>0\bigr\} \cup\bigl\{ (0, \phi_{2},0,\psi_{2})\in\mathcal{S}\bigr\} , \\ &\mathcal{S}_{2}:=\bigl\{ (\phi_{1}, \phi_{2}, \psi_{1},\psi_{2})\in\mathcal{S}\vert G(\phi_{1}, \psi_{1})< 0\bigr\} . \end{aligned} $$
For the sets \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\), we have the following proposition.
Proposition 4.3
\(\mathcal{S}_{1}\)
and
\(\mathcal{S}_{2}\)
are invariant sets under the solution flow generated by the periodic initial value problem (1)-(5).
Proof
We prove the proposition by contradiction. First we prove that \(\mathcal {S}_{1}\) is an invariant set. Let \((u_{0},u_{1},n_{0},V_{0})\in\mathcal{S}_{1}\), and suppose that there exists a time \(t_{1}\) such that \((u(t_{1}),u_{t}(t_{1}),n(t_{1}),V(t_{1}))\notin\mathcal{S}_{1}\), then \(G(u(t_{1}),n(t_{1}))\leq0\) and \((u(t_{1}),n(t_{1}))\neq(0,0)\), which implies \((u(t_{1}),n(t_{1}))\in\Phi^{-}\). Set
$$ s=\inf\bigl\{ 0\leq t\leq t_{1}\vert\bigl(u(t),u_{t}(t),n(t),V(t) \bigr)\notin\mathcal {S}_{1}\bigr\} , $$
(62)
then \(G(u(t),n(t))\geq0\) for all \(0\leq t< s\). Let \(\{s_{k}\}\) be the minimizing sequence for problem (62), then \((u(s_{k}),n(s_{k}))\in\Phi^{-}\). By the weak lower semi-continuity of \(G(u(\cdot),n(\cdot))\), we have
$$ G\bigl(u(s),n(s)\bigr)\leq\lim_{k\rightarrow\infty}\inf G \bigl(u(s_{k}),n(s_{k})\bigr)\leq0, \quad\bigl(u(s),n(s)\bigr) \neq(0,0). $$
(63)
On the other hand, from (45), (49) and (56) in Proposition 4.2, we obtain
$$ \begin{aligned}[b] P\bigl(u(s),n(s)\bigr)&=\lim _{t\rightarrow s^{-}}\inf P\bigl(u(t),n(t)\bigr) \\ &\leq \lim_{t\rightarrow s^{-}}\inf \biggl(P\bigl(u(t),n(t)\bigr)+ \frac{1}{\lambda +1}G\bigl(u(t),n(t)\bigr) \biggr) \\ &=\lim_{t\rightarrow s^{-}}\inf F\bigl(u(t),n(t)\bigr) \\ &\leq\lim_{t\rightarrow s^{-}}\inf E\bigl(u(t),u_{t}(t),n(t),V(t) \bigr) \\ &< d_{\Phi}=d_{\Phi^{-}}, \end{aligned} $$
(64)
which contradicts definition (48). So \(\mathcal{S}_{1}\) is invariant.
Now we turn to proving that \(\mathcal{S}_{2}\) is also invariant. Similarly, let \((u_{0},u_{1},n_{0},V_{0})\in\mathcal{S}_{2}\), and assume that there exists \(t_{2}\) such that \((u(t_{2}),u_{t}(t_{2}),n(t_{2}),V(t_{2}))\notin \mathcal{S}_{2}\), which implies \(G(u(t_{2}),n(t_{2}))\geq0\). From (45) and (49), we have
$$ \begin{aligned}[b] P\bigl(u(t_{2}),n(t_{2}) \bigr)&=F\bigl(u(t_{2}),n(t_{2})\bigr)-\frac{1}{\lambda+1}G \bigl(u(t_{2}),n(t_{2})\bigr) \\ &\leq F\bigl(u(t_{2}),n(t_{2})\bigr)< d_{\Phi}. \end{aligned} $$
(65)
Let
$$ s=\inf\bigl\{ 0\leq t\leq t_{2}\vert\bigl(u(t),u_{t}(t),n(t),V(t) \bigr)\notin\mathcal {S}_{2}\bigr\} , $$
(66)
then \(G(u(t),n(t))>d_{\Phi}\) for all \(0\leq t< s\). However, from (45), there holds
$$ \begin{aligned}[b] G\bigl(u(s),n(s)\bigr)&=\lim _{t\rightarrow s^{-}}\inf(\lambda +1)\bigl[F\bigl(u(t),n(t)\bigr)-P \bigl(u(t),n(t)\bigr)\bigr] \\ &\leq\lim_{t\rightarrow s^{-}}\inf(\lambda +1)\bigl[E\bigl(u(t),u_{t}(t),n(t),V(t) \bigr)-d_{\Phi}\bigr] \\ &\leq(\lambda+1)\bigl[E(u_{0},u_{1},n_{0},V_{0})-d_{\Phi}\bigr] \\ &< 0. \end{aligned} $$
(67)
From Proposition 4.2, we know that if \(G(u(s),n(s))<0\), then \(G(u(s),n(s))>d_{\Phi}\), which makes contradiction. Thus \(\mathcal{S}_{2}\) is also invariant. The proof of Proposition 4.3 is completed. □
Based on the results from previous propositions, we can derive a sharp threshold of global existence and blowup for the solution \((u(x, t),n(x, t), V(x, t))\) to the periodic initial value problem (1)-(5) in terms of the relationship between the initial energy \(E(u_{0}, u_{1},n_{0}, V_{0})\) and \(d_{\Phi}>0\). Here we state the main results as follows.
Theorem 2
Global existence and blowup
Suppose that
\((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\)
and satisfy
$$ E(u_{0}, u_{1}, n_{0},V_{0})< d_{\Phi}. $$
(68)
Then
(1) If
$$ G(u_{0}, n_{0})< 0, $$
(69)
then the solution
\((u(x,t), n(x,t), V(x,t))\)
of the periodic initial value problem (1)-(5) blows up in finite time. That is, there exists
\(T>0\)
such that
$$\lim_{t\rightarrow T}\bigl( \Vert u \Vert ^{2}_{L_{\mathrm{per}}^{2}(\Omega )}+ \Vert n \Vert ^{2}_{\dot{H}_{\mathrm{per}}^{-1}(\Omega)}\bigr)=\infty. $$
(2) If
$$ G(u_{0}, n_{0})>0, $$
(70)
then the solution
\((u(x,t), n(x,t), V(x,t))\)
of the periodic initial value problem (1)-(5) exists globally on
\(t\in[0,\infty )\)
and satisfies
$$ \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac{1}{2} \Vert u \Vert _{L^{4}}^{4}+\frac {1}{2} \Vert n \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \frac{1}{2} \Vert V \Vert ^{2}< d_{\Phi}, $$
(71)
or
$$ \Vert u_{t} \Vert ^{2}+\frac{4}{3} \Vert u \Vert ^{2}+\frac {4}{3} \Vert \nabla u \Vert ^{2}+\frac{1}{6} \Vert n \Vert ^{2}+ \frac{2}{3}H^{2} \Vert \nabla n \Vert ^{2}+ \frac {1}{2} \Vert V \Vert ^{2} \leq2d_{\Phi}+ \frac{1}{6}C^{\star}d^{2}_{\Phi}, $$
(72)
where
\(C^{\star}\)
is a positive constant which satisfies the Gagliardo-Nirenberg inequality.
Proof
First we prove (1) of Theorem 2. From (39), (49) and condition (68), we have
$$ F(u_{0},n_{0})\leq E(u_{0}, u_{1},n_{0},V_{0})< d_{\Phi}. $$
(73)
(1) When \(G(u_{0}, n_{0})<0\), as the initial values \((u_{0}, u_{1},n_{0},V_{0})\in \mathcal{S}_{2}\), and it follows that \((u(x,t), u_{t}(x,t), n(x,t), V(x,t)) \in\mathcal{S}_{2}\) by Proposition 4.3. Thus
$$ G\bigl(u(x,t),n(x,t)\bigr)< 0 \mbox{ for } t\in[0, T). $$
(74)
And from (49) and (73), there holds
$$ F\bigl(u(x,t),n(x,t)\bigr)< d_{\Phi}. $$
(75)
On the other hand, since \((u(x,t), u_{t}(x,t),n(x,t),V(x,t))\) is a solution of periodic initial value problem (1)-(5), under assumption (42), we set
$$ Y(t)=2 \bigl\Vert u(x,t) \bigr\Vert ^{2}+ \bigl\Vert g(x,t) \bigr\Vert ^{2}. $$
(76)
Thus we get
$$ Y^{\prime}(t)=2 \int_{\Omega}(u_{t}\overline{u}+u\overline{u}_{t})\,dx+2 \int _{\Omega}gg_{t}\,dx, $$
(77)
where u̅ is the complex conjugate of u. By computing \(F^{\prime\prime}(t)\), one can obtain
$$ \begin{aligned}[b] Y^{\prime\prime}(t)&=2 \int_{\Omega}(u_{tt}\overline{u}+u_{t} \overline {u}_{t}+u_{t}\overline{u}_{t}+u \overline{u}_{tt})\,dx+2 \int_{\Omega}(g_{t}g_{t}+gg_{tt})\,dx \\ &=4 \Vert u_{t} \Vert ^{2}+4 \int_{\Omega}\operatorname {Re}(u_{tt}\overline{u})\,dx+2 \Vert g_{t} \Vert ^{2}+2 \int_{\Omega}gg_{tt}\,dx \\ &=4 \Vert u_{t} \Vert ^{2}+4 \int_{\Omega}\operatorname{Re}\bigl(\bigl(\Delta u-u-nu- \vert u \vert ^{2}u\bigr)\overline{u}\bigr)\,dx+2 \Vert g_{t} \Vert ^{2} \\ &\quad{}+2 \int_{\Omega}g\bigl(-\nabla n-\nabla \vert u \vert ^{2}+H^{2}\nabla\Delta n\bigr)\,dx \\ &=4 \Vert u_{t} \Vert ^{2}-4 \Vert u \Vert ^{2}-4 \Vert \nabla u \Vert ^{2}-4 \Vert u \Vert _{L^{4}}^{4}-4 \int_{\Omega}n \vert u \vert ^{2}\,dx+2 \Vert V \Vert ^{2} \\ &\quad{}+2 \int_{\Omega}(\nabla\cdot g) \bigl(n+ \vert u \vert ^{2}-H^{2}\Delta n\bigr)\,dx \\ &=4 \Vert u_{t} \Vert ^{2}+2 \Vert V \Vert ^{2}-4 \Vert u \Vert ^{2}-4 \Vert \nabla u \Vert ^{2}-4 \Vert u \Vert _{L^{4}}^{4}-4 \int_{\Omega}n \vert u \vert ^{2}\,dx \\ &\quad{}-2 \Vert n \Vert ^{2}-2 \int_{\Omega}n \vert u \vert ^{2}\,dx-2H^{2} \Vert \nabla n \Vert ^{2} \\ &=2\bigl(2 \Vert u_{t} \Vert ^{2}+ \Vert V \Vert ^{2}\bigr)-4 \Vert u \Vert ^{2}-4 \Vert \nabla u \Vert ^{2}-4 \Vert u \Vert _{L^{4}}^{4}-2 \Vert n \Vert ^{2} \\ &\quad{}-2H^{2} \Vert \nabla n \Vert ^{2}-6 \int_{\Omega}n \vert u \vert ^{2}\,dx \\ &=2\bigl(2 \Vert u_{t} \Vert ^{2}+ \Vert V \Vert ^{2}\bigr)-2G(u,n). \end{aligned} $$
(78)
Combining (74), it yields that
$$ Y^{\prime\prime}(t)>0,\quad \mbox{for } t\in[0,T). $$
(79)
Before continuing to complete the proof, we give some property about the function \(Y(t)\) defined in (76).
Proposition 4.4
The function
\(Y^{\prime}(t)\)
in (77) is positive for some
t. That is, there exists
\(t_{1}>0\)
such that
\(Y^{\prime}(t)>0\)
for all
\(t>t_{1}\).
Proof
We prove it by contradiction. Suppose that, for all \(t>0\), there holds
$$ Y^{\prime}(t)\leq0. $$
(80)
Combining (79), \(Y(t)\) must tend to a finite and nonnegative limit \(Y_{0}\) as \(t\rightarrow\infty\). From Proposition 4.3, it concludes \(Y_{0}>0\). And as \(t\rightarrow\infty\), there holds \(Y(t)\rightarrow Y_{0}\), \(Y^{\prime}(t)\rightarrow0\) and \(Y^{\prime\prime}(t)\rightarrow0\). Then, from (78) and \(G(u,n)<0\), it follows that
$$ \lim_{t\rightarrow\infty}2\bigl(2 \Vert u_{t} \Vert ^{2}+ \Vert V \Vert ^{2}\bigr)=0 $$
(81)
and
$$ \lim_{t\rightarrow\infty}G(u,n)=0. $$
(82)
Now, for any fixed \(t>0\), and since \(G(u,n)<0\), there exists \(0<\theta <1\) such that \(G(\theta^{\frac{1}{2}}u,\theta n)=0\) and \((\theta^{\frac{1}{2}}u,\theta n)\neq0\). Thus from (44) we can get
$$ -\frac{3}{2}\theta^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx =\theta \bigl( \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}\bigr)+ \theta ^{2} \Vert u \Vert _{L^{4}}^{4}+ \frac{1}{2}\theta^{2} \Vert n \Vert ^{2}+ \frac{1}{2}H^{2}\theta^{2} \Vert \nabla n \Vert ^{2} $$
(83)
and
$$ \int_{\Omega}n \vert u \vert ^{2}\,dx< 0. $$
(84)
Thus by (48) we get
$$ F\bigl(\theta^{\frac{1}{2}}u,\theta n\bigr)\geq d_{\Phi}. $$
(85)
From (43) and (83) it follows that
$$ \begin{aligned}[b] F\bigl(\theta^{\frac{1}{2}}u,\theta n\bigr)&=\theta\bigl( \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}\bigr)+\frac{1}{2}\theta^{2} \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \theta^{2} \Vert n \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2}\theta^{2} \Vert \nabla u \Vert ^{2}+\theta ^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx \\ &=-\frac{3}{2}\theta^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx- \frac {1}{2}\theta^{2} \Vert u \Vert _{L^{4}}^{4}+ \theta^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx \\ &=-\frac{1}{2}\theta^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx- \frac {1}{2}\theta^{2} \Vert u \Vert _{L^{4}}^{4}. \end{aligned} $$
(86)
Thus
$$ \begin{aligned}[b] F(u, n)-F\bigl( \theta^{\frac{1}{2}}u,\theta n\bigr) &= \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac {1}{2} \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n \Vert ^{2}+ \frac{H^{2}}{2} \Vert \nabla n \Vert ^{2} \\ &\quad{}+ \int_{\Omega}n \vert u \vert ^{2}\,dx+ \frac{1}{2}\theta ^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx+ \frac{1}{2}\theta^{2} \Vert u \Vert _{L^{4}}^{4} \\ &= \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n \Vert ^{2}+\frac {H^{2}}{2} \Vert \nabla n \Vert ^{2} \\ &\quad{}+ \int_{\Omega}n \vert u \vert ^{2}\,dx+ \frac{1}{2}\theta ^{2} \int_{\Omega}n \vert u \vert ^{2}\,dx+ \frac{1}{2}\bigl(\theta^{2}-1\bigr) \Vert u \Vert _{L^{4}}^{4}. \end{aligned} $$
(87)
On the other hand, from definition (44) and (82), we can obtain as \(t\rightarrow\infty\),
$$ \Vert n \Vert ^{2}+2 \Vert u \Vert _{L^{4}}^{4}+3 \int _{\Omega}n \vert u \vert ^{2}\,dx\leq0. $$
(88)
Together with
$$ -\frac{3}{2} \Vert n \Vert ^{2}- \frac{3}{2} \int_{\Omega} \vert u \vert ^{4}\,dx\leq3 \int_{\Omega}n \vert u \vert ^{2}\,dx\leq \frac{3}{2} \Vert n \Vert ^{2}+\frac{3}{2} \int_{\Omega} \vert u \vert ^{4}\,dx, $$
(89)
there holds
$$ \int_{\Omega} \vert u \vert ^{4}\,dx\leq \Vert n \Vert ^{2}, $$
(90)
and
$$ \int_{\Omega} \vert u \vert ^{4}\,dx\leq \Vert n \Vert ^{2}\leq-2 \Vert u \Vert _{L^{4}}^{4}-3 \int_{\Omega}n \vert u \vert ^{2}\,dx, $$
(91)
which shows
$$ \int_{\Omega} \vert u \vert ^{4}\,dx\leq- \int_{\Omega}n \vert u \vert ^{2}\,dx. $$
(92)
Since \(0<\theta<1\), we have as \(t\rightarrow\infty\),
$$ \frac{1}{2}\bigl(\theta^{2}-1\bigr) \int_{\Omega} \vert u \vert ^{4}\,dx\geq \frac{1}{2}\bigl(\theta^{2}-1\bigr) \biggl(- \int_{\Omega}n \vert u \vert ^{2}\,dx\biggr). $$
(93)
Combining (87) and (93), we get
$$ \begin{aligned}[b] F(u, n)-F\bigl( \theta^{\frac{1}{2}}u,\theta n\bigr)&\geq \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \int_{\Omega}n \vert u \vert ^{2}\,dx+ \frac{1}{2} \int_{\Omega}n \vert u \vert ^{2}\,dx \\ &= \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \frac{3}{2} \int _{\Omega}n \vert u \vert ^{2}\,dx \\ &=\frac{1}{2}G(u,n). \end{aligned} $$
(94)
In all, from (82), (85) and (94), we can conclude that as \(t\rightarrow\infty\),
$$ F(u, n)\geq F\bigl(\theta^{\frac{1}{2}}u,\theta n\bigr) \geq d_{\Phi}, $$
(95)
which contradicts \(F(u, n)< d_{\Phi}\) from (75). So supposition (80) is not true. Thus \(Y^{\prime}(t)>0\) for some t. Thus the proof of Proposition 4.4 is completed. □
Corollary 1
Under the conditions of Proposition
4.4, the function
\(Y(t)\)
defined in (76) and
\(Z(t)= \Vert u(x,t) \Vert ^{2}\)
are both increasing for all
\(t>t_{1}\).
Proof
We can compute that
$$Z^{\prime}(t)= \int_{\Omega}(u_{t}\overline{u}+u\overline{u}_{t})\,dx $$
and
$$ \begin{aligned}[b] Z^{\prime\prime}(t)&=2 \Vert u_{t} \Vert ^{2}-2 \Vert u \Vert ^{2}-2 \Vert \nabla u \Vert ^{2}-2 \Vert u \Vert _{L^{4}}^{4}-2 \int_{\Omega}n \vert u \vert ^{2}\,dx \\ &=2 \Vert u_{t} \Vert ^{2}+ \Vert n \Vert ^{2}+H^{2} \Vert \nabla n \Vert ^{2}-\biggl(- \int_{\Omega}n \vert u \vert ^{2}\,dx\biggr)-G(u,n) \\ &\geq2 \Vert u_{t} \Vert ^{2}+ \Vert n \Vert ^{2}+H^{2} \Vert \nabla n \Vert ^{2}- \biggl( \frac{1}{2} \Vert n \Vert ^{2}+\frac{1}{2} \Vert u \Vert _{L^{4}}^{4} \biggr)-G(u,n) \\ &\geq2 \Vert u_{t} \Vert ^{2}+\frac{1}{2} \Vert n \Vert ^{2}+H^{2} \Vert \nabla n \Vert ^{2}- \frac{1}{2} \Vert u \Vert _{L^{4}}^{4}-G(u,n). \end{aligned} $$
(96)
Since \(G(u,n)<0\), we have
$$2 \Vert u \Vert ^{2}+2 \Vert \nabla u \Vert ^{2}+2 \Vert u \Vert _{L^{4}}^{4}+ \Vert n \Vert ^{2}+H^{2} \Vert \nabla n \Vert ^{2} \leq-3 \int_{\Omega}n \vert u \vert ^{2}\,dx\leq \frac{3}{2} \Vert u \Vert _{L^{4}}^{4}+ \frac{3}{2} \Vert n \Vert ^{2}, $$
which implies that
$$ 0< 2 \Vert u \Vert ^{2}+2 \Vert \nabla u \Vert ^{2}+2 \Vert u \Vert _{L^{4}}^{4}+H^{2} \Vert \nabla n \Vert ^{2}\leq \frac{1}{2} \Vert n \Vert ^{2}-\frac{1}{2} \Vert u \Vert _{L^{4}}^{4}. $$
(97)
Combining \(G(u,n)<0\), (96) and (97), we have \(Z^{\prime\prime}(t)>0\). So \(Z^{\prime}(t)\) is strictly increasing for all \(t>0\). Thus if we choose \((u_{0}, u_{1})\) properly such that \(Z^{\prime }(0)=\int_{\Omega}(u_{0}\overline{u_{1}}+u_{1}\overline{u_{0}})\,dx\geq0\), then for all \(t>0\), \(Z^{\prime}(t)>0\). Therefore, \(Z(t)= \Vert u(x,t) \Vert ^{2}\) is strictly increasing for all \(t>0\). Without loss of generality and for simplicity, we omit the condition in the present paper and assume that if \(Y(t)\) is increasing for all \(t>t_{1}\), then \(\Vert u \Vert ^{2}\) is increasing for all \(t>t_{1}\). □
Now we go back to the proof. First from (39), (40) and (78), one gets
$$\begin{aligned}[b] -6 \int_{\Omega}n \vert u \vert ^{2}\,dx&=-6E(u_{0},u_{1},n_{0},V_{0})+6 \Vert u_{t} \Vert ^{2}+6 \Vert u \Vert ^{2}+6 \Vert \nabla u \Vert ^{2} \\ &\quad{}+3 \Vert u \Vert _{L^{4}}^{4} +3 \Vert n \Vert ^{2}+3H^{2} \Vert \nabla n \Vert ^{2}+3 \Vert V \Vert ^{2} \end{aligned} $$
and
$$ \begin{aligned}[b] Y^{\prime\prime}(t)&=5\bigl(2 \Vert u_{t} \Vert ^{2}+ \Vert V \Vert ^{2} \bigr)+2 \Vert u \Vert ^{2}+2 \Vert \nabla u \Vert ^{2} \\ &\quad{}+H^{2} \Vert \nabla n \Vert ^{2}+ \Vert n \Vert ^{2}- \Vert u \Vert _{L^{4}}^{4}-6E(u_{0},u_{1},n_{0},V_{0}). \end{aligned} $$
(98)
Since \(E(u_{0},u_{1},n_{0},V_{0})\) is a fixed value, and by Corollary 1, the term \(2 \Vert u \Vert ^{2}-6E(u_{0},u_{1},n_{0},V_{0})\) will eventually become positive and still remain positive thereafter. Meanwhile, combining (97), we know that the quantity
$$2 \Vert u \Vert ^{2}+2 \Vert \nabla u \Vert ^{2}+H^{2} \Vert \nabla n \Vert ^{2}+ \Vert n \Vert ^{2}- \Vert u \Vert _{L^{4}}^{4}-6E(u_{0},u_{1},n_{0},V_{0}) $$
will eventually become positive and will remain positive thereafter. Thus
$$ Y^{\prime\prime}(t)\geq5\bigl(2 \Vert u_{t} \Vert ^{2}+ \Vert V \Vert ^{2}\bigr). $$
(99)
From (76), (77) and (99), we obtain
$$ Y^{\prime}(t)=2 \int_{\Omega}(u_{t}\overline{u}+u\overline{u}_{t})\,dx+2 \int _{\Omega}gg_{t}\,dx\leq4 \Vert u_{t} \Vert \Vert u \Vert +2 \Vert g \Vert \Vert V \Vert $$
(100)
and
$$ \begin{aligned}[b] \bigl(Y^{\prime}(t) \bigr)^{2}&\leq 16 \Vert u_{t} \Vert ^{2} \Vert u \Vert ^{2}+16 \Vert u_{t} \Vert \Vert u \Vert \Vert g \Vert \Vert V \Vert +4 \Vert g \Vert ^{2} \Vert V \Vert ^{2} \\ &=4\bigl(4 \Vert u_{t} \Vert ^{2} \Vert u \Vert ^{2}+\bigl(2 \Vert u_{t} \Vert \Vert g \Vert \bigr) \cdot\bigl(2 \Vert u \Vert \Vert V \Vert \bigr)+ \Vert g \Vert ^{2} \Vert V \Vert ^{2}\bigr) \\ &\leq4\bigl(4 \Vert u_{t} \Vert ^{2} \Vert u \Vert ^{2}+2 \Vert u_{t} \Vert ^{2} \Vert g \Vert ^{2}+2 \Vert u \Vert ^{2} \Vert V \Vert ^{2}+ \Vert g \Vert ^{2} \Vert V \Vert ^{2} \bigr) \\ &=\frac{4}{5}\times5\bigl(2 \Vert u \Vert ^{2}+ \Vert g \Vert ^{2}\bigr) \bigl(2 \Vert u_{t} \Vert ^{2}+ \Vert V \Vert ^{2}\bigr) \\ &\leq\frac{4}{5}Y(t)Y^{\prime\prime}(t), \end{aligned} $$
(101)
which implies
$$ Y(t)Y^{\prime\prime}(t)-\frac{5}{4}\bigl(Y^{\prime}(t) \bigr)^{2}\geq0. $$
(102)
On the other hand, we have
$$ \bigl(Y^{-\frac{1}{4}}(t)\bigr)^{\prime\prime}=- \frac{1}{4}Y^{-\frac {9}{4}}(t) \biggl(Y(t)Y^{\prime\prime}(t)- \frac{5}{4}\bigl(Y^{\prime}(t)\bigr)^{2}\biggr). $$
(103)
Thus from (102) we have
$$ \bigl(Y^{-\frac{1}{4}}(t)\bigr)^{\prime\prime}\leq0. $$
(104)
Therefore \(F^{-\frac{1}{4}}(t)\) is convex for sufficiently large t, and \(Y(t)\geq0\), thus there exists a finite time \(T^{\star}\) such that
$$\lim_{t\rightarrow T^{\star}}Y^{-\frac{1}{4}}(t)=0, $$
which implies
$$ \lim_{t\rightarrow T^{\star}}Y(t)=\lim_{t\rightarrow T^{\star}} \bigl(2 \Vert u \Vert ^{2}+ \Vert g \Vert ^{2}\bigr)= \infty. $$
(105)
Thus one gets \(T<\infty\) and
$$ \lim_{t\rightarrow T}\bigl( \Vert u \Vert ^{2}_{L_{\mathrm{per}}^{2}(\Omega )}+ \Vert n \Vert ^{2}_{\dot{H}_{\mathrm{per}}^{-1}(\Omega)} \bigr)=\infty. $$
(106)
The proof (1) of Theorem 2 is completed.
(2) Now we turn to proving (2) of Theorem 2. When \(G(u_{0},n_{0})>0\), (73) and Proposition 4.3 imply that \((u(x,t), u_{t}(x,t),n(x,t),V(x,t))\in\mathcal{S}_{1}\) and \(E(u, u_{t}, n,V)< d_{\Phi}\). There will be two cases to be discussed: \(\int_{\Omega}n \vert u \vert ^{2}\,dx\geq0\) and \(\int_{\Omega}n \vert u \vert ^{2}\,dx<0\), respectively.
For case (i) \(\int_{\Omega}n \vert u \vert ^{2}\,dx\geq0\), from (39), (40) and (73) we have
$$ \begin{aligned}[b] & \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac{1}{2} \Vert u \Vert _{L^{4}}^{4}+ \frac {1}{2} \Vert n \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+\frac{1}{2} \Vert V \Vert ^{2} \\ &\quad \leq E(u_{0},u_{1},n_{0},V_{0}) < d_{\Phi}. \end{aligned} $$
(107)
Thus we established the a priori estimates of \(u_{t}\) in \(L^{2}(\Omega)\), u in \(H^{1}(\Omega)\), n in \(H^{1}(\Omega)\) and V in \(L^{2}(\Omega)\) for \(t\in[0, T)\). Thus it must be \(T=\infty\). Then the solution \((u,n,V)\) of the periodic initial value problem (1)-(5) exists globally on \(t\in[0,\infty)\). Furthermore, (107) implies estimate (71).
For case (ii) \(\int_{\Omega}n \vert u \vert ^{2}\,dx<0\). First, from Hölder’s inequality, we have
$$ - \int_{\Omega}n \vert u \vert ^{2}\,dx\leq \frac{1}{2} \Vert n \Vert ^{2}+\frac{1}{2} \Vert u \Vert _{L^{4}}^{4}. $$
(108)
Thus we can get
$$ \begin{aligned}[b] E(u,u_{t},n,V)&= \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac{1}{2} \Vert u \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \frac{1}{2} \Vert V \Vert ^{2}-\biggl(- \int_{\Omega}n \vert u \vert ^{2}\,dx\biggr) \\ &\geq \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+\frac{1}{2} \Vert V \Vert ^{2}, \end{aligned} $$
(109)
which leads to
$$ \begin{aligned}[b] \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \frac{1}{2} \Vert V \Vert ^{2}&\leq E(u,u_{t},n,V) \\ &=E(u_{0},u_{1},n_{0},V_{0})< d_{\Phi}. \end{aligned} $$
(110)
Meanwhile, from \(G(u,n)>0\) and \(F(u,n)< d_{\Phi}\), we obtain
$$ \int_{\Omega}n \vert u \vert ^{2}\,dx>- \frac{2}{3} \Vert u \Vert ^{2}-\frac{2}{3} \Vert \nabla u \Vert ^{2}-\frac{2}{3} \Vert u \Vert _{L^{4}}^{4}-\frac{1}{3} \Vert n \Vert ^{2}-\frac {1}{3}H^{2} \Vert \nabla n \Vert ^{2} $$
(111)
and
$$ \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+\frac {1}{2} \Vert u \Vert _{L^{4}}^{4}+ \frac{1}{2} \Vert n \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla n \Vert ^{2}+ \int_{\Omega}n \vert u \vert ^{2}\,dx< d_{\Phi}. $$
(112)
Thus from (111) and (112) there holds
$$ \frac{1}{3} \Vert u \Vert ^{2}+ \frac{1}{3} \Vert \nabla u \Vert ^{2}+\frac{1}{6} \Vert n \Vert ^{2}+\frac {1}{6}H^{2} \Vert \nabla n \Vert ^{2}< d_{\Phi}+\frac{1}{6} \Vert u \Vert _{L^{4}}^{4}. $$
(113)
According to inequality (10) in Lemma 1 and (110), we have
$$ \Vert u \Vert _{L^{4}}^{4}\leq C^{\star}\bigl( \Vert u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}\bigr)^{2}\leq C^{\star}d^{2}_{\Phi}, $$
(114)
where \(C^{\star}\) is a positive constant which satisfies the Gagliardo-Nirenberg inequality. Combining (113) and (114) yields
$$ \frac{1}{3} \Vert u \Vert ^{2}+ \frac{1}{3} \Vert \nabla u \Vert ^{2}+\frac{1}{6} \Vert n \Vert ^{2}+\frac {1}{6}H^{2} \Vert \nabla n \Vert ^{2}< d_{\Phi}+\frac{1}{6}C^{\star}d^{2}_{\Phi}. $$
(115)
Therefore, by (110) and (115), we have
$$ \begin{aligned}[b] & \Vert u_{t} \Vert ^{2}+\frac{4}{3} \Vert u \Vert ^{2}+ \frac{4}{3} \Vert \nabla u \Vert ^{2}+\frac{1}{6} \Vert n \Vert ^{2}+\frac{2}{3}H^{2} \Vert \nabla n \Vert ^{2}+\frac {1}{2} \Vert V \Vert ^{2} \\ &\quad \leq2d_{\Phi}+\frac{1}{6}C^{\star}d^{2}_{\Phi}. \end{aligned} $$
(116)
Similarly, we have established the a priori estimates of \(u_{t}\) in \(L^{2}(\Omega)\), u in \(H^{1}(\Omega)\), n in \(H^{1}(\Omega)\) and V in \(L^{2}(\Omega)\) for \(t\in[0, T)\). Thus it must be \(T=\infty\). Then the solution \((u,n,V)\) of the periodic initial value problem (1)-(5) exists globally on \(t\in[0,\infty)\). Furthermore, (116) implies estimate (72).
From the discussions of case (i) and case (ii), we complete the proof of (2) of Theorem 2. In sum, the proof of Theorem 2 is completed. □
Based on the results in Theorem 2, we give two more specific conditions of how small the initial data are for the solutions to exist globally.
Theorem 3
Small initial values criterions
Suppose that
\((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\)
and satisfy
$$ (1)\quad \textstyle\begin{cases} \int_{\Omega}n_{0} \vert u_{0} \vert ^{2}< 0, \\ \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+\frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4}+\frac{1}{2} \Vert n_{0} \Vert ^{2}\\ \quad{} +\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2}< d_{\Phi}, \end{cases} $$
(117)
or
$$ (2)\quad \textstyle\begin{cases} \int_{\Omega}n_{0} \vert u_{0} \vert ^{2}>0, \\ \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \Vert u_{0} \Vert _{L^{4}}^{4}+ \Vert n_{0} \Vert ^{2}\\ \quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2}< d_{\Phi}. \end{cases} $$
(118)
Then the solution
\((u(x,t), n(x,t), V(x,t))\)
of the periodic initial value problem (1)-(5) exists globally.
Proof
(1) If \(\int_{\Omega}n_{0} \vert u_{0} \vert ^{2}<0\), and from (117), we have
$$ \begin{aligned}[b] E(u_{0},u_{1},n_{0},V_{0})&= \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4}+ \frac{1}{2} \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2}+ \int_{\Omega}n_{0} \vert u_{0} \vert ^{2}\,dx \\ &\leq \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4}+ \frac{1}{2} \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2} \\ &< d_{\Phi}. \end{aligned} $$
(119)
Next we will prove \(G(u_{0},n_{0})>0\). If it is not true, there holds \(G(u_{0},n_{0})\leq0\). Similar to Proposition 4.2, there exists \(0<\theta\leq 1\) such that \(G(\theta^{\frac{1}{2}}u_{0},\theta n_{0})= 0\) and \((\theta ^{\frac{1}{2}}u_{0},\theta n_{0})\neq 0\). Since \((u_{0},n_{0})\neq(0,0)\), so \((\theta^{\frac{1}{2}}u_{0},\theta n_{0})\in\Phi\) and
$$ F\bigl(\theta^{\frac{1}{2}}u_{0},\theta n_{0}\bigr)\geq d_{\Phi}. $$
(120)
Meanwhile, for \(0<\theta\leq1\), \((\theta^{\frac{1}{2}}u_{0},u_{1},\theta n_{0},V_{0})\) satisfy condition (117), so we arrive at
$$ \begin{aligned}[b] F\bigl(\theta^{\frac{1}{2}}u_{0}, \theta n_{0}\bigr)&=\theta \Vert u_{1} \Vert ^{2}+\theta \Vert u_{0} \Vert ^{2}+\theta \Vert \nabla u_{0} \Vert ^{2}+\frac{1}{2} \theta^{2} \Vert u_{0} \Vert _{L^{4}}^{4}+ \frac {1}{2}\theta \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2}\theta \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2}\theta \Vert V_{0} \Vert ^{2}+\theta^{2} \int_{\Omega}n_{0} \vert u_{0} \vert ^{2}\,dx \\ &\leq \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4}+ \frac{1}{2} \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2} \\ &< d_{\Phi}, \end{aligned} $$
(121)
which is contradictory to (120). So there must be \(G(u_{0},n_{0})>0\). Combining (119) and Theorem 2, we obtain the first result.
(2) If \(\int_{\Omega}n_{0} \vert u_{0} \vert ^{2}>0\), and from (118), we have
$$\begin{aligned} E(u_{0},u_{1},n_{0},V_{0})&= \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4}+ \frac{1}{2} \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2}+ \int_{\Omega}n_{0} \vert u_{0} \vert ^{2}\,dx \\ &\leq \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4}+ \frac{1}{2} \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2}+\frac{1}{2} \Vert n_{0} \Vert ^{2}+ \frac{1}{2} \Vert u_{0} \Vert _{L^{4}}^{4} \\ &\leq \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \Vert u_{0} \Vert _{L^{4}}^{4}+ \Vert n_{0} \Vert ^{2} \\ &\quad{}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2} \\ &< d_{\Phi}. \end{aligned}$$
(122)
While from (44), there holds
$$ \begin{aligned}[b] G(u_{0},n_{0})&=2 \Vert u_{0} \Vert ^{2}+2 \Vert \nabla u_{0} \Vert ^{2}+2 \Vert u_{0} \Vert _{L^{4}}^{4}+ \Vert n_{0} \Vert ^{2} \\ &\quad{}+H^{2} \Vert \nabla n_{0} \Vert ^{2}+3 \int_{\Omega}n_{0} \vert u_{0} \vert ^{2}\,dx \\ &\geq0. \end{aligned} $$
(123)
Thus combining (122), (123) and Theorem 2, we obtain the second result. In sum, the proof of Theorem 3 is completed. □
Remark 1
From Theorem 3, one could see that if \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\), no matter \(\int_{\Omega}n_{0} \vert u_{0} \vert ^{2}<0\) or \(\int _{\Omega}n_{0} \vert u_{0} \vert ^{2}>0\), the solution \((u(x,t), n(x,t), V(x,t))\) of the periodic initial value problem (1)-(5) still exists globally only if
$$ \Vert u_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}+ \Vert u_{0} \Vert _{L^{4}}^{4}+ \Vert n_{0} \Vert ^{2}+\frac{H^{2}}{2} \Vert \nabla n_{0} \Vert ^{2}+\frac {1}{2} \Vert V_{0} \Vert ^{2}< d_{\Phi}. $$
(124)