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Asymptotic stability of standing waves for the coupled nonlinear Schrödinger system
- Yang Liao1Email author,
- Quanbao Sun2,
- Xin Zhao3 and
- Ming Cheng4
Boundary Value Problems20152015:3
https://doi.org/10.1186/s13661-014-0266-4
© Liao et al.; licensee Springer 2014
Received: 21 October 2014
Accepted: 8 December 2014
Published: 10 January 2015
Abstract
This paper considers the asymptotic stability of standing waves of a coupled nonlinear Schrödinger system with an attractive potential. Meanwhile, the existence of a center manifold is obtained.
Keywords
asymptotic stability center manifold Schrödinger equation1 Introduction and main results
The two-component system of time-dependent nonlinear Schrödinger equations arises in the binary mixture of Bose-Einstein condensates with two different hyperfine states (see [1]):
where \(u(x,t)\) and \(v(x,t)\) denote the wave functions, ħ is the Planck constant divided by 2π, m is atom mass, \(V_{i}\) is the trapping potential for the ith hyperfine state, \(a_{i}\geq0\), \(i=1,2,3,4\), is the associated axial frequency.
$$\begin{aligned} &i\hbar\partial_{t}u+\frac{\hbar^{2}}{2m}\triangle u-V_{1}(x)u-a_{1}|u|^{2}u-a_{2}|v|^{2}u=0, \\ &i\hbar\partial_{t}v+\frac{\hbar^{2}}{2m}\triangle v-V_{2}(x)v-a_{3}|v|^{2}v-a_{4}|u|^{2}v=0, \end{aligned}$$
By rescaling and some simple assumptions, the above system could be viewed as the coupled nonlinear Schrödinger system:
with initial data
In the present paper, we assume that \(a_{2}=a_{4}\).
$$\begin{aligned} &i\partial_{t}u+\triangle u-V_{1}(x)u-a_{1}|u|^{2}u-a_{2}|v|^{2}u=0, \end{aligned}$$
(1.1)
$$\begin{aligned} &i\partial_{t}v+\triangle v-V_{2}(x)v-a_{3}|v|^{2}v-a_{4}|u|^{2}v=0, \quad x\in\mathbb{R}^{2},t>0, \end{aligned}$$
(1.2)
$$\begin{aligned} u(0,x)=u_{0}(x),\qquad v(0,x)=v_{0}(x). \end{aligned}$$
In the last decades, there has been a lot of interest in the study of localized modes in the coupled nonlinear Schrödinger system. In particular, for the existence of standing wave (periodic in time and exponentially localized in space) for various coupled nonlinear Schrödinger system, one may refer to [2–6] for more details. As we know, there are few results on the asymptotic stability of standing waves for the coupled nonlinear Schrödinger system (1.1)-(1.2).
Assume that the standing wave of system (1.1)-(1.2) has the form as follows:
where \(E_{1},E_{2}\in\mathbb{R}\) and \(\psi_{E_{1}},\phi_{E_{2}}\in\mathbf {H}^{2}(\mathbb{R}^{2})\) satisfy the time independent equations:
$$\begin{aligned} u_{E_{1}}(t,x)=e^{-iE_{1}t}\psi_{E_{1}}(x),\qquad v_{E_{2}}(t,x)=e^{-iE_{2}t}\phi_{E_{2}}(x), \end{aligned}$$
$$\begin{aligned} &[-\triangle+V_{1}]\psi_{E_{1}}+a_{1}| \psi_{E_{1}}|^{2}\psi_{E_{1}}+a_{2}|\phi _{E_{2}}|^{2}\psi_{E_{2}}=E_{1} \psi_{E_{1}}, \end{aligned}$$
(1.3)
$$\begin{aligned} &[-\triangle+V_{2}]\phi_{E_{2}}+a_{3}| \phi_{E_{2}}|^{2}\phi_{E_{2}}+a_{4}|\psi _{E_{1}}|^{2}\phi_{E_{2}}=E_{2} \phi_{E_{2}}. \end{aligned}$$
(1.4)
Here, we give some notations. Assume that \(\langle x\rangle=(1+|x|^{2})^{\frac {1}{2}}\). For \(\sigma\in\mathbb{R}\), the Sobolev space \(L_{\sigma }^{2}\) with the norm \(\|f(x)\|_{L^{2}_{\sigma}}=\|\langle x\rangle ^{\sigma }f(x)\|_{L^{2}}\) denotes the space of functions \(f(x)\) such that \(\langle x\rangle ^{\sigma}f(x)\) are square integrable. \(C_{a,b,\ldots}\) denotes a constant depending on \(a,b,\ldots\) .
Define
and
Then the system (1.3)-(1.4) can be rewritten as
It is obvious that \(X\equiv0\) is a trivial solution of (1.5). In order to make a bifurcation with a nontrivial, one parameter family of solutions, we need (iii) in the following assumptions:
$$\begin{aligned} &X=(\psi,\phi)^{T}, \\ &F(\psi_{E_{1}},\phi_{E_{2}})=\bigl(a_{1}| \psi_{E_{1}}|^{2}\psi _{E_{1}}+a_{2}| \phi_{E_{2}}|^{2}\psi_{E_{2}},a_{3}| \phi_{E_{2}}|^{2}\phi_{E_{2}}+a_{4}|\psi _{E_{1}}|^{2}\phi_{E_{2}}\bigr)^{T}, \\ &\mathcal{A}= \left ( \begin{array}{@{}c@{\quad}c@{}} -\triangle+V_{1}& 0 \\ 0 & -\triangle+V_{2} \end{array} \right ), \end{aligned}$$
$$\begin{aligned} \mathcal{E}= \left ( \begin{array}{@{}c@{\quad}c@{}} E_{1}& 0 \\ 0& E_{2} \end{array} \right ). \end{aligned}$$
$$\begin{aligned} \mathcal{A}X-\mathcal{E}X=F. \end{aligned}$$
(1.5)
- (A)Assume that:
- (i)There exist C and α such thatNote that potential \(V_{1}\), \(V_{2}\) could be different in our paper.$$\begin{aligned} \bigl|V_{i}(x)\bigr|\leq C\langle x\rangle ^{-\alpha},\quad i=1,2, \forall x\in\mathbb{R}^{2}. \end{aligned}$$
- (ii)
0 is a regular point of the spectrum of the linear operator \(-\triangle+V_{i}\) acting on \(\mathbf{L}^{2}\), \(i=1,2\).
- (iii)
For \(i=1,2\), \(-\triangle+V_{i}\) acting on \(\mathbf{L}^{2}\) has exactly two negative eigenvalues \(\omega_{i}\) with corresponding normalized eigenvectors \(\psi_{0}\) and \(\phi_{0}\), respectively. It is well known that \(\psi_{0}\) and \(\phi_{0}\) are exponentially decaying as \(|x|\rightarrow \infty\), and they could be chosen strictly positive.
- (i)
The main result of this paper is the following.
Theorem 1.1
Assume that hypothesis (A) holds. Then there exists an
\(\varepsilon_{0}>0\)
such that for any fixed
\(\sigma>1\)
and the initial data
\(u_{0}\), \(v_{0}\)
satisfying
the initial value problem (1.1)-(1.2) is globally well-posed in
\(\mathbf{H}^{1}\times\mathbf{H}^{1}\).
$$\begin{aligned} &\max\bigl\{ \|u_{0}\|_{\mathbf{L}_{\sigma}^{2}},\|u_{0}\|_{\mathbf{H}^{1}}\bigr\} \leq \varepsilon_{0}, \\ &\max\bigl\{ \|v_{0}\|_{\mathbf{L}_{\sigma}^{2}},\|v_{0}\|_{\mathbf{H}^{1}}\bigr\} \leq \varepsilon_{0}, \end{aligned}$$
Moreover, the solution of the initial value problem (1.1)-(1.2) has the form
with
where
\(2\leq p\leq p_{0}\).
$$\begin{aligned} &u(t,x)=a(t)\psi_{0}(x)+h_{1}\bigl(a(t) \bigr)+r_{1}(t,x), \end{aligned}$$
(1.6)
$$\begin{aligned} &v(t,x)=b(t)\phi_{0}(x)+h_{2}\bigl(b(t) \bigr)+r_{2}(t,x), \end{aligned}$$
(1.7)
$$\begin{aligned} &\bigl\| r_{i}(t)\bigr\| _{\mathbf{L}_{-\sigma}^{2}}\leq C_{p_{0}}\varepsilon _{0}\bigl(1+|t|\bigr)^{2p_{0}^{-1}-1}, \\ &\bigl\| r_{i}(t)\bigr\| _{\mathbf{L}^{p}}\leq C_{p,p_{0}}\varepsilon _{0}\bigl(1+|t|\bigr)^{2p_{0}^{-1}-1}\log^{\frac {1-2p_{0}^{-1}}{1-2p_{0}^{-1}}}\bigl(2+|t|\bigr), \quad i=1,2, \end{aligned}$$
This paper is organized as follows. In the next section, we show the existence of a center manifold of the coupled nonlinear Schrödinger system (1.1)-(1.2). We investigate the asymptotic stability of the system (1.1)-(1.2) in Section 3.
2 The center manifold
This section is devoted to the proof of the existence of a center manifold. The method of constructing the center manifold is based on the standard bifurcation argument in Banach spaces for (1.3)-(1.4) at \((\omega_{1},\omega_{2})^{T}\) (see [7]). Since the spectrum of the operator \(-\triangle+V_{i}\) has a discrete and continuous part, we follow the idea of [8, 9] and decompose the solution of (1.5) in its projection onto the discrete and continuous part:
where \(P_{c}\) denotes the projector onto the continuous spectrum of \(-\triangle+V_{i}\) in \(L^{2}\).
$$\begin{aligned} &\psi_{E_{1}}=a\psi_{0}+h_{1},\qquad a=\langle \psi_{0},\psi_{E_{1}}\rangle, \qquad h_{1}=P_{c} \psi_{E_{1}}, \\ &\phi_{E_{1}}=b\phi_{0}+h_{2}, \qquad b=\langle \phi_{0},\phi_{E_{2}}\rangle, \qquad h_{2}=P_{c} \phi_{E_{2}}, \end{aligned}$$
Now, projecting (1.5) onto \(X_{0}=(\psi_{0},\phi_{0})^{T}\) and its orthogonal complement, i.e. range \(P_{c}\), we have
where
and
Now we have the following result on the center manifold of the system (1.1)-(1.2):
$$\begin{aligned} &X=(\mathcal{A}-\mathcal{E})^{-1}P_{c}F_{1}(a,b,X), \end{aligned}$$
(2.1)
$$\begin{aligned} &\mathcal{E}_{0}-\mathcal{E}=-\mathcal{B}^{-1}F_{2}(a,b,X), \end{aligned}$$
(2.2)
$$\begin{aligned} X=\left ( \begin{array}{@{}c@{}} h_{1} \\ h_{2} \end{array} \right ),\qquad \mathcal{B}= \left ( \begin{array}{@{}c@{\quad}c@{}} a& 0 \\ 0& b \end{array} \right ),\qquad \mathcal{E}_{0}= \left ( \begin{array}{@{}c@{\quad}c@{}} \omega_{1}& 0 \\ 0& \omega_{2} \end{array} \right ),\qquad \mathcal{E}= \left ( \begin{array}{@{}c@{\quad}c@{}} E_{1}& 0 \\ 0& E_{2} \end{array} \right ), \end{aligned}$$
$$\begin{aligned}& F_{1}(a,b,X)=F(a\psi_{0}+h_{1},b \phi_{0}+h_{2}), \\& \begin{aligned}[b] F_{2}(a,b,X)&= \bigl( \bigl\langle \psi_{0},a_{1}|a\psi_{0}+h_{1}|^{2}(a \psi _{0}+h_{1})+a_{2}|b\phi_{0}+h_{2}|^{2}(a \psi_{0}+h_{1}) \bigr\rangle , \\ &{} \bigl\langle \phi_{0},a_{1}|b\phi_{0}+h_{2}|^{2}(b \phi_{0}+h_{2})+a_{2}|a\psi _{0}+h_{1}|^{2}(b \phi_{0}+h_{2}) \bigr\rangle \bigr)^{T}. \end{aligned} \end{aligned}$$
Theorem 2.1
Let
\(\delta_{E_{1}},\delta_{E_{2}},\delta>0\). Then for
\(|E_{1}-\omega _{1}|<\delta_{E_{1}}\), \(|E_{2}-\omega_{2}|<\delta_{E_{2}}\), \(\|\psi_{E_{1}}\| _{\mathbf{L}^{2}_{\sigma}\cap\mathbf{H}^{2}}<\delta\), and
\(\|\phi_{E_{2}}\| _{\mathbf{L}^{2}_{\sigma}\cap\mathbf{H}^{2}}<\delta\), there exist
\(\mathbf{C}^{1}\)
functions
such that the eigenvalue problem (1.3)-(1.4) has a unique solution up to multiplication with
\(e^{i\theta}\), \(\theta\in(0,2\pi)\), which can be represented as
$$\begin{aligned} &h_{1}:\bigl\{ a\in\mathbb{C}:|a|<\delta\bigr\} \mapsto\mathbf{L}^{2}_{\sigma} \cap \mathbf{H}^{2}, \\ &h_{2}:\bigl\{ b\in\mathbb{C}:|b|<\delta\bigr\} \mapsto\mathbf{L}^{2}_{\sigma} \cap \mathbf{H}^{2} \end{aligned}$$
$$\begin{aligned} &\psi_{E_{1}}=a\psi_{0}+h_{1}(a), \qquad \langle \psi_{0},h_{1}\rangle=0, \qquad |a|<\delta, \\ &\phi_{E_{2}}=b\phi_{0}+h_{2}(b), \qquad\langle \phi_{0},h_{2}\rangle=0, \qquad |b|<\delta. \end{aligned}$$
Proof
We follow the idea of [8, 9] and use the implicit function theorem to solve (2.1)-(2.2). We define a \(\mathbf{C}^{1}\) function \(\mathcal{F}:(-\infty,0)^{2}\times \mathbb{C}^{2}\times(\mathbf{L}^{2}_{\sigma}\cap\mathbf{H}^{2})^{2}\) as
$$\begin{aligned} \mathcal{F}(\mathcal{E},a,b,X)=X+(\mathcal{A}-\mathcal{E})^{-1}P_{c}F_{1}. \end{aligned}$$
It is easy to see that
Therefore, from the implicit function theorem, we know that (2.1) has a unique solution \(X=\bar{X}(\mathcal{E},a,b)\) with \(\|X\| _{(\mathbf{L}^{2}_{\sigma}\cap\mathbf{H}^{2})^{2}}<\delta_{1}\), where \(\bar{X}(\mathcal{E},a,b)\) is a \(\mathbf{C}^{1}\) function from \((\omega _{1}-\delta_{1},\omega_{1}+\delta_{1})\times(\omega_{2}-\delta_{1},\omega _{2}+\delta_{1})\times\{(a,b)\in\mathbb{C}^{2}:|a|,|b|<\delta_{1}\}\) to \((\mathbf{L}^{2}_{\sigma}\cap\mathbf{H}^{2})^{2}\) for \(\delta_{1}>0\). By direct computation, we know that \((e^{i\theta}a,e^{i\theta}b,e^{i\theta}X)\) is also a solution of (2.1) for \(\theta\in(0,2\pi)\). By uniqueness, we have \(\bar{X}(\mathcal{E},a,b)=\tilde{\mathcal{A}}\bar{X}(\mathcal {E},|a|,|b|)\), where \(\tilde{\mathcal{A}}=\operatorname{diag} (\frac{a}{|a|},\frac {b}{|b|} )\).
$$\begin{aligned} \mathcal{F}(\mathcal{E}_{0},0,0,0)=0,\qquad D_{X} \mathcal{F}(\mathcal{E}_{0},0,0,0)=I. \end{aligned}$$
Substituting \(X=\bar{X}(\mathcal{E},a,b)\) into (2.2). From \(\bar {X}(\mathcal{E},a,b)=\tilde{\mathcal{A}}\bar{X}(\mathcal{E},|a|,|b|)\), we have
where
Define a \(\mathbf{C}^{1}\) function \(\mathcal{G}\) on \((\omega_{1}-\delta _{1},\omega_{1}+\delta_{1})\times(\omega_{2}-\delta_{1},\omega_{2}+\delta _{1})\times(-\delta_{1},\delta_{1})^{2}\) as follows:
We can see that
From the implicit function theorem, we find that (2.2) has a unique solution \(\mathcal{E}=\tilde{\mathcal{E}}(|a|,|b|)\) with \(X=\bar {X}(\mathcal{E},a,b)\), where \(\tilde{\mathcal{E}}(|a|,|b|)\) is a \(\mathbf{C}^{1}\) function from \((-\delta,\delta)\times(-\delta,\delta)\) to \((\omega_{1}-\delta _{E_{1}},\omega_{1}+\delta_{E_{1}})\times(\omega_{2}-\delta_{E_{2}},\omega _{2}+\delta_{E_{2}})\) for \(0<\delta\leq\delta_{1}\) and \(0<\delta _{E_{1}},\delta_{E_{2}}\leq\delta_{1}\). □
$$\begin{aligned} \mathcal{E}_{0}-\mathcal{E}=-\bar{\mathcal{B}}^{-1}F_{2} \bigl(|a|,|b|,\bar {X}\bigl(\mathcal{E},|a|,|b|\bigr)\bigr), \end{aligned}$$
$$\begin{aligned} \bar{\mathcal{B}}= \left ( \begin{array}{@{}c@{\quad}c} |a|& 0 \\ 0& |b| \end{array} \right ). \end{aligned}$$
$$\begin{aligned} \mathcal{G}(\mathcal{E},a,b)=\mathcal{E}_{0}-\mathcal{E}+\bar{ \mathcal {B}}^{-1}F_{2}\bigl(|a|,|b|,\bar{X}\bigl( \mathcal{E},|a|,|b|\bigr)\bigr). \end{aligned}$$
$$\begin{aligned} \mathcal{G}(\mathcal{E}_{0},0,0)=0,\qquad D_{\mathcal{E}} \mathcal{F}(\mathcal {E}_{0},0,0)=-I. \end{aligned}$$
3 Proof of Theorem 1.1
We consider the asymptotic stability of the coupled Schrödinger system (1.1)-(1.2) in this section. The global well-posedness for the system (1.1)-(1.2) with small initial data is obtained by Cazenave [10].
Define
By Theorem 2.1, we can define \(h_{1}(a(t))\) and \(h_{2}(b(t))\in\mathbb{C}\) by choosing \(\varepsilon_{0}<\delta\). Therefore,
We can see that the solution is described by the scalar \(a(t),b(t)\in \mathbb{C}\) and \(r_{1}(t),r_{2}(t)\in\mathbf{C}(\mathbb{R},\mathbf{H}^{1})\). More precisely, from (1.6)-(1.7), we have
Note that \(h_{1}\), \(h_{2}\), and \(Dh_{1}\), \(Dh_{2}\) have range orthogonal to \(\psi_{0}\), \(\phi_{0}\). \(r_{1}\), \(r_{2}\) and \(\frac {dr_{1}}{dt}\), \(\frac{dr_{2}}{dt}\) are orthogonal to \(\psi_{0}\), \(\phi_{0}\), respectively. Therefore, from (3.1)-(3.2), we get
Using \(Dh_{1}|_{a}(iE_{1}a)=-E_{1}ih_{1}(a)\) and \(Dh_{2}|_{b}(iE_{2}b)=-E_{2}ih_{2}(b)\), we have
The linear part of (3.5)-(3.6) is
Define the operator \(\mathcal{S}(t,s)Y\) as the solution of the linear equation (3.7)-(3.8):
The following lemma can be obtained by a small modification of the proof of Theorem 4.1 in [8], so we omit the proof.
$$\begin{aligned} a(t)=\langle\psi_{0},u\rangle,\qquad b(t)=\langle \phi_{0},v\rangle,\quad \mbox{for }t\in \mathbb{R}. \end{aligned}$$
$$\begin{aligned} &r_{1}(t)=u(t)-h_{1}\bigl(a(t)\bigr)-a(t) \psi_{0},\qquad \bigl\langle \psi_{0},r_{1}(t)\bigr\rangle \equiv 0, \\ &r_{2}(t)=v(t)-h_{2}\bigl(b(t)\bigr)-b(t) \phi_{0},\qquad \bigl\langle \phi_{0},r_{2}(t)\bigr\rangle \equiv0. \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] &i\frac{\mathrm{d}a}{\mathrm{d}t}\psi_{0}+iDh_{1}\Big|_{a} \frac{\mathrm {d}a}{\mathrm{d}t}+i\frac{\mathrm{d}r_{1}}{\mathrm{d}t} \\ &\quad=E_{1}a\psi_{0}+E_{1}h_{1}(a)+(- \triangle+V_{1})r_{1}+a_{1}\bigl(2| \psi_{E_{1}}|^{2}r_{1}+\psi _{E_{1}}^{2} \bar{r}_{1} \\ &\qquad{}+2\psi_{E_{1}}|r_{1}|^{2}+ \psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr)+a_{2}\bigl(\phi_{E_{2}}\psi _{E_{1}} \bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &\qquad{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+ \psi_{E_{1}}\bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2} \psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr), \end{aligned} \end{aligned}$$
(3.1)
$$\begin{aligned}& \begin{aligned}[b] &i\frac{\mathrm{d}b}{\mathrm{d}t}\phi_{0}+iDh_{2}\Big|_{b} \frac{\mathrm {d}b}{\mathrm{d}t}+i\frac{\mathrm{d}r_{2}}{\mathrm{d}t} \\ &\quad=E_{2}b\phi_{0}+E_{2}h_{2}(b)+(- \triangle+V_{3})r_{2}+a_{3}\bigl(2| \phi_{E_{2}}|^{2}r_{2}+\phi _{E_{2}}^{2} \bar{r}_{2} \\ &\qquad{}+2\phi_{E_{2}}|r_{2}|^{2}+ \phi_{E_{2}}r_{2}^{2}+|r_{2}|^{2}r_{2} \bigr)+a_{4}\bigl(\phi_{E_{2}}\psi _{E_{1}} \bar{r}_{1}+|\psi_{E_{1}}|^{2}r_{2} \\ &\qquad{}+\psi_{E_{1}}r_{2}\bar{r}_{1}+ \phi_{E_{2}}\bar{\psi}_{E_{1}}r_{1}+|r_{1}|^{2} \phi _{E_{2}}+\bar{\psi}_{E_{1}}r_{1}r_{2}+|r_{1}|^{2}r_{2} \bigr). \end{aligned} \end{aligned}$$
(3.2)
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}a}{\mathrm{d}t}={}&E_{1}\bigl(|a|\bigr)a+a_{1} \bigl\langle \psi_{0},2|\psi _{E_{1}}|^{2}r_{1}+ \psi_{E_{1}}^{2}\bar{r}_{1}+2\psi_{E_{1}}|r_{1}|^{2} \\ &{}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr\rangle +a_{2}\bigl\langle \psi_{0},\phi_{E_{2}} \psi _{E_{1}}\bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+\psi_{E_{1}} \bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2}\psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr\rangle , \end{aligned} \end{aligned}$$
(3.3)
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}b}{\mathrm{d}t}={}&E_{2}\bigl(|b|\bigr)b+a_{3} \bigl\langle \phi_{0},2|\phi _{E_{2}}|^{2}r_{2}+ \phi_{E_{2}}^{2}\bar{r}_{2}+2\phi_{E_{2}}|r_{2}|^{2} \\ &{}+\phi_{E_{2}}r_{2}^{2}+|r_{2}|^{2}r_{2} \bigr\rangle +a_{4}\bigl\langle \phi_{0},\phi_{E_{2}} \psi _{E_{1}}\bar{r}_{1}+|\psi_{E_{1}}|^{2}r_{2} \\ &{}+\psi_{E_{1}}r_{2}\bar{r}_{1}+\phi_{E_{2}} \bar{\psi}_{E_{1}}r_{1}+|r_{1}|^{2}\phi _{E_{2}}+\bar{\psi}_{E_{1}}r_{1}r_{2}+|r_{1}|^{2}r_{2} \bigr\rangle . \end{aligned} \end{aligned}$$
(3.4)
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}r_{1}}{\mathrm{d}t}={}&(-\triangle+V_{1})r_{1}+a_{1}P_{c} \bigl(2|\psi _{E_{1}}|^{2}r_{1}+\psi_{E_{1}}^{2} \bar{r}_{1} \\ &{}+2\psi_{E_{1}}|r_{1}|^{2}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr)+a_{2}P_{c}\bigl(\phi_{E_{2}}\psi _{E_{1}}\bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+\psi_{E_{1}} \bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2}\psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr), \\ &{}-Dh_{1}|_{a}\bigl(a_{1}\bigl\langle \psi_{0},2|\psi_{E_{1}}|^{2}r_{1}+ \psi_{E_{1}}^{2}\bar {r}_{1}+2\psi_{E_{1}}|r_{1}|^{2} \\ &{}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr\rangle +a_{2}\bigl\langle \psi_{0},\phi_{E_{2}} \psi _{E_{1}}\bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+\psi_{E_{1}} \bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2}\psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr\rangle \bigr), \end{aligned} \end{aligned}$$
(3.5)
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}r_{2}}{\mathrm{d}t}={}&(-\triangle+V_{3})r_{2}+a_{3}P_{c} \bigl(2|\phi _{E_{2}}|^{2}r_{2}+\phi_{E_{2}}^{2} \bar{r}_{2} \\ &{}+2\phi_{E_{2}}|r_{2}|^{2}+\phi_{E_{2}}r_{2}^{2}+|r_{2}|^{2}r_{2} \bigr)+a_{4}P_{c}\bigl(\phi_{E_{2}}\psi _{E_{1}}\bar{r}_{1}+|\psi_{E_{1}}|^{2}r_{2} \\ &{}+\psi_{E_{1}}r_{2}\bar{r}_{1}+\phi_{E_{2}} \bar{\psi}_{E_{1}}r_{1}+|r_{1}|^{2}\phi _{E_{2}}+\bar{\psi}_{E_{1}}r_{1}r_{2}+|r_{1}|^{2}r_{2} \bigr) \\ &{}-Dh_{2}|_{b}\bigl(a_{3}\bigl\langle \phi_{0},2|\phi_{E_{2}}|^{2}r_{2}+ \phi_{E_{2}}^{2}\bar {r}_{2}+2\phi_{E_{2}}|r_{2}|^{2} \\ &{}+\phi_{E_{2}}r_{2}^{2}+|r_{2}|^{2}r_{2} \bigr\rangle +a_{4}\bigl\langle \phi_{0},\phi_{E_{2}} \psi _{E_{1}}\bar{r}_{1}+|\psi_{E_{1}}|^{2}r_{2} \\ &{}+\psi_{E_{1}}r_{2}\bar{r}_{1}+\phi_{E_{2}} \bar{\psi}_{E_{1}}r_{1}+|r_{1}|^{2}\phi _{E_{2}}+\bar{\psi}_{E_{1}}r_{1}r_{2}+|r_{1}|^{2}r_{2} \bigr\rangle \bigr). \end{aligned} \end{aligned}$$
(3.6)
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}r'_{1}}{\mathrm{d}t}={}&(-\triangle+V_{1})r'_{1}+a_{1}P_{c} \bigl(2|\psi _{E_{1}}|^{2}r'_{1}+ \psi_{E_{1}}^{2}\bar{r}'_{1}\bigr) \\ &{}+a_{2}P_{c}\bigl(\phi_{E_{2}}\psi _{E_{1}} \bar{r}'_{2}+|\phi_{E_{2}}|^{2}r'_{1}+ \psi_{E_{1}}\bar{\phi }_{E_{2}}r'_{2}\bigr) \\ &{}-Dh_{1}|_{a}\bigl\langle \psi_{0},a_{1} \bigl(2|\psi_{E_{1}}|^{2}r'_{1}+ \psi_{E_{1}}^{2}\bar{r}'_{1}\bigr) \\ &{}+a_{2}\bigl(\phi_{E_{2}}\psi_{E_{1}} \bar{r}'_{2}+|\phi _{E_{2}}|^{2}r'_{1}+ \psi_{E_{1}}\bar{\phi}_{E_{2}}r'_{2}\bigr) \bigr\rangle , \end{aligned} \end{aligned}$$
(3.7)
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}r'_{2}}{\mathrm{d}t}={}&(-\triangle+V_{3})r'_{2}+a_{3}P_{c} \bigl(2|\phi _{E_{2}}|^{2}r'_{2}+ \phi_{E_{2}}^{2}\bar{r}'_{2}\bigr) \\ &{}+a_{4}P_{c}\bigl(\phi_{E_{2}}\psi_{E_{1}} \bar{r}'_{1}+|\psi _{E_{1}}|^{2}r'_{2}+ \phi_{E_{2}}\bar{\psi}_{E_{1}}r'_{1}\bigr) \\ &{}-Dh_{2}|_{a}\bigl\langle \phi_{0},a_{3} \bigl(2|\phi_{E_{2}}|^{2}r'_{2}+ \phi_{E_{2}}^{2}\bar {r}'_{2}\bigr) \\ &{}+a_{4}\bigl(\phi_{E_{2}}\psi_{E_{1}} \bar{r}'_{1}+|\psi _{E_{1}}|^{2}r_{2}+ \phi_{E_{2}}\bar{\psi}_{E_{1}}r'_{1}\bigr) \bigr\rangle . \end{aligned} \end{aligned}$$
(3.8)
$$\begin{aligned} \mathcal{S}(t,s)Y=R' \quad \mbox{with } R'= \bigl(r'_{1},r'_{2} \bigr)^{T}. \end{aligned}$$
Lemma 3.1
There exists
\(\varepsilon>0\)
such that if
then there exist
\(C, C_{p}>0\)
with the property that for any
\(t,s\in \mathbb{R}\), we have
where
\(p\geq2\)
and
\(\frac{1}{p'}+\frac{1}{p}=1\).
$$\begin{aligned} \bigl\| \langle x\rangle^{\sigma}\psi_{E_{1}}\bigr\| _{\mathbf{H}^{2}}< \varepsilon, \qquad \bigl\| \langle x\rangle^{\sigma }\phi_{E_{2}} \bigr\| _{\mathbf{H}^{2}}<\varepsilon, \end{aligned}$$
$$\begin{aligned} &\bigl\| \mathcal{S}(t,s)\bigr\| _{\mathbf{L}^{2}_{\sigma} \times\mathbf{L}^{2}_{\sigma }\rightarrow\mathbf{L}^{2}_{-\sigma}\times\mathbf{L}^{2}_{-\sigma}}\leq C\bigl(1+|t-s|\bigr)^{-1} \log^{-2}\bigl(2+|t-s|\bigr), \end{aligned}$$
(3.9)
$$\begin{aligned} &\bigl\| \mathcal{S}(t,s)\bigr\| _{\mathbf{L}^{p'}\times\mathbf{L}^{p'}\rightarrow \mathbf{L}^{2}_{-\sigma}\times\mathbf{L}^{2}_{-\sigma}}\leq C_{p}|t-s|^{1-\frac{2}{p}}, \end{aligned}$$
(3.10)
$$\begin{aligned} &\bigl\| \mathcal{S}(t,s)\bigr\| _{\mathbf{L}^{2}_{\sigma}\times\mathbf{L}^{2}_{\sigma }\rightarrow\mathbf{L}^{p}\times\mathbf{L}^{p}}\leq C_{p}|t-s|^{1-\frac{2}{p}}, \end{aligned}$$
(3.11)
By Duhamel’s principle, it follows from (3.5)-(3.6) that
where \(R=(r_{1},r_{2})^{T}\), \(G_{1}=(g_{1},g_{2})^{T}\), \(G_{2}=(g_{3},g_{4})^{T}\), and \(DH=\operatorname{diag}(Dh_{1}|_{a},Dh_{2}|_{b})\) with
For fixed \(p\geq6\), we define the Banach space
endowed with the norm
Consider the nonlinear part in (3.12):
$$\begin{aligned} R(t)=\mathcal{S}(t,0)R(0)-i\int_{0}^{t} \mathcal{S}(t,s)P_{c}G_{1}\,ds+i\int_{0}^{t} \mathcal{S}(t,s) (DH)G_{2}\,ds, \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned}[b] g_{1}={}&a_{1}\bigl(2 \psi_{E_{1}}|r_{1}|^{2}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr) \\ &{}+a_{2}\bigl(\phi_{E_{2}}r_{1}\bar{r}_{2}+|r_{2}|^{2} \psi_{E_{1}}+\bar{\phi }_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr), \end{aligned} \end{aligned}$$
(3.13)
$$\begin{aligned}& \begin{aligned}[b] g_{2}={}&a_{3}\bigl(2 \phi_{E_{2}}|r_{2}|^{2}+\phi_{E_{2}}r_{2}^{2}+|r_{2}|^{2}r_{2} \bigr) \\ &{}+a_{4}\bigl(\psi_{E_{1}}r_{2} \bar{r}_{1}+|r_{1}|^{2}\phi_{E_{2}}+\bar{\psi }_{E_{1}}r_{1}r_{2}+|r_{1}|^{2}r_{2} \bigr), \end{aligned} \end{aligned}$$
(3.14)
$$\begin{aligned}& \begin{aligned}[b] g_{3}={}&a_{1}\bigl\langle \psi_{0},2\psi_{E_{1}}|r_{1}|^{2}+\psi _{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr\rangle \\ &{}+a_{2}\bigl\langle \psi_{0},\phi_{E_{2}}r_{1} \bar{r}_{2}+|r_{2}|^{2}\psi_{E_{1}}+\bar{\phi }_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr\rangle , \end{aligned} \end{aligned}$$
(3.15)
$$\begin{aligned}& \begin{aligned}[b] g_{4}={}&a_{3}\bigl\langle \phi_{0},2\phi_{E_{2}}|r_{2}|^{2}+\phi _{E_{2}}r_{2}^{2}+|r_{2}|^{2}r_{2} \bigr\rangle \\ &{}+a_{4}\bigl\langle \phi_{0},\psi_{E_{1}}r_{2} \bar{r}_{1}+|r_{1}|^{2}\phi_{E_{2}}+\bar{\psi }_{E_{1}}r_{1}r_{2}+|r_{1}|^{2}r_{2} \bigr\rangle . \end{aligned} \end{aligned}$$
(3.16)
$$\begin{aligned} \mathbb{B} =& \biggl\{ u:\mathbb{R}\rightarrow\mathbf{L}^{2}_{-\sigma} \cap \mathbf{L}^{p}\cap\mathbf{L}^{2} \Bigm|\sup _{t\geq0}(1+t)^{1-\frac{2}{p}}\|u\| _{\mathbf{L}^{2}_{-\sigma}}, \\ &{}\sup_{t\geq0}\frac{(1+|t|)^{1-\frac{2}{p}}}{\log(2+|t|)}\|u\|_{\mathbf {L}^{p}}, \sup _{t\geq0}\|u\|_{\mathbf{L}^{2}}<\infty \biggr\} \end{aligned}$$
$$\begin{aligned} \|u\|_{\mathbb{B}}=\max \biggl\{ \sup_{t\geq0}\bigl(1+|t|\bigr)^{1-\frac{2}{p}} \|u\| _{\mathbf{L}^{2}_{-\sigma}}, \sup_{t\geq0}\frac{(1+|t|)^{1-\frac {2}{p}}}{\log(2+|t|)}\|u \|_{\mathbf{L}^{p}}, \sup_{t\geq0}\|u\|_{\mathbf {L}^{2}} \biggr\} . \end{aligned}$$
$$\begin{aligned} (\mathcal{N}R) (t):=-i\int_{0}^{t} \mathcal{S}(t,s)P_{c}G_{1}\,ds+i\int_{0}^{t} \mathcal {S}(t,s) (DH)G_{2}\,ds. \end{aligned}$$
(3.17)
Lemma 3.2
- (1)
For \(R\in\mathbb{B}\times\mathbb{B}\), the nonlinear operator \(\mathcal{N}:\mathbb{B}\times\mathbb{B}\rightarrow\mathbb{B}\times \mathbb{B}\) is well defined.
- (2)We have$$\begin{aligned} \|\mathcal{N}R_{1}-\mathcal{N}R_{2}\|_{\mathbb{B}\times\mathbb{B}} \leq &C_{a_{1},a_{2},a_{3},a_{4},p}\bigl(\|R_{1}\|_{\mathbb{B}\times\mathbb{B}}+\|R_{2}\| _{\mathbb{B}\times\mathbb{B}}+\|R_{1}\|_{\mathbb{B}\times\mathbb{B}}^{2}+\| R_{2}\|_{\mathbb{B}\times\mathbb{B}}^{2}\bigr) \\ &{}\times\bigl(\|R_{1}-R_{2}\|_{\mathbb{B}\times\mathbb{B}}\bigr). \end{aligned}$$
Proof
It is obvious that the part (1) can be obtained by part (2) choosing \(R_{2}=0\). Now, we only need to prove (2). This proof is based on these estimates of \(\mathcal{N}R_{1}-\mathcal{N}R_{2}\): the \(\mathbf{L}_{-\sigma }^{2}\times\mathbf{L}_{-\sigma}^{2}\) estimate, the \(\mathbf{L}^{p}\times \mathbf{L}^{p}\) estimate and the \(\mathbf{L}^{2}\times\mathbf{L}^{2}\) estimate. In fact, with a similar argument of getting the \(\mathbf {L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma}^{2}\) estimate, we could obtain the \(\mathbf{L}^{p}\times\mathbf{L}^{p}\) estimate and the \(\mathbf {L}^{2}\times\mathbf{L}^{2}\) estimate from (3.10) and (3.11). Here, we consider the \(\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}\) estimate.
Let \(R_{1}=(r_{1},r_{2})^{T}, R_{2}=(\tilde{r}_{1},\tilde{r}_{2})^{T}\in\mathbb{B}\times \mathbb{B}\). One obtains
where \(G_{1}-\tilde{G}_{1}=(g_{1}-\tilde{g}_{1},g_{2}-\tilde{g}_{2})^{T}\) and \(G_{2}-\tilde{G}_{2}=(g_{3}-\tilde{g}_{3},g_{4}-\tilde{g}_{4})^{T}\) with
Let \(4< p<\infty\) and \(\frac{1}{p'}+\frac{1}{p}=1\). We have
where \(A=(A_{1},A_{2})^{T}\) and \(B=(B_{1},B_{2})^{T}\) with
In what follows, we consider the integral terms on A, B, and \(|G_{2}-\tilde{G}_{2}|\) in (3.23). We first estimate the integral term on A. For \(\frac{1}{\alpha}+\frac{2}{p}=\frac{1}{2}\), we have
From (3.9) in Lemma 3.1, we have
where the constants
and
$$\begin{aligned} &(\mathcal{N}R_{1}-\mathcal{N}R_{2}) (t) \\ &\quad=-i\int_{0}^{t}\mathcal{S}(t,s)P_{c}(G_{1}- \tilde{G}_{1})\,ds+i\int_{0}^{t}\mathcal {S}(t,s) (DH) (G_{2}-\tilde{G}_{2})\,ds, \end{aligned}$$
(3.18)
$$\begin{aligned} g_{1}-\tilde{g}_{1} =&2a_{1} \psi_{E_{1}}\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)+a_{1} \psi_{E_{1}}(r_{1}-\tilde{r}_{1}) (r_{1}+ \tilde{r}_{1}) \\ &{}+a_{1}\bigl(|r_{1}|^{2}(r_{1}- \tilde{r}_{1})+\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(\tilde {r}_{1}|r_{1}|+\tilde{r}_{1}|\tilde{r}_{1}|\bigr) \bigr) \\ &{}+a_{2}\phi_{E_{2}}\bigl((r_{1}- \tilde{r}_{1})\bar{r}_{2}+\tilde{r}_{1}( \bar{r}_{2}-\tilde {\bar{r}}_{2})\bigr)+a_{2} \psi_{E_{1}}\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde {r}_{2}|\bigr) \\ &{}+a_{2}\bar{\phi}_{E_{2}}\bigl((r_{1}- \tilde{r}_{1})r_{2}+\tilde{r}_{1}(r_{2}- \tilde {r}_{2})\bigr) \\ &{}+a_{2}\bigl(\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr)r_{1}+|\tilde {r}_{2}|^{2}(r_{1}-\tilde{r}_{1})\bigr), \end{aligned}$$
(3.19)
$$\begin{aligned} g_{2}-\tilde{g}_{2} =&2a_{3} \phi_{E_{2}}\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde {r}_{2}|\bigr)+a_{3} \phi_{E_{2}}(r_{2}-\tilde{r}_{2}) (r_{2}+ \tilde{r}_{2}) \\ &{}+a_{3}\bigl(|r_{2}|^{2}(r_{2}- \tilde{r}_{2})+\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(\tilde {r}_{2}|r_{2}|+\tilde{r}_{2}|\tilde{r}_{2}|\bigr) \bigr) \\ &{}+a_{4}\psi_{E_{1}}\bigl((r_{2}- \tilde{r}_{2})\bar{r}_{1}+\tilde{r}_{2}( \bar{r}_{1}-\tilde {\bar{r}}_{1})\bigr)+a_{4} \phi_{E_{2}}\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr) \\ &{}+a_{4}\bar{\psi}_{E_{1}}\bigl((r_{2}- \tilde{r}_{2})r_{1}+\tilde{r}_{2}(r_{1}- \tilde {r}_{1})\bigr) \\ &{}+a_{4}\bigl(\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr)r_{2}+|\tilde {r}_{1}|^{2}(r_{2}-\tilde{r}_{2})\bigr), \end{aligned}$$
(3.20)
$$\begin{aligned} g_{3}-\tilde{g}_{3} =&a_{1}\bigl\langle \psi_{0},2\psi_{E_{1}}\bigl(|r_{1}|-|\tilde {r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr)+ \psi_{E_{1}}(r_{1}-\tilde{r}_{1}) (r_{1}+ \tilde {r}_{1})\bigr\rangle \\ &{}+a_{1}\bigl\langle \psi_{0},|r_{1}|^{2}(r_{1}- \tilde{r}_{1})+\bigl(|r_{1}|-|\tilde {r}_{1}|\bigr) \bigl( \tilde{r}_{1}|r_{1}|+\tilde{r}_{1}| \tilde{r}_{1}|\bigr)\bigr\rangle \\ &{}+a_{2}\bigl\langle \psi_{0},\phi_{E_{2}} \bigl((r_{1}-\tilde{r}_{1})\bar{r}_{2}+\tilde {r}_{1}(\bar{r}_{2}-\tilde{\bar{r}}_{2})\bigr)+ \psi_{E_{1}}\bigl(|r_{2}|-|\tilde {r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr)\bigr\rangle \\ &{}+a_{2}\bigl\langle \psi_{0},\bar{\phi}_{E_{2}} \bigl((r_{1}-\tilde{r}_{1})r_{2}+\tilde {r}_{1}(r_{2}-\tilde{r}_{2})\bigr)\bigr\rangle \\ &{}+a_{2}\bigl\langle \psi_{0},\bigl(|r_{2}|-| \tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde {r}_{2}|\bigr)r_{1}+| \tilde{r}_{2}|^{2}(r_{1}-\tilde{r}_{1}) \bigr\rangle , \end{aligned}$$
(3.21)
$$\begin{aligned} g_{4}-\tilde{g}_{4} =&a_{3}\bigl\langle \phi_{0},2\phi_{E_{2}}\bigl(|r_{2}|-|\tilde {r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr)+ \phi_{E_{2}}(r_{2}-\tilde{r}_{2}) (r_{2}+ \tilde {r}_{2})\bigr\rangle \\ &{}+a_{3}\bigl\langle \phi_{0},|r_{2}|^{2}(r_{2}- \tilde{r}_{2})+\bigl(|r_{2}|-|\tilde {r}_{2}|\bigr) \bigl( \tilde{r}_{2}|r_{2}|+\tilde{r}_{2}| \tilde{r}_{2}|\bigr)\bigr\rangle \\ &{}+a_{4}\bigl\langle \phi_{0},\psi_{E_{1}} \bigl((r_{2}-\tilde{r}_{2})\bar{r}_{1}+\tilde {r}_{2}(\bar{r}_{1}-\tilde{\bar{r}}_{1})\bigr)+ \phi_{E_{2}}\bigl(|r_{1}|-|\tilde {r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr)\bigr\rangle \\ &{}+a_{4}\bigl\langle \phi_{0},\bar{\psi}_{E_{1}} \bigl((r_{2}-\tilde{r}_{2})r_{1}+\tilde {r}_{2}(r_{1}-\tilde{r}_{1})\bigr)\bigr\rangle \\ &{}+a_{4}\bigl\langle \phi_{0},\bigl(|r_{1}|-| \tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)r_{2}+| \tilde{r}_{1}|^{2}(r_{2}-\tilde{r}_{2}) \bigr\rangle . \end{aligned}$$
(3.22)
$$\begin{aligned} &\|\mathcal{N}R_{1}-\mathcal{N}R_{2} \|_{\mathbf{L}_{-\sigma}^{2}\times\mathbf {L}_{-\sigma}^{2}} \\ &\quad\leq\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma }^{2}\times\mathbf{L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times \mathbf{L}_{-\sigma}^{2}}\|A\|_{\mathbf{L}^{2}\times\mathbf{L}^{2}}\,ds \\ &\qquad{}+\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{p'}\times\mathbf {L}_{\sigma}^{p'}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf {L}_{-\sigma}^{2}}\|B\|_{\mathbf{L}^{p'}\times\mathbf{L}^{p'}}\,ds \\ &\qquad{}+\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{2}\times\mathbf {L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}}\|DH\|_{\mathbf{L}^{2}_{\sigma}\times\mathbf{L}^{2}_{\sigma}}|G_{2}-\tilde {G}_{2}|\,ds, \end{aligned}$$
(3.23)
$$\begin{aligned} A_{1} =&2a_{1}\psi_{E_{1}}\langle x \rangle^{\sigma}\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)+a_{1}\psi_{E_{1}} \langle x\rangle^{\sigma}(r_{1}-\tilde{r}_{1}) (r_{1}+\tilde {r}_{1}) \\ &{}+a_{2}\phi_{E_{2}}\langle x\rangle^{\sigma} \bigl((r_{1}-\tilde{r}_{1})\bar{r}_{2}+ \tilde{r}_{1}(\bar {r}_{2}-\tilde{\bar{r}}_{2}) \bigr)+a_{2}\psi_{E_{1}}\langle x\rangle^{\sigma}\bigl(|r_{2}|-| \tilde {r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr) \\ &{}+a_{2}\bar{\phi}_{E_{2}}\langle x\rangle^{\sigma} \bigl((r_{1}-\tilde{r}_{1})r_{2}+\tilde {r}_{1}(r_{2}-\tilde{r}_{2})\bigr), \end{aligned}$$
(3.24)
$$\begin{aligned} A_{2} =&2a_{3}\phi_{E_{2}}\langle x \rangle^{\sigma}\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde {r}_{2}|\bigr)+a_{3} \phi_{E_{2}}\langle x\rangle ^{\sigma}(r_{2}- \tilde{r}_{2}) (r_{2}+\tilde {r}_{2}) \\ &+a_{4}\psi_{E_{1}}\langle x\rangle ^{\sigma} \bigl((r_{2}-\tilde{r}_{2})\bar{r}_{1}+ \tilde{r}_{2}(\bar {r}_{1}-\tilde{\bar{r}}_{1}) \bigr)+a_{4}\phi_{E_{2}}\langle x\rangle ^{\sigma}\bigl(|r_{1}|-| \tilde {r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr) \\ &+a_{4}\bar{\psi}_{E_{1}}\langle x\rangle ^{\sigma} \bigl((r_{2}-\tilde{r}_{2})r_{1}+\tilde {r}_{2}(r_{1}-\tilde{r}_{1})\bigr), \end{aligned}$$
(3.25)
$$\begin{aligned} B_{1} =&a_{1}\bigl(|r_{1}|^{2}(r_{1}- \tilde{r}_{1})+\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(\tilde {r}_{1}|r_{1}|+\tilde{r}_{1}|\tilde{r}_{1}|\bigr) \bigr) \\ &+a_{2}\bigl(\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr)r_{1}+|\tilde {r}_{2}|^{2}(r_{1}-\tilde{r}_{1})\bigr), \end{aligned}$$
(3.26)
$$\begin{aligned} B_{2} =&a_{3}\bigl(|r_{2}|^{2}(r_{2}- \tilde{r}_{2})+\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(\tilde {r}_{2}|r_{2}|+\tilde{r}_{2}|\tilde{r}_{2}|\bigr) \bigr) \\ &+a_{4}\bigl(\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr)r_{2}+|\tilde {r}_{1}|^{2}(r_{2}-\tilde{r}_{2})\bigr). \end{aligned}$$
(3.27)
$$\begin{aligned} \bigl\| \psi_{E_{1}}\langle x\rangle ^{\sigma}\bigl(|r_{1}|-| \tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr)\bigr\| _{\mathbf{L}^{2}}\leq\bigl\| \psi_{E_{1}}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}} \bigl\| |r_{1}|-|\tilde{r}_{1}| \bigr\| _{\mathbf{L}^{2}} \bigl\| |r_{1}|+|\tilde{r}_{1}| \bigr\| _{\mathbf{L}^{2}}. \end{aligned}$$
$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{2}\times\mathbf {L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}}\|A\|_{\mathbf{L}^{2}\times\mathbf{L}^{2}}\,ds \\ &\quad\leq 3C_{a_{1},a_{2},a_{3},a_{4}}\int_{0}^{t}\bigl(1+|t-s|\bigr)^{-1} \log ^{-2}\bigl(2+|t-s|\bigr) \\ &\qquad{}\times\bigl\| \psi_{E_{1}}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}}\bigl\| \phi _{E_{2}}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}} \bigl( \bigl\| |r_{1}|-|\tilde {r}_{1}| \bigr\| _{\mathbf{L}^{2}} \bigl\| |r_{1}|+|\tilde{r}_{1}| \bigr\| _{\mathbf {L}^{2}} \\ &\qquad{}+ \bigl\| |r_{2}|-|\tilde{r}_{2}| \bigr\| _{\mathbf{L}^{2}} \bigl\| |r_{2}|+|\tilde {r}_{2}| \bigr\| _{\mathbf{L}^{2}} \bigr)\,ds \\ &\quad\leq3C_{a_{1},a_{2},a_{3},a_{4}}C_{1}\int_{0}^{t} \frac{\log^{2}(2+|s|)}{(1+|t-s|)\log (2+|t-s|)}\frac{\||r_{1}|-|\tilde{r}_{1}|\|_{\mathbb{B}}+\||r_{2}|-|\tilde {r}_{2}|\|_{\mathbb{B}}}{(1+|s|)^{1-\frac{2}{p}}} \\ &\qquad{}\times\frac{\||r_{1}|+|\tilde{r}_{1}|\|_{\mathbb{B}}+\||r_{2}|+|\tilde {r}_{2}|\|_{\mathbb{B}}}{(1+|s|)^{1-\frac{2}{p}}}\,ds \\ &\quad\leq 3C_{a_{1},a_{2},a_{3},a_{4}}C_{1}C_{2}\bigl(\|r_{1} \|_{\mathbb{B}}+\|\tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2} \|_{\mathbb{B}}+\|\tilde{r}_{2}\|_{\mathbb{B}}\bigr) \\ &\qquad{}\times\frac {\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}-\tilde{r}_{2}\|_{\mathbb {B}}}{(1+|t|)\log^{2}(2+|t|)}, \end{aligned}$$
(3.28)
$$C_{1}=\max \Bigl\{ \sup_{t>0}\bigl\| \psi_{E_{1}} \langle x\rangle^{\sigma}\bigr\| _{\mathbf{L}^{\alpha }}, \sup_{t>0}\bigl\| \phi_{E_{2}}\langle x\rangle^{\sigma}\bigr\| _{\mathbf{L}^{\alpha}} \Bigr\} $$
$$C_{2}=\sup_{t>0}\bigl(1+|t|\bigr)\log^{2}\bigl(2+|t|\bigr)\int _{0}^{t}\frac{\log ^{2}(2+|s|)}{(1+|t-s|)\log^{2}(2+|t-s|)(1+|s|)^{2-\frac{4}{p}}}\,ds<\infty. $$
Using the interpolate inequality \(\|r\|_{\mathbf{L}^{\alpha}}\leq\|r\| _{\mathbf{L}^{2}}^{1-b}\|r\|^{b}_{\mathbf{L}^{p}}\) for \(\frac{1}{\alpha}=\frac {1-b}{2}+\frac{b}{p}\) with \(p\geq4\) and \(2\leq\alpha\leq p\), one obtains
where \(\frac{2}{\alpha}+\frac{1}{p}=\frac{1}{p'}\) and \(\frac {2(1-b)}{2}+\frac{2b}{p}+\frac{1}{p}=\frac{1}{p'}\) with \((1-\frac {2}{p})(1+2b)=1+\frac{2}{p}>1\).
$$\begin{aligned}& \begin{aligned}[b] \bigl\| |r_{1}|^{2}(r_{1}- \tilde{r}_{1})\bigr\| _{\mathbf{L}^{p'}}&\leq\|r_{1}- \tilde{r}_{1}\| _{\mathbf{L}^{p}}\|r_{1}\|^{2}_{\mathbf{L}^{\alpha}}\\ &\leq\|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}} \|r_{1}\|_{\mathbf{L}^{2}}^{2(1-b)}\| r_{1} \|_{\mathbf{L}^{p}}^{2b}, \end{aligned} \end{aligned}$$
(3.29)
$$\begin{aligned}& \begin{aligned}[b] &\bigl\| \bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr)r_{1}\bigr\| _{\mathbf{L}^{p'}}\\ &\quad\leq \|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}}\bigl(\|r_{1} \|_{\mathbf{L}^{\alpha}}+\| \tilde{r}_{1}\|_{\mathbf{L}^{\alpha}}\bigr)\|r_{1} \|_{\mathbf{L}^{\alpha }}\\ &\quad\leq\|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}}\bigl( \|r_{1}\|_{\mathbf{L}^{2}}^{1-b}\| r_{1} \|_{\mathbf{L}^{p}}^{b}+\|\tilde{r}_{1}\|_{\mathbf{L}^{2}}^{1-b} \|\tilde {r}_{1}\|_{\mathbf{L}^{p}}^{b}\bigr)\|r_{1} \|_{\mathbf{L}^{2}}^{1-b}\|r_{1}\|_{\mathbf {L}^{p}}^{b}, \end{aligned} \end{aligned}$$
(3.30)
$$\begin{aligned}& \begin{aligned}[b] &\bigl\| \bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl( \tilde{r}_{1}|r_{1}|+\tilde{r}_{1}| \tilde{r}_{1}|\bigr)\bigr\| _{\mathbf{L}^{p'}} \\ &\quad\leq\|r_{1}- \tilde{r}_{1}\|_{\mathbf{L}^{p}}\|\tilde{r}_{1} \|_{\mathbf {L}^{\alpha}}\bigl(\|r_{1}\|_{\mathbf{L}^{\alpha}}+\|\tilde{r}_{1} \|_{\mathbf {L}^{\alpha}}\bigr)\\ &\quad\leq\|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}}\bigl( \|r_{1}\|_{\mathbf{L}^{2}}^{1-b}\| r_{1} \|_{\mathbf{L}^{p}}^{b}+\|\tilde{r}_{1}\|_{\mathbf{L}^{2}}^{1-b} \|\tilde {r}_{1}\|_{\mathbf{L}^{p}}^{b}\bigr)\| \tilde{r}_{1}\|_{\mathbf{L}^{2}}^{1-b}\|\tilde {r}_{1} \|_{\mathbf{L}^{p}}^{b}, \end{aligned} \end{aligned}$$
(3.31)
Note that \((1-\frac{2}{p})(1+2b)>1\). Therefore, from (3.10), we have
where the constant
$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}(t,s)\bigr\| _{\mathbf{L}_{\sigma}^{p'}\times\mathbf {L}_{\sigma}^{p'}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf {L}_{-\sigma}^{2}} \|B\|_{\mathbf{L}^{p'}\times\mathbf{L}^{p'}}\,ds \\ &\quad\leq C_{p,a_{1},a_{2},a_{3},a_{4}}\bigl(\|r_{1}\|^{2}_{\mathbb{B}}+ \|\tilde{r}_{1}\| ^{2}_{\mathbb{B}}+\|r_{2} \|^{2}_{\mathbb{B}}+\|\tilde{r}_{2}\|^{2}_{\mathbb {B}} \bigr) \\ &\qquad{}\times\bigl(\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+ \|r_{2}-\tilde{r}_{2}\|_{\mathbb {B}}\bigr)\int _{0}^{t}\frac{\log(2+|s|)^{1+2b}}{|t-s|^{1-\frac {2}{p}}(1+|s|)^{(1-\frac{2}{p})(1+2b)}}\,ds \\ &\quad\leq C_{p,a_{1},a_{2},a_{3},a_{4}}C_{3} \bigl(\|r_{1} \|^{2}_{\mathbb{B}}+\|\tilde{r}_{1}\| ^{2}_{\mathbb{B}}+ \|r_{2}\|^{2}_{\mathbb{B}}+\|\tilde{r}_{2} \|^{2}_{\mathbb {B}} \bigr)\frac{\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}-\tilde{r}_{2}\| _{\mathbb{B}}}{(1+|s|)^{1-\frac{2}{p}}}, \end{aligned}$$
$$C_{3}=\sup_{t>0}\bigl(1+|t|\bigr)^{1-\frac{2}{p}}\int_{0}^{t}\frac {\log(2+|s|)^{1+2b}}{|t-s|^{1-\frac{2}{p}}(1+|s|)^{(1-\frac {2}{p})(1+2b)}}\,ds<\infty. $$
It is easy to obtain
for \(\frac{1}{\alpha}+\frac{2}{p}=\frac{1}{2}\). From (3.9) in Lemma 3.1, we get
where we have the constant
$$\begin{aligned} & \bigl|\bigl\langle \psi_{0},2\psi_{E_{1}}\bigl(|r_{1}|-| \tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)\bigr\rangle \bigr|\leq2\|\psi_{0}\|_{\mathbf{L}^{\infty}}\|\psi_{E_{1}}\| _{\mathbf{L}^{\alpha}}\|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}} \bigl(\|r_{1}\| _{\mathbf{L}^{p}}+\|\tilde{r}_{1}\|_{\mathbf{L}^{p}} \bigr), \\ & \bigl|\bigl\langle \psi_{0},|r_{1}|^{2}(r_{1}- \tilde{r}_{1})\bigr\rangle \bigr|\leq\bigl\| \psi _{0}\langle x\rangle ^{\sigma}\bigr\| _{\mathbf{L}^{\alpha}}\|r_{1}-\tilde{r}_{1} \|_{\mathbf {L}^{2}_{-\sigma}}\|r_{1}\|^{2}_{\mathbf{L}^{p}} \end{aligned}$$
$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{2}\times\mathbf {L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}}|G_{2}-\tilde{G}_{2}|\,ds \\ &\quad\leq C_{a_{1},a_{2},a_{3},a_{4}} \bigl(\|r_{1}\|_{\mathbb{B}}+\| \tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2}\|_{\mathbb{B}}+\| \tilde{r}_{2}\|_{\mathbb{B}}+\|r_{1}\| _{\mathbb{B}}^{2}+ \|\tilde{r}_{1}\|_{\mathbb{B}}^{2}+\|r_{2} \|_{\mathbb{B}}^{2}+\| \tilde{r}_{2}\|_{\mathbb{B}}^{2} \bigr) \\ &\qquad{}\times \bigl(\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+ \|r_{2}-\tilde{r}_{2}\| _{\mathbb{B}} \bigr)\int _{0}^{t}\frac{(\|\psi_{0}\langle x\rangle ^{\sigma}\|_{\mathbf {L}^{\alpha}}+\|\phi_{0}\langle x\rangle ^{\sigma}\|_{\mathbf{L}^{\alpha }})}{(1+|t-s|)\log^{2}(2+|t-s|)}\cdot\frac{\log ^{2}(2+|s|)}{(1+|s|)^{3-\frac{6}{p}}}\,ds \\ &\quad\leq C_{a_{1},a_{2},a_{3},a_{4}}C_{5} \bigl(\|r_{1} \|_{\mathbb{B}}+\|\tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2} \|_{\mathbb{B}}+\|\tilde{r}_{2}\|_{\mathbb{B}}+\|r_{1}\| _{\mathbb{B}}^{2}+\|\tilde{r}_{1}\|_{\mathbb{B}}^{2}+ \|r_{2}\|_{\mathbb{B}}^{2}+\| \tilde{r}_{2} \|_{\mathbb{B}}^{2} \bigr) \\ &\qquad{}\times\frac{\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}-\tilde{r}_{2}\| _{\mathbb{B}}}{(1+|t|)\log^{2}(2+|t|)}, \end{aligned}$$
$$C_{5}= \bigl(\bigl\| \psi_{0}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}}+\bigl\| \phi _{0}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}} \bigr)\int_{0}^{t} \frac{\log ^{2}(2+|s|)}{(1+|s|)^{3-\frac{6}{p}}(1+|t-s|)\log^{2}(2+|t-s|)}\,ds<\infty. $$
Hence, we conclude that
This completes the proof. □
$$\begin{aligned} &\|\mathcal{N}R_{1}-\mathcal{N}R_{2}\|_{\mathbb{B}\times\mathbb {B}} \\ &\quad\leq C_{a_{1},a_{2},a_{3},a_{4},p} \bigl(\|r_{1}\|_{\mathbb{B}}+\| \tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2}\|_{\mathbb{B}}+\| \tilde{r}_{2}\|_{\mathbb{B}}+\|r_{1}\| _{\mathbb{B}}^{2}+ \|\tilde{r}_{1}\|_{\mathbb{B}}^{2} \\ &\qquad{}+\|r_{2}\|_{\mathbb{B}}^{2}+\| \tilde{r}_{2}\|_{\mathbb{B}}^{2} \bigr)\times \bigl( \|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}- \tilde{r}_{2}\|_{\mathbb{B}} \bigr). \end{aligned}$$
Proof of Theorem 1.1
Define a closed ball \(B(R_{0},\mathbf {r})\subset\mathbb{B}\times\mathbb{B}\) with center \(R_{0}=\mathcal {S}(t,0)R(0)\) and radius \(\mathbf{r}=\frac{L\|R_{0}\|_{\mathbb{B}\times \mathbb{B}}}{2-\operatorname{Lip}}\). By Lemma 3.1, there exists a constant \(C_{6}\) such that
Choose \(\varepsilon_{0}\) such that \(C_{6}\varepsilon_{0}<\frac{1}{2}(\sqrt {1+2C^{-1}_{a_{1},a_{2},a_{3},a_{4},p}}-1)\). Then there exists a constant \(0<\operatorname{Lip}<1\) such that
It is easy to conclude that the right hand side of (3.12) leaves \(B(R_{0},\mathbf{r})\) invariant, and it is a contraction with Lipschitz constant Lip on \(B(R_{0},\mathbf{r})\). From the contraction mapping theorem, (3.12) has a unique solution in \(B(R_{0},\mathbf{r})\). If we have two solutions of (3.12), one in \(\mathbf{C}(\mathbb {R},\mathbf{H}^{1}\times\mathbf{H}^{1})\) from classical well-posedness theory and one in \(\mathbf{C}(\mathbb{R},(\mathbf{L}^{2}_{-\sigma}\cap \mathbf{L}^{2}\cap \mathbf{L}^{p})\times(\mathbf{L}^{2}_{-\sigma}\cap\mathbf{L}^{2}\cap\mathbf {L}^{p}))\) from the above argument for \(p\geq6\). By uniqueness and the continuous embedding of \(\mathbf{H}^{1}\) in \(\mathbf{L}^{2}_{-\sigma}\cap \mathbf{L}^{2}\cap\mathbf{L}^{p}\), we infer that the two solutions must coincide. Therefore, the time decaying estimates also hold for the \(\mathbf{H}^{1}\times\mathbf{H}^{1}\) solutions. This completes the proof of Theorem 1.1. □
$$\begin{aligned} \|R_{0}\|_{\mathbb{B}\times\mathbb{B}}\leq C_{6}\bigl\| R(0)\bigr\| _{\mathbf {L}^{2}_{\sigma}\times\mathbf{L}^{2}_{\sigma}}. \end{aligned}$$
$$\begin{aligned} \|R_{0}\|_{\mathbb{B}\times\mathbb{B}}\leq\frac{2-\operatorname{Lip}}{4}\Bigl(\sqrt {1+2\operatorname{Lip}C^{-1}_{a_{1},a_{2},a_{3},a_{4},p}}-1\Bigr). \end{aligned}$$
Declarations
Acknowledgements
The authors express their sincere thanks to the anonymous referees for the comments.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
(1)
School of Mathematics and Information Science, Henan University of Economics and Law
(2)
School of Mathematics and Information Science, Henan Economics and Technical School
(3)
College of Information Technology, Jilin Agricultural University
(4)
College of Mathematics, Jilin University
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