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
$$\begin{aligned} a(t)=\langle\psi_{0},u\rangle,\qquad b(t)=\langle \phi_{0},v\rangle,\quad \mbox{for }t\in \mathbb{R}. \end{aligned}$$
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,
$$\begin{aligned} &r_{1}(t)=u(t)-h_{1}\bigl(a(t)\bigr)-a(t) \psi_{0},\qquad \bigl\langle \psi_{0},r_{1}(t)\bigr\rangle \equiv 0, \\ &r_{2}(t)=v(t)-h_{2}\bigl(b(t)\bigr)-b(t) \phi_{0},\qquad \bigl\langle \phi_{0},r_{2}(t)\bigr\rangle \equiv0. \end{aligned}$$
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
$$\begin{aligned}& \begin{aligned}[b] &i\frac{\mathrm{d}a}{\mathrm{d}t}\psi_{0}+iDh_{1}\Big|_{a} \frac{\mathrm {d}a}{\mathrm{d}t}+i\frac{\mathrm{d}r_{1}}{\mathrm{d}t} \\ &\quad=E_{1}a\psi_{0}+E_{1}h_{1}(a)+(- \triangle+V_{1})r_{1}+a_{1}\bigl(2| \psi_{E_{1}}|^{2}r_{1}+\psi _{E_{1}}^{2} \bar{r}_{1} \\ &\qquad{}+2\psi_{E_{1}}|r_{1}|^{2}+ \psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr)+a_{2}\bigl(\phi_{E_{2}}\psi _{E_{1}} \bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &\qquad{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+ \psi_{E_{1}}\bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2} \psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr), \end{aligned} \end{aligned}$$
(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)
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
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}a}{\mathrm{d}t}={}&E_{1}\bigl(|a|\bigr)a+a_{1} \bigl\langle \psi_{0},2|\psi _{E_{1}}|^{2}r_{1}+ \psi_{E_{1}}^{2}\bar{r}_{1}+2\psi_{E_{1}}|r_{1}|^{2} \\ &{}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr\rangle +a_{2}\bigl\langle \psi_{0},\phi_{E_{2}} \psi _{E_{1}}\bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+\psi_{E_{1}} \bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2}\psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr\rangle , \end{aligned} \end{aligned}$$
(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)
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
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}r_{1}}{\mathrm{d}t}={}&(-\triangle+V_{1})r_{1}+a_{1}P_{c} \bigl(2|\psi _{E_{1}}|^{2}r_{1}+\psi_{E_{1}}^{2} \bar{r}_{1} \\ &{}+2\psi_{E_{1}}|r_{1}|^{2}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr)+a_{2}P_{c}\bigl(\phi_{E_{2}}\psi _{E_{1}}\bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+\psi_{E_{1}} \bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2}\psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr), \\ &{}-Dh_{1}|_{a}\bigl(a_{1}\bigl\langle \psi_{0},2|\psi_{E_{1}}|^{2}r_{1}+ \psi_{E_{1}}^{2}\bar {r}_{1}+2\psi_{E_{1}}|r_{1}|^{2} \\ &{}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr\rangle +a_{2}\bigl\langle \psi_{0},\phi_{E_{2}} \psi _{E_{1}}\bar{r}_{2}+|\phi_{E_{2}}|^{2}r_{1} \\ &{}+\phi_{E_{2}}r_{1}\bar{r}_{2}+\psi_{E_{1}} \bar{\phi}_{E_{2}}r_{2}+|r_{2}|^{2}\psi _{E_{1}}+\bar{\phi}_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr\rangle \bigr), \end{aligned} \end{aligned}$$
(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)
The linear part of (3.5)-(3.6) is
$$\begin{aligned}& \begin{aligned}[b] i\frac{\mathrm{d}r'_{1}}{\mathrm{d}t}={}&(-\triangle+V_{1})r'_{1}+a_{1}P_{c} \bigl(2|\psi _{E_{1}}|^{2}r'_{1}+ \psi_{E_{1}}^{2}\bar{r}'_{1}\bigr) \\ &{}+a_{2}P_{c}\bigl(\phi_{E_{2}}\psi _{E_{1}} \bar{r}'_{2}+|\phi_{E_{2}}|^{2}r'_{1}+ \psi_{E_{1}}\bar{\phi }_{E_{2}}r'_{2}\bigr) \\ &{}-Dh_{1}|_{a}\bigl\langle \psi_{0},a_{1} \bigl(2|\psi_{E_{1}}|^{2}r'_{1}+ \psi_{E_{1}}^{2}\bar{r}'_{1}\bigr) \\ &{}+a_{2}\bigl(\phi_{E_{2}}\psi_{E_{1}} \bar{r}'_{2}+|\phi _{E_{2}}|^{2}r'_{1}+ \psi_{E_{1}}\bar{\phi}_{E_{2}}r'_{2}\bigr) \bigr\rangle , \end{aligned} \end{aligned}$$
(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)
Define the operator \(\mathcal{S}(t,s)Y\) as the solution of the linear equation (3.7)-(3.8):
$$\begin{aligned} \mathcal{S}(t,s)Y=R' \quad \mbox{with } R'= \bigl(r'_{1},r'_{2} \bigr)^{T}. \end{aligned}$$
The following lemma can be obtained by a small modification of the proof of Theorem 4.1 in [8], so we omit the proof.
Lemma 3.1
There exists
\(\varepsilon>0\)
such that if
$$\begin{aligned} \bigl\| \langle x\rangle^{\sigma}\psi_{E_{1}}\bigr\| _{\mathbf{H}^{2}}< \varepsilon, \qquad \bigl\| \langle x\rangle^{\sigma }\phi_{E_{2}} \bigr\| _{\mathbf{H}^{2}}<\varepsilon, \end{aligned}$$
then there exist
\(C, C_{p}>0\)
with the property that for any
\(t,s\in \mathbb{R}\), we have
$$\begin{aligned} &\bigl\| \mathcal{S}(t,s)\bigr\| _{\mathbf{L}^{2}_{\sigma} \times\mathbf{L}^{2}_{\sigma }\rightarrow\mathbf{L}^{2}_{-\sigma}\times\mathbf{L}^{2}_{-\sigma}}\leq C\bigl(1+|t-s|\bigr)^{-1} \log^{-2}\bigl(2+|t-s|\bigr), \end{aligned}$$
(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)
where
\(p\geq2\)
and
\(\frac{1}{p'}+\frac{1}{p}=1\).
By Duhamel’s principle, it follows from (3.5)-(3.6) that
$$\begin{aligned} R(t)=\mathcal{S}(t,0)R(0)-i\int_{0}^{t} \mathcal{S}(t,s)P_{c}G_{1}\,ds+i\int_{0}^{t} \mathcal{S}(t,s) (DH)G_{2}\,ds, \end{aligned}$$
(3.12)
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
$$\begin{aligned}& \begin{aligned}[b] g_{1}={}&a_{1}\bigl(2 \psi_{E_{1}}|r_{1}|^{2}+\psi_{E_{1}}r_{1}^{2}+|r_{1}|^{2}r_{1} \bigr) \\ &{}+a_{2}\bigl(\phi_{E_{2}}r_{1}\bar{r}_{2}+|r_{2}|^{2} \psi_{E_{1}}+\bar{\phi }_{E_{2}}r_{1}r_{2}+|r_{2}|^{2}r_{1} \bigr), \end{aligned} \end{aligned}$$
(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)
For fixed \(p\geq6\), we define the Banach space
$$\begin{aligned} \mathbb{B} =& \biggl\{ u:\mathbb{R}\rightarrow\mathbf{L}^{2}_{-\sigma} \cap \mathbf{L}^{p}\cap\mathbf{L}^{2} \Bigm|\sup _{t\geq0}(1+t)^{1-\frac{2}{p}}\|u\| _{\mathbf{L}^{2}_{-\sigma}}, \\ &{}\sup_{t\geq0}\frac{(1+|t|)^{1-\frac{2}{p}}}{\log(2+|t|)}\|u\|_{\mathbf {L}^{p}}, \sup _{t\geq0}\|u\|_{\mathbf{L}^{2}}<\infty \biggr\} \end{aligned}$$
endowed with the norm
$$\begin{aligned} \|u\|_{\mathbb{B}}=\max \biggl\{ \sup_{t\geq0}\bigl(1+|t|\bigr)^{1-\frac{2}{p}} \|u\| _{\mathbf{L}^{2}_{-\sigma}}, \sup_{t\geq0}\frac{(1+|t|)^{1-\frac {2}{p}}}{\log(2+|t|)}\|u \|_{\mathbf{L}^{p}}, \sup_{t\geq0}\|u\|_{\mathbf {L}^{2}} \biggr\} . \end{aligned}$$
Consider the nonlinear part in (3.12):
$$\begin{aligned} (\mathcal{N}R) (t):=-i\int_{0}^{t} \mathcal{S}(t,s)P_{c}G_{1}\,ds+i\int_{0}^{t} \mathcal {S}(t,s) (DH)G_{2}\,ds. \end{aligned}$$
(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
$$\begin{aligned} &(\mathcal{N}R_{1}-\mathcal{N}R_{2}) (t) \\ &\quad=-i\int_{0}^{t}\mathcal{S}(t,s)P_{c}(G_{1}- \tilde{G}_{1})\,ds+i\int_{0}^{t}\mathcal {S}(t,s) (DH) (G_{2}-\tilde{G}_{2})\,ds, \end{aligned}$$
(3.18)
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
$$\begin{aligned} g_{1}-\tilde{g}_{1} =&2a_{1} \psi_{E_{1}}\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)+a_{1} \psi_{E_{1}}(r_{1}-\tilde{r}_{1}) (r_{1}+ \tilde{r}_{1}) \\ &{}+a_{1}\bigl(|r_{1}|^{2}(r_{1}- \tilde{r}_{1})+\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(\tilde {r}_{1}|r_{1}|+\tilde{r}_{1}|\tilde{r}_{1}|\bigr) \bigr) \\ &{}+a_{2}\phi_{E_{2}}\bigl((r_{1}- \tilde{r}_{1})\bar{r}_{2}+\tilde{r}_{1}( \bar{r}_{2}-\tilde {\bar{r}}_{2})\bigr)+a_{2} \psi_{E_{1}}\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde {r}_{2}|\bigr) \\ &{}+a_{2}\bar{\phi}_{E_{2}}\bigl((r_{1}- \tilde{r}_{1})r_{2}+\tilde{r}_{1}(r_{2}- \tilde {r}_{2})\bigr) \\ &{}+a_{2}\bigl(\bigl(|r_{2}|-|\tilde{r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr)r_{1}+|\tilde {r}_{2}|^{2}(r_{1}-\tilde{r}_{1})\bigr), \end{aligned}$$
(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)
Let \(4< p<\infty\) and \(\frac{1}{p'}+\frac{1}{p}=1\). We have
$$\begin{aligned} &\|\mathcal{N}R_{1}-\mathcal{N}R_{2} \|_{\mathbf{L}_{-\sigma}^{2}\times\mathbf {L}_{-\sigma}^{2}} \\ &\quad\leq\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma }^{2}\times\mathbf{L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times \mathbf{L}_{-\sigma}^{2}}\|A\|_{\mathbf{L}^{2}\times\mathbf{L}^{2}}\,ds \\ &\qquad{}+\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{p'}\times\mathbf {L}_{\sigma}^{p'}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf {L}_{-\sigma}^{2}}\|B\|_{\mathbf{L}^{p'}\times\mathbf{L}^{p'}}\,ds \\ &\qquad{}+\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{2}\times\mathbf {L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}}\|DH\|_{\mathbf{L}^{2}_{\sigma}\times\mathbf{L}^{2}_{\sigma}}|G_{2}-\tilde {G}_{2}|\,ds, \end{aligned}$$
(3.23)
where \(A=(A_{1},A_{2})^{T}\) and \(B=(B_{1},B_{2})^{T}\) with
$$\begin{aligned} A_{1} =&2a_{1}\psi_{E_{1}}\langle x \rangle^{\sigma}\bigl(|r_{1}|-|\tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)+a_{1}\psi_{E_{1}} \langle x\rangle^{\sigma}(r_{1}-\tilde{r}_{1}) (r_{1}+\tilde {r}_{1}) \\ &{}+a_{2}\phi_{E_{2}}\langle x\rangle^{\sigma} \bigl((r_{1}-\tilde{r}_{1})\bar{r}_{2}+ \tilde{r}_{1}(\bar {r}_{2}-\tilde{\bar{r}}_{2}) \bigr)+a_{2}\psi_{E_{1}}\langle x\rangle^{\sigma}\bigl(|r_{2}|-| \tilde {r}_{2}|\bigr) \bigl(|r_{2}|+|\tilde{r}_{2}|\bigr) \\ &{}+a_{2}\bar{\phi}_{E_{2}}\langle x\rangle^{\sigma} \bigl((r_{1}-\tilde{r}_{1})r_{2}+\tilde {r}_{1}(r_{2}-\tilde{r}_{2})\bigr), \end{aligned}$$
(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)
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
$$\begin{aligned} \bigl\| \psi_{E_{1}}\langle x\rangle ^{\sigma}\bigl(|r_{1}|-| \tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde{r}_{1}|\bigr)\bigr\| _{\mathbf{L}^{2}}\leq\bigl\| \psi_{E_{1}}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}} \bigl\| |r_{1}|-|\tilde{r}_{1}| \bigr\| _{\mathbf{L}^{2}} \bigl\| |r_{1}|+|\tilde{r}_{1}| \bigr\| _{\mathbf{L}^{2}}. \end{aligned}$$
From (3.9) in Lemma 3.1, we have
$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{2}\times\mathbf {L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}}\|A\|_{\mathbf{L}^{2}\times\mathbf{L}^{2}}\,ds \\ &\quad\leq 3C_{a_{1},a_{2},a_{3},a_{4}}\int_{0}^{t}\bigl(1+|t-s|\bigr)^{-1} \log ^{-2}\bigl(2+|t-s|\bigr) \\ &\qquad{}\times\bigl\| \psi_{E_{1}}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}}\bigl\| \phi _{E_{2}}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}} \bigl( \bigl\| |r_{1}|-|\tilde {r}_{1}| \bigr\| _{\mathbf{L}^{2}} \bigl\| |r_{1}|+|\tilde{r}_{1}| \bigr\| _{\mathbf {L}^{2}} \\ &\qquad{}+ \bigl\| |r_{2}|-|\tilde{r}_{2}| \bigr\| _{\mathbf{L}^{2}} \bigl\| |r_{2}|+|\tilde {r}_{2}| \bigr\| _{\mathbf{L}^{2}} \bigr)\,ds \\ &\quad\leq3C_{a_{1},a_{2},a_{3},a_{4}}C_{1}\int_{0}^{t} \frac{\log^{2}(2+|s|)}{(1+|t-s|)\log (2+|t-s|)}\frac{\||r_{1}|-|\tilde{r}_{1}|\|_{\mathbb{B}}+\||r_{2}|-|\tilde {r}_{2}|\|_{\mathbb{B}}}{(1+|s|)^{1-\frac{2}{p}}} \\ &\qquad{}\times\frac{\||r_{1}|+|\tilde{r}_{1}|\|_{\mathbb{B}}+\||r_{2}|+|\tilde {r}_{2}|\|_{\mathbb{B}}}{(1+|s|)^{1-\frac{2}{p}}}\,ds \\ &\quad\leq 3C_{a_{1},a_{2},a_{3},a_{4}}C_{1}C_{2}\bigl(\|r_{1} \|_{\mathbb{B}}+\|\tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2} \|_{\mathbb{B}}+\|\tilde{r}_{2}\|_{\mathbb{B}}\bigr) \\ &\qquad{}\times\frac {\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}-\tilde{r}_{2}\|_{\mathbb {B}}}{(1+|t|)\log^{2}(2+|t|)}, \end{aligned}$$
(3.28)
where the constants
$$C_{1}=\max \Bigl\{ \sup_{t>0}\bigl\| \psi_{E_{1}} \langle x\rangle^{\sigma}\bigr\| _{\mathbf{L}^{\alpha }}, \sup_{t>0}\bigl\| \phi_{E_{2}}\langle x\rangle^{\sigma}\bigr\| _{\mathbf{L}^{\alpha}} \Bigr\} $$
and
$$C_{2}=\sup_{t>0}\bigl(1+|t|\bigr)\log^{2}\bigl(2+|t|\bigr)\int _{0}^{t}\frac{\log ^{2}(2+|s|)}{(1+|t-s|)\log^{2}(2+|t-s|)(1+|s|)^{2-\frac{4}{p}}}\,ds<\infty. $$
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
$$\begin{aligned}& \begin{aligned}[b] \bigl\| |r_{1}|^{2}(r_{1}- \tilde{r}_{1})\bigr\| _{\mathbf{L}^{p'}}&\leq\|r_{1}- \tilde{r}_{1}\| _{\mathbf{L}^{p}}\|r_{1}\|^{2}_{\mathbf{L}^{\alpha}}\\ &\leq\|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}} \|r_{1}\|_{\mathbf{L}^{2}}^{2(1-b)}\| r_{1} \|_{\mathbf{L}^{p}}^{2b}, \end{aligned} \end{aligned}$$
(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)
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 \((1-\frac {2}{p})(1+2b)=1+\frac{2}{p}>1\).
Note that \((1-\frac{2}{p})(1+2b)>1\). Therefore, from (3.10), we have
$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}(t,s)\bigr\| _{\mathbf{L}_{\sigma}^{p'}\times\mathbf {L}_{\sigma}^{p'}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf {L}_{-\sigma}^{2}} \|B\|_{\mathbf{L}^{p'}\times\mathbf{L}^{p'}}\,ds \\ &\quad\leq C_{p,a_{1},a_{2},a_{3},a_{4}}\bigl(\|r_{1}\|^{2}_{\mathbb{B}}+ \|\tilde{r}_{1}\| ^{2}_{\mathbb{B}}+\|r_{2} \|^{2}_{\mathbb{B}}+\|\tilde{r}_{2}\|^{2}_{\mathbb {B}} \bigr) \\ &\qquad{}\times\bigl(\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+ \|r_{2}-\tilde{r}_{2}\|_{\mathbb {B}}\bigr)\int _{0}^{t}\frac{\log(2+|s|)^{1+2b}}{|t-s|^{1-\frac {2}{p}}(1+|s|)^{(1-\frac{2}{p})(1+2b)}}\,ds \\ &\quad\leq C_{p,a_{1},a_{2},a_{3},a_{4}}C_{3} \bigl(\|r_{1} \|^{2}_{\mathbb{B}}+\|\tilde{r}_{1}\| ^{2}_{\mathbb{B}}+ \|r_{2}\|^{2}_{\mathbb{B}}+\|\tilde{r}_{2} \|^{2}_{\mathbb {B}} \bigr)\frac{\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}-\tilde{r}_{2}\| _{\mathbb{B}}}{(1+|s|)^{1-\frac{2}{p}}}, \end{aligned}$$
where the constant
$$C_{3}=\sup_{t>0}\bigl(1+|t|\bigr)^{1-\frac{2}{p}}\int_{0}^{t}\frac {\log(2+|s|)^{1+2b}}{|t-s|^{1-\frac{2}{p}}(1+|s|)^{(1-\frac {2}{p})(1+2b)}}\,ds<\infty. $$
It is easy to obtain
$$\begin{aligned} & \bigl|\bigl\langle \psi_{0},2\psi_{E_{1}}\bigl(|r_{1}|-| \tilde{r}_{1}|\bigr) \bigl(|r_{1}|+|\tilde {r}_{1}|\bigr)\bigr\rangle \bigr|\leq2\|\psi_{0}\|_{\mathbf{L}^{\infty}}\|\psi_{E_{1}}\| _{\mathbf{L}^{\alpha}}\|r_{1}-\tilde{r}_{1}\|_{\mathbf{L}^{p}} \bigl(\|r_{1}\| _{\mathbf{L}^{p}}+\|\tilde{r}_{1}\|_{\mathbf{L}^{p}} \bigr), \\ & \bigl|\bigl\langle \psi_{0},|r_{1}|^{2}(r_{1}- \tilde{r}_{1})\bigr\rangle \bigr|\leq\bigl\| \psi _{0}\langle x\rangle ^{\sigma}\bigr\| _{\mathbf{L}^{\alpha}}\|r_{1}-\tilde{r}_{1} \|_{\mathbf {L}^{2}_{-\sigma}}\|r_{1}\|^{2}_{\mathbf{L}^{p}} \end{aligned}$$
for \(\frac{1}{\alpha}+\frac{2}{p}=\frac{1}{2}\). From (3.9) in Lemma 3.1, we get
$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}(t,s) \bigr\| _{\mathbf{L}_{\sigma}^{2}\times\mathbf {L}_{\sigma}^{2}\rightarrow\mathbf{L}_{-\sigma}^{2}\times\mathbf{L}_{-\sigma }^{2}}|G_{2}-\tilde{G}_{2}|\,ds \\ &\quad\leq C_{a_{1},a_{2},a_{3},a_{4}} \bigl(\|r_{1}\|_{\mathbb{B}}+\| \tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2}\|_{\mathbb{B}}+\| \tilde{r}_{2}\|_{\mathbb{B}}+\|r_{1}\| _{\mathbb{B}}^{2}+ \|\tilde{r}_{1}\|_{\mathbb{B}}^{2}+\|r_{2} \|_{\mathbb{B}}^{2}+\| \tilde{r}_{2}\|_{\mathbb{B}}^{2} \bigr) \\ &\qquad{}\times \bigl(\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+ \|r_{2}-\tilde{r}_{2}\| _{\mathbb{B}} \bigr)\int _{0}^{t}\frac{(\|\psi_{0}\langle x\rangle ^{\sigma}\|_{\mathbf {L}^{\alpha}}+\|\phi_{0}\langle x\rangle ^{\sigma}\|_{\mathbf{L}^{\alpha }})}{(1+|t-s|)\log^{2}(2+|t-s|)}\cdot\frac{\log ^{2}(2+|s|)}{(1+|s|)^{3-\frac{6}{p}}}\,ds \\ &\quad\leq C_{a_{1},a_{2},a_{3},a_{4}}C_{5} \bigl(\|r_{1} \|_{\mathbb{B}}+\|\tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2} \|_{\mathbb{B}}+\|\tilde{r}_{2}\|_{\mathbb{B}}+\|r_{1}\| _{\mathbb{B}}^{2}+\|\tilde{r}_{1}\|_{\mathbb{B}}^{2}+ \|r_{2}\|_{\mathbb{B}}^{2}+\| \tilde{r}_{2} \|_{\mathbb{B}}^{2} \bigr) \\ &\qquad{}\times\frac{\|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}-\tilde{r}_{2}\| _{\mathbb{B}}}{(1+|t|)\log^{2}(2+|t|)}, \end{aligned}$$
where we have the constant
$$C_{5}= \bigl(\bigl\| \psi_{0}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}}+\bigl\| \phi _{0}\langle x\rangle ^{\sigma} \bigr\| _{\mathbf{L}^{\alpha}} \bigr)\int_{0}^{t} \frac{\log ^{2}(2+|s|)}{(1+|s|)^{3-\frac{6}{p}}(1+|t-s|)\log^{2}(2+|t-s|)}\,ds<\infty. $$
Hence, we conclude that
$$\begin{aligned} &\|\mathcal{N}R_{1}-\mathcal{N}R_{2}\|_{\mathbb{B}\times\mathbb {B}} \\ &\quad\leq C_{a_{1},a_{2},a_{3},a_{4},p} \bigl(\|r_{1}\|_{\mathbb{B}}+\| \tilde{r}_{1}\| _{\mathbb{B}}+\|r_{2}\|_{\mathbb{B}}+\| \tilde{r}_{2}\|_{\mathbb{B}}+\|r_{1}\| _{\mathbb{B}}^{2}+ \|\tilde{r}_{1}\|_{\mathbb{B}}^{2} \\ &\qquad{}+\|r_{2}\|_{\mathbb{B}}^{2}+\| \tilde{r}_{2}\|_{\mathbb{B}}^{2} \bigr)\times \bigl( \|r_{1}-\tilde{r}_{1}\|_{\mathbb{B}}+\|r_{2}- \tilde{r}_{2}\|_{\mathbb{B}} \bigr). \end{aligned}$$
This completes the proof. □
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
$$\begin{aligned} \|R_{0}\|_{\mathbb{B}\times\mathbb{B}}\leq C_{6}\bigl\| R(0)\bigr\| _{\mathbf {L}^{2}_{\sigma}\times\mathbf{L}^{2}_{\sigma}}. \end{aligned}$$
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
$$\begin{aligned} \|R_{0}\|_{\mathbb{B}\times\mathbb{B}}\leq\frac{2-\operatorname{Lip}}{4}\Bigl(\sqrt {1+2\operatorname{Lip}C^{-1}_{a_{1},a_{2},a_{3},a_{4},p}}-1\Bigr). \end{aligned}$$
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. □