Almost Global Solutions to Hamiltonian Derivative Nonlinear Schrödinger Equations on the Circle

Abstract

Consider a class of 1-d Hamiltonian derivative nonlinear Schrödinger equations

$$\begin{aligned} \mathbf{i}{\psi }_t=\partial _{xx}\psi + V* \psi + \mathbf{i}\partial _{x}\big ({\partial _{\bar{\psi }} F(x,\psi ,\bar{\psi })}\big ),\quad x\in \mathbb {T}, \end{aligned}$$

where $V\in \Theta _m$ [$\Theta _m$ is defined in (1.5)]. The nonlinearity of these equations includes $(\psi _x,\bar{\psi }_x)$ and depends on space variable x periodically, which means that the nonlinearity is unbounded (see Definition 1.1) and the momentum set (see Definition 2.2) of the corresponding Hamiltonian function is unbounded. In this paper, we obtain that for any potential V outside a small measure subset of ${\Theta }_m$, if the initial value is smaller than $R\ll 1$ in p-Sobolev norm, then the corresponding solution to this equation is also smaller than 2R during a time interval $(-\,cR^{-r_*},cR^{-r_*})$ (for any given positive $r_*$). The main methods are constructing Birkhoff normal forms to unbounded Hamiltonian systems which have unbounded momentum sets and using the special symmetry of the Hamiltonian functions to control p-Sobolev norms of the solutions.

Appendix 1

1.1 Proof of Theorem 3

Proof

Denote

$$\begin{aligned} g^{(-1)}(u,{\bar{u}}):=\sum _{r=3}^{r_*+3} P_r(u,{\bar{u}}),\quad {{\mathcal {R}}}^{N(-1)}(u,{\bar{u}}):=0,\quad {{\mathcal {R}}}^{T(-1)}(u,{\bar{u}}):=\sum _{r=r_*+4}^{\infty } P_r(u,{\bar{u}}), \end{aligned}$$

where $P_r(u,{\bar{u}})$ is an r-degree homogeneous polynomial of $P(u,{\bar{u}})$. Thus (2.8) can be rewritten as

$$\begin{aligned} H^{(-1)}:= H =H_0 +g^{(-1)} + {{\mathcal {R}}}^{N(-1)}+{{\mathcal {R}}}^{T(-1)},\ \text{ defined } \text{ in } \ B_p(R_*). \end{aligned}$$

(7.1)

To start with, the results hold at rank $r=0$. For any $R<R_*$ and any N satisfying (4.22), we will look for a bounded Lie-transformation ${{\mathcal {T}}}_0$ to eliminate the non-normalized monomials of $\Gamma ^N_{\le 2}g^{(-1)}_{3}$. The Lie-transformation ${{\mathcal {T}}}_0$ is constructed from 1-time flow $\Phi _{S^{(0)}}^t$ of the following equations,

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {\dot{u}}_j= -\,\mathbf{i} \text{ sgn }(j)\cdot \nabla _{{\bar{u}}_j}S^{(0)} (u,{\bar{u}}),\\ \dot{{\bar{u}}}_j=\ \ \mathbf{i} \text{ sgn }(j) \cdot \nabla _{u_j} S^{(0)}(u,{\bar{u}}), \end{array}\right. \quad j\in \mathbb {Z}^*, \end{aligned}$$

where $S^{(0)}$ is undetermined. Under transformation ${{\mathcal {T}}}_0$ the new Hamiltonian $H^{(0)}$ has the following form,

$$\begin{aligned} H^{(0)}= & {} H^{(-1)}\circ {{\mathcal {T}}}_0=(H_0 +g^{(-1)}+{{\mathcal {R}}}^{T(-1)}) \circ {\Phi _{S^{(0)}}^{1}}\nonumber \\= & {} H_0 \nonumber \\&+ \,\{H_0, S^{(0)}\}+g^{(-1)}_3\end{aligned}$$

(7.2)

$$\begin{aligned}&+\!\sum _{t=4}^{r_*+3}g^{(-1)}_{t}\!+\! \sum _{\nu \ge 2}(H_0 )_{(\nu ,S^{(0)})} + \sum _{\nu \ge 1}\sum _{t=3}^{r_*+3}(g_t^{(-1)})_{(\nu , S^{(0)})}\nonumber \\&+\sum _{\nu \ge 0}\sum _{t=r_*+4}^{\infty }({{\mathcal {R}}}_t^{T(-1)})_{(\nu , S^{(0)})}, \end{aligned}$$

(7.3)

where $({\cdot })_{(\nu ,S^{(0)})}$ is defined in (4.25). The auxiliary Hamiltonian function $S^{(0)}$ is obtained by solving the following homological equation

$$\begin{aligned} \{H_0, S^{(0)}\}+\Gamma ^N_{\le 2 } P_{3}=Z_{3}. \end{aligned}$$

(7.4)

Using Remark 3.2, $\Gamma ^N_{>2} P_3$ and $\Gamma ^N_{\le 2}P_3$ are still of $\beta$-type symmetric coefficients semi-bounded by C(r) with $r=-1$, where C(r) is defined in (4.23). From Lemma 4.2, $Z_{3}$ is a 3-degree $(\gamma ,\alpha ,N)$-normal form of $\Gamma ^N_{\le 2}P_{3}$ and the Hamiltonian vector field of $S^{(0)}$ satisfies

$$\begin{aligned} \Vert X_{S^{(0)}}\Vert _{p}\le & {} \frac{N^{\alpha }}{\gamma }\Vert \lfloor X_{\Gamma _{\le 2}^{N}P_3}\rceil \Vert _{p-1}\nonumber \\\le & {} 8 c^2 C 3^{p+1} \frac{ N^{\alpha }}{{\gamma }} \Vert u\Vert _{p}\Vert u\Vert _{2}, \quad \text{ for } \text{ any } \ {(u,{\bar{u}})\in B_p(2R)}. \end{aligned}$$

(7.5)

From (7.4), the following holds true

$$\begin{aligned} (7.2)=Z_{3}(u,{\bar{u}})+ \Gamma ^N_{> 2}P_{3}. \end{aligned}$$

The Lie-transformation ${{\mathcal {T}}}_0$ satisfies

$$\begin{aligned}&\sup _{(u,{\bar{u}})\in B_p(R)}\Vert {{\mathcal {T}}}_0 (u,{\bar{u}})-(u,{\bar{u}})\Vert _{p} =\sup _{(u,{\bar{u}})\in B_p(R)} \Vert \Phi _{S^{(0)}}^1(u,{\bar{u}})-(u,{\bar{u}})\Vert _{p}\nonumber \\&\quad = \sup _{(u,{\bar{u}})\in B_p(R)}\big \Vert \int _{t=0}^1 X_{S^{(0)}}\circ \Phi _{S^{(0)}}^{\tau }(u,{\bar{u}})(\tau ) d\tau \big \Vert _{p}. \end{aligned}$$

(7.6)

Use the bootstrap method to estimate ${{\mathcal {T}}}_0$. First, assume that

$$\begin{aligned} \Phi _{S^{(0)}}^t:B_p(R)\rightarrow B_p(2R),\quad \text{ for } \text{ any } \quad t\in [0,1]. \end{aligned}$$

(7.7)

By (7.5)–(7.7), the following inequality holds true

$$\begin{aligned}&\sup _{(u,{\bar{u}})\in B_p(R)}\Vert {{\mathcal {T}}}_0 (u,{\bar{u}})-(u,{\bar{u}})\Vert _{p}\le \sup _{(u,{\bar{u}})\in B_p(2R)} \big \Vert \int _{t=0}^1 X_{S^{(0)}}(\tau ) d\tau \big \Vert _{p}\nonumber \\&\quad \quad \le 8 C c^2 \frac{ N^{\alpha }}{{\gamma }} 3^{p+1} (2R)^2. \end{aligned}$$

(7.8)

Since R is small enough, from (4.22) and (7.8), the transformation ${{\mathcal {T}}}_0$ satisfies

$$\begin{aligned} \sup _{(u,{\bar{u}})\in B_p(R)}\Vert {{\mathcal {T}}}_0 (u,{\bar{u}})-(u,{\bar{u}})\Vert _{p} \le R, \end{aligned}$$

which means

$$\begin{aligned} {{\mathcal {T}}}_0:B_p(R)\rightarrow B_p(2R). \end{aligned}$$

(7.9)

Denote ${{\mathcal {T}}}^{ (0)}:={{\mathcal {T}}}_0$. By (4.22), (7.6) and (7.8), it is verified that (4.24) holds for rank $r=0$:

$$\begin{aligned} \sup _{(u,{\bar{u}})\in B_p(R)}\Vert {{\mathcal {T}}}^{(0)}(u,{\bar{u}})-(u,{\bar{u}}) \Vert _{p} \le C(p,r_*) R^{2-\frac{1}{r_*+1}}. \end{aligned}$$

Set

$$\begin{aligned} Z^{(0)}:=Z^{(-1)}+Z_{3}, \quad {{\mathcal {R}}}^{N(0)}:=\mathcal{R}^{N(-1)}+ \Gamma ^N_{>2}g^{(-1)}_{3}. \end{aligned}$$

(7.10)

Since $Z_3$ and ${g}^{(-1)}_3$ have $\beta$-type symmetric coefficients, by Remark Remark 3.2 and Lemma 3.1, $Z^{(0)}$ and ${{\mathcal {R}}}^{N(0)}$ are still having $\beta$-type symmetric coefficients. Denote the $r_*+3$-degree polynomial of power series (7.3) as $g^{(0)}$ and the remainder as ${{\mathcal {R}}}^{T(0)}$, i.e.,

$$\begin{aligned} g^{(0)}:=\sum _{t= 4}^{r_*+3}g^{(0)}_t,\quad {{\mathcal {R}}}^{T(0)}:=\sum _{t>r_*+3}{{\mathcal {R}}}^{T(0)}_t, \end{aligned}$$

where for any $4\le t\le r_*+3$,

$$\begin{aligned} g^{(0)}_t:=g^{(-1)}_{t}+ (H_0)_{(t-2,S^{(0)})}+ \sum _{n'=1}^{t-3}\left( g^{(-1)}_{t-n'}\right) _{(n',S^{(0)})} \end{aligned}$$

and for any $t> r_*+3$

$$\begin{aligned} {{\mathcal {R}}}^{T(0)}_t:=(H_0)_{(t-2,S^{(0)})}+ \sum _{n'=1}^{t-3}\left( g^{(-1)}_{t-n'}\right) _{(n',S^{(0)})}+ \sum _{n'=1}^{t-3}\left( {{\mathcal {R}}}^{T(-1)}_{t-n'}\right) _{(n',S^{(0)})}. \end{aligned}$$

From Remark 4.4 and Lemma 3.1, $g^{(0)}$ and ${{\mathcal {R}}}^{T(0)}$ also have $\beta$-type symmetric coefficients. In order to estimate the coefficients of them, one needs to estimate the coefficients of $(H_0)_{(t-2,S^{(0)})}$, $\sum _{n'=1}^{t-3}(g^{(-1)}_{t-n'})_{(n',S^{(0)})}$ and $\sum _{n'=1}^{t-3}({{\mathcal {R}}}^{T(-1)}_{t-n'})_{(n',S^{(0)})}$. By Remark 4.4, we get that

$$\begin{aligned} \big | \big (({\tilde{g}}_{t-n'}^{(-1)})_{(n',S^{(0)})}\big )^{i}_{t,lk}\big | \le \frac{C_1^{t-2}}{\langle i\rangle ^{\beta } }\left( \frac{72cN^{\alpha +1}2^{\beta }}{\gamma }\right) ^{n'} \frac{1}{n'!}\prod _{n=0}^{n'-1}(t-n'+n+1). \end{aligned}$$

(7.11)

and

$$\begin{aligned} |\big ({({\tilde{Z}}_3)}_{(t-3,S^{(0)})} \big )^{i}_{t,lk} |\le \frac{ C_1^{t-2}}{\langle i\rangle ^{\beta }} \left( 72c\frac{N^{1+\alpha }2^{\beta }}{\ \gamma \ }\right) ^{t-3} \frac{1}{(t-3)!}\prod _{n=0}^{t-4}(4+n). \end{aligned}$$

(7.12)

When N satisfies (4.22), together with (7.11) and (7.12), it follows

$$\begin{aligned} \big | {({\widetilde{g}}^{(0)})}^{i}_{t,lk}\big |\le & {} |{({\tilde{g}}^{(-1)})}^{i}_{t,lk}|+ |{\big ((\tilde{Z_3})_{(t-3,S^{(0)})} \big )}^{i}_{t,lk} |+ \sum _{n'=1}^{t-3}\big |\big ({({\tilde{g}}^{(-1)}_{t-n'})}_{(n',S^{(0)}}\big )^{i}_{t,lk}\big |\\\le & {} \frac{ \big (C(\frac{2^{\beta } N^{2\alpha }}{\gamma })\big )^{t-2}}{\langle i\rangle ^{\beta } }:= \frac{\big (C(0)\big )^{t-2}}{\langle i\rangle ^{\beta }}. \end{aligned}$$

Similarly, ${{\mathcal {R}}}^{T(0)}(u,{\bar{u}})$ has still $\beta$-type symmetric coefficients semi-bounded by C(0).

Now assume that the results hold for rank $r< r_*$. By these assumptions, there exist a real number ${\tilde{R}}<R_*$ and a Lie-transformation which changes Hamiltonian (7.1) into the following form

$$\begin{aligned} H^{(r)}: = H \circ {{\mathcal {T}}}^{(r)}:=H_0 +Z^{(r)}+{{\mathcal {R}}}^{N(r)}+g^{(r)}+{{\mathcal {R}}}^{T(r)}, \end{aligned}$$

which is defined in $B_p(R_r)$ ($R<{\tilde{R}}<R_*$), where $R_r:=\frac{2r_*-r}{2r_*}R$. One should construct a bounded Lie-transformation ${{\mathcal {T}}}_r$ to eliminate the non-normalized monomials of $\Gamma ^N_{\le 2}g^{(r)}_{r+4}$. Because $g_{r+4}^{(r)}$ has $\beta$-type symmetric coefficients, by Remark 3.2, the coefficients of $\Gamma ^N_{\le 2}g^{(r)}_{r+4}$ and $\Gamma ^N_{>2}g^{(r)}_{r+4}$ are of $\beta$-type symmetric coefficients semi-bounded by C(r). Make use of the 1-time flow of the following equation, for any $j\in \mathbb {Z}^*$

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {\dot{u}}_j= \ \ \mathbf{i} \text{ sgn }(j)\cdot \partial _{{\bar{u}}_j}S^{(r)} (u,{\bar{u}}),\\ \dot{{\bar{u}}}_j=-\mathbf{i} \text{ sgn }(j)\cdot \partial _{u_j} S^{(r)}(u,{\bar{u}}), \end{array}\right. \end{aligned}$$

to define a Lie-transformation ${{\mathcal {T}}}_r$, under which the new Hamiltonian has the following form formally,

$$\begin{aligned}&H^{(r+1)}:=H^{(r)}\circ {{\mathcal {T}}}_r\nonumber \\&\quad = H_0 +Z^{(r)}+{{\mathcal {R}}}^{N(r)}\nonumber \\&\quad \quad +\, \{H_0, S^{(r)}\}+g^{(r)}_{r+4} \end{aligned}$$

(7.13)

$$\begin{aligned}&\qquad + \sum _{t=r+5}^{r_*+3}g^{(r)}_t + \sum _{\nu \ge 2}(H_0 )_{(\nu ,S^{(r)})} + \sum _{\nu \ge 1}(Z^{(r)}+g^{(r)} +\mathcal{R}^{N(r)})_{(\nu ,S^{(r)})}\nonumber \\&\qquad +\sum _{\nu \ge 0}({{\mathcal {R}}}^{T(r)})_{(\nu ,S^{(r)})}. \end{aligned}$$

(7.14)

The auxiliary Hamiltonian $S^{(r)}$ can be obtained by solving the following homological equation

$$\begin{aligned} \{H_0, S^{(r)}\}+\Gamma ^N_{\le 2}g^{(r)}_{r+4}=Z_{r+4}. \end{aligned}$$

(7.15)

From Lemma 4.2, $Z_{r+4}$ is an $(r+4)$-degree $(\gamma ,\alpha ,N)$-normal form of $\Gamma ^N_{\le 2}g^{(r)}_{r+4}$ and

$$\begin{aligned} (7.13)=Z_{r+4}+\Gamma ^N_{>2}g^{(r)}_{r+4}. \end{aligned}$$

The Hamiltonian vector field $X_{S^{(r)}}$ satisfies

$$\begin{aligned}&\sup _{(u,{\bar{u}})\in B_p(R_r)}\Vert X_{S^{(r)}} \Vert _{p}\le \sup _{(u,{\bar{u}})\in B_p(R_r)}\frac{N^{\alpha }}{\gamma }\Vert \lfloor X_{\Gamma _{\le 2}^N g^{(r)}_{r+4} } \rceil \Vert _{p} \nonumber \\&\qquad \le 8(C(r))^{r+2} (r+4)^{p+1}c^{r+3} \frac{N^{\alpha }}{\gamma }R^{r+3}. \end{aligned}$$

(7.16)

Using (7.16) and bootstrap method, suppose that

$$\begin{aligned} \Phi _{S^{(r)}}^t:B_p(R_{r+1})\rightarrow B_p(R_r), \end{aligned}$$

(7.17)

for any $t\in [0,1]$. Then we obtain

$$\begin{aligned}&\sup _{(u,{\bar{u}})\in B_p (R_r)}\Vert {{\mathcal {T}}}_r(u,{\bar{u}})-(u,{\bar{u}})\Vert _{p} = \sup _{(u,{\bar{u}})\in B_p(R_r)}\Vert \Phi _{S^{(r)}}^1(u,{\bar{u}})-(u,{\bar{u}})\Vert _{p}\nonumber \\&\quad = \sup _{(u,{\bar{u}})\in B_p(R_r)}\big \Vert \int _{t=0}^1 X_{S^{(r)}}\circ \Phi _{S^{(r)}}^{\tau }(u,{\bar{u}})(\tau ) d\tau \big \Vert _{p}\nonumber \\&\quad \le 16 (C(r))^{r+2} (r+4)^{p+1}c^{r+3} \frac{N^{\alpha }}{\gamma }R^{r+3}. \end{aligned}$$

(7.18)

By (4.22) and (7.18), the transformation ${{\mathcal {T}}}_r$ satisfies

$$\begin{aligned} \sup _{(u,{\bar{u}})\in B_p(R)}\Vert {{\mathcal {T}}}_r(u,{\bar{u}})-(u,{\bar{u}})\Vert _{p} \le \delta /2=(R_{r}-R_{r+1})/2, \end{aligned}$$

which verifies (7.17). Denote ${{\mathcal {T}}}^{ (r+1)}:={{\mathcal {T}}}^{ (r)}\circ {{\mathcal {T}}}_r$. By (7.17) and (7.18), noting that $R< {\tilde{R}}<R_*<1$, it holds

$$\begin{aligned}&\sup _{(u,{\bar{u}})\in B_p\big (({R_{r}+R_{r+1}})/2\big )}\Vert {{\mathcal {T}}}^{(r+1)}(u,{\bar{u}})-(u,{\bar{u}})\ \Vert _{p}\nonumber \\&\qquad \le \sup _{(u,{\bar{u}})\in B_p\big (\frac{R_{r}+R_{r+1}}{2}\big )} \big (\Vert {{\mathcal {T}}}^{ (r)}\circ {{\mathcal {T}}}_r (u,{\bar{u}})-{{\mathcal {T}}}_r (u,{\bar{u}})\Vert _{p}+\Vert \mathcal{T}_r (u,{\bar{u}})-(u,{\bar{u}})\Vert _{p}\big )\nonumber \\&\qquad \le \sup _{(u,{\bar{u}})\in B_p (R_{r})}\Vert \mathcal{T}^{(r)}(u,{\bar{u}})-(u,{\bar{u}})\Vert _{p}+16\big (C(r)\big )^{r+2} (r+4)^{p+1}c^{r+3} \frac{N^{\alpha }}{\gamma } R^{r+3}.\nonumber \\ \end{aligned}$$

(7.19)

Because $C(t)\le C(t+1)$ for any positive integer t, from (7.19), one has that

$$\begin{aligned}&\sup _{(u,{\bar{u}})\in B_p\big (({R_{r}+R_{r+1}})/2\big )}\Vert {{\mathcal {T}}}^{ (r+1)}(u,{\bar{u}})-(u,{\bar{u}})\ \Vert _{p}\\&\quad \le 16\sum _{t=3}^{r+3}\frac{N^{\alpha }}{\gamma }\big (C(t-3)\big )^{t-2}t^{p+1}c^{t-1} R^{t-1} +16\frac{N^{\alpha }}{\gamma }\big (C(r+1)\big )^{r+2} (r+4)^{p+1}c^{r+3} R_{r}^{r+3} \\&\quad \le 16\frac{N^{\alpha }}{\gamma }\sum _{t=3}^{r+4} \big (C(t-3)\big )^{t-2} \ t^{p+1}c^{t-1} R^{t-1} \le R^{2-\frac{1}{r_*+1}}. \end{aligned}$$

Denote

$$\begin{aligned} Z^{(r+1)}:=Z^{(r)}+Z_{r+4}, \quad {{\mathcal {R}}}^{N(r+1)}:=\mathcal{R}^{N(r)}+\Gamma ^N_{>2}g^{(r)}_{r+4}. \end{aligned}$$

(7.20)

By Remark 4.4 and Lemma 3.1, $Z^{(r+1)}$ and ${{\mathcal {R}}}^{N(r+1)}$ have $\beta$-type symmetric coefficients. Denote

$$\begin{aligned} g^{(r+1)}=\sum _{t=r+5}^{r_*+3}g^{(r+1)}_t,\quad {{\mathcal {R}}}^{T(r+1)}=\sum _{t>r_*+3}{{\mathcal {R}}}^{T(r+1)}_t, \end{aligned}$$

where

$$\begin{aligned}&g^{(r+1)}_t:=\\&\left\{ \begin{array}{lr} g^{(r)}_{t}+ \left( Z_{r+4}-\Gamma _{N\le 2}g_{r+4}^{(r)}\right) _{(\frac{t-r-4}{r+2},S^{(r)})} +\sum _{n'=0}^{[\frac{t-3}{r+2}]}(\mathcal{R}^{N(r)}_{t-n'(r+2)})_{(n',S^{(r)})}&{} \\ +\sum _{n'=0}^{[\frac{t-(r+4)}{r+2}]} \left( g^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})} +\sum _{n'=2}^{[\frac{t-3}{r+2}]}\left( Z^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})},&{} (r+2) | (t-2);\\ g^{(r)}_{t} +\sum _{n'=0}^{[\frac{t-3}{r+2}]}(\mathcal{R}^{N(r)}_{t-n'(r+2)})_{(n',S^{(r)})}+\sum _{n'=0}^{[\frac{t-(r+4)}{r+2}]} \left( g^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})}&{}\\ +\sum _{n'=2}^{[\frac{t-3}{r+2}]}\left( Z^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})},&{} (r+2) \not \mid (t-2); \end{array} \right. \end{aligned}$$

and

$$\begin{aligned}&{{\mathcal {R}}}^{T(r+1)}_{t}:= \\&\left\{ \begin{array}{l@{\quad }l} \left( Z_{r+4}-\Gamma ^N_{\le 2}g_{r+4}^{(r)}\right) _{ \left( \frac{t-r-4}{r+2},S^{(r)}\right) } +\sum _{n'=1}^{[\frac{t-3}{r+2}]}\left( \mathcal{R}^{N(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})}&{} \\ +\sum _{n'=1}^{[\frac{t-(r+4)}{r+2}]} \left( g^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})} +\sum _{n'=1}^{[\frac{t-3}{r+2}]} \left( Z^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})} &{} \\ +\sum _{n'=0}^{[\frac{t-r_*-4}{r+2}]}\left( \mathcal{R}^{T(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})},&{} \text{ when }\ (r+2)|(t-2);\\ \sum _{n'=1}^{[\frac{t-3}{r+2}]}\left( \mathcal{R}^{N(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})}+\sum _{n'=1}^{[\frac{t-(r+4)}{r+2}]} \left( g^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})}&{} \\ +\sum _{n'=1}^{[\frac{t-3}{r+2}]} \left( Z^{(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})} +\sum _{n'=0}^{[\frac{t-r_*-4}{r+2}]}\left( \mathcal{R}^{T(r)}_{t-n'(r+2)}\right) _{(n',S^{(r)})},&{} \text{ when }\ (r+2)\not \mid (t-2); \end{array} \right. \end{aligned}$$

where [a] denotes the integer part of the real number a. Using Lemma 3.1 and Remark 4.4, from the fact that $g^{(r)}, \ {{\mathcal {R}}}^{T(r)}, \ {{\mathcal {R}}}^{N(r)}$ and $Z^{(r)}$ have $\beta$-type symmetric coefficients semi-bounded by C(r), then $g^{(r+1)}$ and ${{\mathcal {R}}}^{T(r+1)}$ also have $\beta$-type symmetric coefficients.

Using Remark 4.4, the following estimates hold: for any $|l+k|=t$ with ${{\mathcal {M}}}(l,k)=i\in M_{(g^{(r)}_{r+4})_{(\frac{t-r-4}{r+2},S^{(r)})}}$,

$$\begin{aligned}&\big |\widetilde{\big ((Z_{r+4}-\Gamma ^N_{\le 2 }g_{r+4}^{(r)})_{(\frac{t-r-4}{r+2},S^{(r)})} \big )}^{i}_{r,lk}\big |\nonumber \\&\quad \le \frac{\big (C(r)\big )^{r+2}}{ \langle i\rangle ^{\beta }} \left( 2^{\beta +2}(r+4)^2\big (C(r)\big )^{r+2}c \frac{N^{\alpha +1}}{\gamma }\right) ^{\frac{t-r-4}{r+2}}\frac{ (2N)^{(t-r-4)(r+2)}}{(\frac{t-r-4}{r+2})!}\prod _{n=1}^{\frac{t-r-4}{r+2}}\nonumber \\&\quad \quad \big (t+1-n(r+2)\big ); \end{aligned}$$

(7.21)

for any $|l+k|=t$ with ${{\mathcal {M}}}(l,k)=i\in M_{(g^{(r)}_{t-n'(r+2)})_{(n',S^{(r)})}}$,

$$\begin{aligned}&\big |\big (\widetilde{(g^{(r)}_{t-n'(r+2)})_{(n',S^{(r)})}\big )}^{i}_{t,lk}\big | \nonumber \\&\quad \le \!\frac{ \big (\!C(r)\!\big )^{t\! -\! n'\! (r+2\!)-2}}{\langle i\rangle ^{\beta } }\left( \! 2^{\beta +2}(r+4)^2 \big (C(r)\big )^{r+2} c \frac{N^{\alpha +1}}{\gamma }\right) ^{n'}\frac{(2N)^{(r+2)n'} }{n'!} \!\prod _{n=1}^{n'} \nonumber \\&\quad \quad \big (t + 1- n(r+2)\big ); \end{aligned}$$

(7.22)

for any $|l+k|=t$ with ${{\mathcal {M}}}(l,k)=i\in M_{(Z^{(r)}_{t-n'(r+2)})_{(n',S^{(r)})}},$

$$\begin{aligned}&\Big |\left( \widetilde{(Z^{(r)}_{t-n'(r+2)})}_{(n',S^{(r)})}\right) ^{i}_{t,lk}\Big | \nonumber \\&\quad \le \!\frac{\big (C(r)\big )^{\! t\!-\!n'\!(r+2\!)\!-2}}{\langle i\rangle ^{\beta } }\left( 2^{\beta +2}(r+4)^2\left( \!C(r)\right) ^{r+2}c \frac{N^{\alpha +1}}{\gamma }\right) ^{n'}\frac{(2N)^{(r+2)n'}}{n'!}\prod _{n=1}^{n'} \nonumber \\&\quad \quad \big (t + 1 - n(r+2)\big ), \end{aligned}$$

(7.23)

for any $|l+k|=t$ with ${{\mathcal {M}}}(l,k)= i\in M_{({{\mathcal {R}}}^{N(r)}_{t-n'(r+2)})_{(n',S^{(r)})}}$

$$\begin{aligned}&\big |\left( \widetilde{(\mathcal{R}^{N(r)}_{t-n'(r+2)})}_{(n',S^{(r)})}\right) ^{i}_{t,lk}\big | \nonumber \\&\quad \le \frac{\big (C(r)\big )^{\!t\!-\!n'\!(\!r\!+\!2\!)\!-\!2}}{\langle i\rangle ^{\beta } }\left( 2^{\beta +2} (r+4)^2\big (C(r)\big )^{r+2}c \frac{N^{\alpha +1}}{\gamma }\right) ^{n'}\frac{(2N)^{(r+2)n'}}{n'!} \prod _{n=1}^{n'} \nonumber \\&\quad \quad \big (t+1-n(r+2)\big ). \end{aligned}$$

(7.24)

and for any $|l+k|=t$ with ${{\mathcal {M}}}(l,k)= i\in M_{({{\mathcal {R}}}^{T(r)}_{t-n'(r+2)})_{(n',S^{(r)})}}$

$$\begin{aligned}&\big |\left( \widetilde{ (\mathcal{R}^{T(r)}_{t-n'(r+2)})}_{(n',S^{(r)})}\right) ^{i}_{t,lk}\big |\nonumber \\&\quad \le \frac{\big (C(r)\big )^{t-n'(r+2)-2}}{\langle i\rangle ^{\beta } }\left( 2^{\beta +2} (r+4)^2\big (C(r)\big )^{r+2}c \frac{N^{\alpha +1}}{\gamma }\right) ^{n'}\frac{(2N)^{(r+2)n'}}{n'!} \prod _{n=1}^{n'} \nonumber \\&\quad \quad \big (t +1-n(r+2)\big ). \end{aligned}$$

(7.25)

By (7.21)–(7.25) and assumption (4.22), for any $r+5\le t\le r_*+3$, $|l+k|=t$ and $i\in M_{g^{(r+1)}_{t}}$, the following estimate holds

$$\begin{aligned} |({\tilde{g}}^{(r+1)})^{i}_{t,lk}|\le \big (C(r+1)\big )^{t-2} \frac{1}{\langle i \rangle ^{\beta } }, \end{aligned}$$

(7.26)

which means that $g^{(r+1)}(u,{\bar{u}})$ has $\beta$-type symmetric coefficients semi-bounded by $C(r+1)>0$.

Similarly, ${{\mathcal {R}}}^{T(r+1)}(u,{\bar{u}})$ and ${{\mathcal {R}}}^{N(r+1)}(u,{\bar{u}})$ are also of $\beta$-type symmetric coefficients semi-bounded by $C(r+1)>0$. $\square$

1.2 The Result to DNLS Eq. (1.8)

Suppose that the function $F(x,\psi ,\bar{\psi })$ in Eq. (1.8) satisfies the followings.

$\mathbf{B}_\mathbf{1}$:

$F(x,\xi , \eta )$ is a polynomial about $(\xi , \eta )$ in a neighborhood of the origin and satisfies

$$\begin{aligned} \overline{F(x,\psi ,\bar{\psi })}=F(x,\psi ,\bar{\psi }), \ F(x,\psi ,\bar{\psi })= F(x+2\pi ,\psi ,\bar{\psi }) \end{aligned}$$

(7.27)

and $F(x,\psi ,\bar{\psi })$ vanishes at least at order 2 in $(\psi ,\bar{\psi })$ at the origin.

$\mathbf{B}_\mathbf{2}$:

For any fixed $(\psi ,\bar{\psi })$ being in a neighborhood of the origin, $F(\cdot , \psi ,\bar{\psi }) \in H^{\beta +1}(\mathbb {T},\mathbb {C})$ ($\beta$ is a big enough positive real number) satisfies

$$\begin{aligned} F(x+2\pi ,\psi ,\bar{\psi })=F(x,\psi ,\bar{\psi }). \end{aligned}$$

Then (1.8) becomes a Hamiltonian PDE with a real value Hamiltonian function³

$$\begin{aligned} H_{(1.8)} (\psi ,\bar{\psi })= & {} \int _{\mathbb {T}} -|\partial _{x} \psi |^2+(V* \psi )\bar{\psi } + \mathbf{i}\frac{1}{2}\partial _x F(x,\psi ,{\overline{\psi }}) \nonumber \\&+ \mathbf{i}\partial _{\psi } F (x,\psi ,{\overline{\psi }})\psi _xdx \end{aligned}$$

(7.29)

on symplectic space $({\widetilde{H}}^{p}(\mathbb {T},\mathbb {C}),$$w^0)$, where

$$\begin{aligned} {\widetilde{H}}^{p}(\mathbb {T},\mathbb {C}):=\{ (\xi ,\eta )\ |\ (\xi ,\eta )\in {H}^{p}(\mathbb {T},\mathbb {C}),\ \bar{\xi }=\eta \} \end{aligned}$$

and

$$\begin{aligned} w^0 = J_0 d \psi \wedge d \bar{\psi },\quad J_0^{-1}= \left( \begin{array}{c@{\quad }c} 0 &{} -\mathbf{i} \\ \mathbf{i} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(7.30)

The corresponding Hamiltonian vector of $H_{(1.8)}(\psi ,\bar{\psi })$ under symplectic form $w^0$ is

$$\begin{aligned} X^{w^0}_{H_{(1.8)}}:=\left( - \mathbf{i}\partial _{\bar{\psi }} H_{(1.8)},\ \mathbf{i}\partial _{{\psi }} H_{(1.8)} \right) ^{T} \end{aligned}$$

and the Eq. (1.8) can be written as

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {\dot{\psi }} = -\,\mathbf{i} \partial _{\bar{\psi }} H_{(1.8)}(\psi ,\bar{\psi }),\\ \dot{{\bar{\psi }}} =\ \mathbf{i} \partial _{\psi } H_{(1.8)}(\psi ,\bar{\psi }). \end{array} \right. \end{aligned}$$

(7.31)

Theorem 5

Suppose that the Eq. (1.8) satisfies assumptions $\mathbf{B_1}$–$\mathbf{B_2}$. For any integer $r_*>1$, there exist an almost full measure set $\widetilde{\Theta }^0_m \subset \Theta ^0_m$ and $p_*>0$ such that for any fixed $V \in \widetilde{\Theta }^0_m$ and any p fulfilling $(\beta -4)/2>p>p_{*}$, if the initial data of the solution to (1.8) satisfies

$$\begin{aligned} \Vert \psi (x,0)\Vert _{H^p(\mathbb {T},\mathbb {C})}\le R<R_{*}, \end{aligned}$$

then one has

$$\begin{aligned} \Vert \psi (x,t)\Vert _{H^p(\mathbb {T},\mathbb {C})}<2R, \qquad \forall ~|t| \prec R^{-r_*-1}. \end{aligned}$$

The proof of Theorem 5 is similar to Theorem 1. We list the main idea.

• 1. Under Fourier transformation, constructing small-norm almost global solutions to Eq. (1.8) becomes into studying the long time stability of the solutions around the origin point to the follow Hamiltonian equation

  $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {\dot{u}}_j= -\,\mathbf{i} \partial _{{{\bar{u}}}_j} H_{(2.8)} (u,{\bar{u}}),\\ \dot{{\bar{u}}}_j=\ \ \mathbf{i} \partial _{ u_j} H_{(2.8)}(u,{\bar{u}}), \end{array}\right. \quad j\in \mathbb {Z} \end{aligned}$$

  (7.32)

  defined in $({{\mathcal {H}}}^p (\mathbb {Z},\mathbb {C}),{\tilde{w}}^0)$, with a symplectic form ${\tilde{w}}^0:=\mathbf{i}\sum _{j\in \mathbb {Z}} d u_j \wedge d{\bar{u}}_j$ and a Hamiltonian function

  $$\begin{aligned} H_{(7.33)}(u,{\bar{u}})=H_0 + P(u,{\bar{u}}), \end{aligned}$$

  (7.33)

  where

  $$\begin{aligned} H_0:=\sum _{j\in {\mathbb {Z}}} \omega ^0(V)|u_j|^2,\quad \omega _j^0(V):=-j^2+\frac{v_j}{\langle j \rangle ^m}\in \mathbb {R}, \ \omega ^0_j(V)=\omega ^0_{-j}(V). \end{aligned}$$

  (7.34)

  The polynomial

  $$\begin{aligned} P (u,{\bar{u}})=\sum _{t= 3}^n P_t(u,{\bar{u}}),\quad P_t(u,{\bar{u}}):= \sum \limits _{|k+l|=t, {{\mathcal {M}}}(l,k)=i\in M_{P_t}\subset \mathbb {Z} } P^i_{t,lk} u^{l}{\bar{u}}^k \end{aligned}$$

  does not have $\beta$-type symmetric coefficients, but its coefficients have the following form

  $$\begin{aligned} P^i_{t,lk}:=\sum _{({l}^0,{k}^0,i^0)\subset {{\mathcal {A}}}_{P^i_{t,lk} }}\ {P}^{i({l}^0,{k}^0,i^0)}_{t,lk}\left( {{\mathcal {M}}}({l}^0, {k}^0)-\frac{i^0}{2}\right) , \end{aligned}$$

  where

  $$\begin{aligned} {{\mathcal {A}}}_{P^i_{t,lk}}\subset \{({\tilde{l}},{\tilde{k}},{\tilde{i}}) \ |\ 0\le {\tilde{l}}_j\le l_j,\ 0\le {\tilde{k}}_j\le k_j,\ \text{ for } \text{ any } j\in \mathbb {Z},\ ({\tilde{l}},{\tilde{k}})\in \mathbb {N}^{{{\mathbb {Z}}}}\times \mathbb {N}^{{{\mathbb {Z}}}},\ {\tilde{i}}\in \mathbb {Z}\}, \end{aligned}$$

  and for any $(l^0,k^0,i^0)\in {{\mathcal {A}}}_{P^i_{t,lk}}$, it holds true

  $$\begin{aligned} (k- k^0,l-{l}^0, i^0-2i)\in {{\mathcal {A}}}_{P^{-i}_{t,kl}},\quad \overline{P^{i(l^0,k^0,i^0)}_{t,lk}}=P^{-i(k-k^0, {l}-l^0, i^0-2i)}_{t,kl}. \end{aligned}$$

  (7.35)

  Moreover, there exists a constant $C>0$ such that for any $l,k\in \mathbb {N}^{{\mathbb {Z}}}$ with $|l+k|=t$ and ${{\mathcal {M}}}(l,k)=i$, the following inequality holds true

  $$\begin{aligned} \sum _{(l^0,k^0,i^0)\in {{\mathcal {A}}}_{P^i_{t,lk}}} |P^{i(l^0,k^0,i^0)}_{t,lk}|\cdot \max \{ \langle i^0\rangle , \langle i^0-2i\rangle \} \le \frac{ C^{t-2}}{\langle i\rangle ^{\beta }}. \end{aligned}$$

  (7.36)

  Similar to Proposition 3.1, we get that the Hamiltonian vector field $X_P:B_p(R)\subset {{\mathcal {H}}}^p(\mathbb {Z},\mathbb {C})\rightarrow {{\mathcal {H}}}^{p-1}(\mathbb {Z},\mathbb {C})$($0<R \ll 1$).

  Fortunately, the condition (7.35) and (7.36) help us to get that the t-degree homogeneous $P_t(u,{\bar{u}})$ holds the inequalities in Proposition 3.2 and Corollary 1 under symplectic form ${\tilde{w}}^0$.

• 2. We establish the high order Birkhoff normal form to system (7.32) under a bounded symplectic transformation around the origin point. From the definitions of symplectic structure ${\tilde{w}}^0$ and $\omega (V):=(\omega _j(V))_{j\in \mathbb {Z}}$, for any $j\in \mathbb {Z}$ and any (l, k) fulfilling $\sum _{j\in \mathbb {Z}} |l_j+l_{-j}-k_j-k_{-j}|=0$ we obtain

  $$\begin{aligned} \{ u^l{\bar{u}}^k, \Vert u\Vert _p^2\}_{{\tilde{w}}^0}=\sum _{j\in \mathbb {Z}} \mathbf{i}(l_j+l_{-j}-k_j-k_{-j})u^l{\bar{u}}^k=0 \end{aligned}$$

  (7.37)

  and for any $V\in \Theta _m^0$

  $$\begin{aligned} \{ u_j{\bar{u}}_{-j}+{\bar{u}}_ju_{-j}, \ \omega _j(V) |q_j|^2+\omega _{-j}(V)|q_j|^2 \}_{{\tilde{w}}^0}=0. \end{aligned}$$

  (7.38)

  From (7.37) and (7.38), although we can not found a symplectic transformation to eliminate the term $u^l{\bar{u}}^k$ in $P(u,{\bar{u}})$ which satisfies $\sum _{j\in \mathbb {Z}} |l_j+l_{-j}-k_j-k_{-j}|=0$, this term will not effect the ${{\mathcal {H}}}^p(\mathbb {Z},\mathbb {C})$-norm of the solution to (7.32). In this case $(\gamma ,\alpha ,N)$ non resonant condition to $\omega (V)$ up to r order has some differences with the one (defined in Definition 4.1). Now it should be hold for any (l, k) fulfilling $\Gamma _{>N}(l+k)\le 2,$$\sum _{j\in \mathbb {Z}} |l_j+l_{-j}-k_j-k_{-j}|\ne 0$ and $3\le |l+k|\le r,$ and satisfies

  $$\begin{aligned} |\langle \omega (V), \ (l-k)\rangle |>\frac{\gamma M_{l,k}}{N^{\alpha }}. \end{aligned}$$

  (7.39)

  By the $(\gamma ,\alpha ,N)$-non resonant condition (7.39) to $\omega (V)$ up to r order, there exists a symplectic transformation under which Hamiltonian system (7.32) is changed into

  $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {\dot{u}}_j= -\,\mathbf{i} \partial _{{{\bar{u}}}_j} H^{(r)} (u,{\bar{u}}),\\ \dot{{\bar{u}}}_j=\ \ \mathbf{i} \partial _{ u_j} H^{(r)} (u,{\bar{u}}), \end{array}\right. \quad j\in \mathbb {Z} \end{aligned}$$

  (7.40)

  with

  $$\begin{aligned} H^{(r+3)}(u,{\bar{u}})=H_0+\underbrace{Z^{(r+3)}(u,{\bar{u}})+{{\mathcal {R}}}^{N(r+3)}(u,{\bar{u}})+{{\mathcal {R}}}^{T(r+3)}(u,{\bar{u}})}_{P^{(r+3)}(u,{\bar{u}})}, \end{aligned}$$

  $P^{(r+3)}(u,{\bar{u}})$ still satisfies (7.35) and (7.36) for a new positive constant $C_*$ taking place C in (7.36). The $(r+3)$-degree $(\gamma ,\alpha ,N)$-normal from $Z^{(r+3)}(u,{\bar{u}})$ depends on not only $|u_j|^2$ but also $(u_j{\bar{u}}_{-j}+{\bar{u}}_ju_{-j})_{j\in \mathbb {Z}}$. From (7.38), the normal form $Z(u,{\bar{u}})$ still satisfies the estimate of Lemma 4.1 under symplectic form ${\tilde{w}}^0$. For any N satisfying (4.22), we have

  $$\begin{aligned} \sup _{\Vert (u,{\bar{u}})\Vert _p\le R} |\{Z^{(r+3)}(u,{\bar{u}})+{{\mathcal {R}}}^{N(r+3)}(u,{\bar{u}})+{{\mathcal {R}}}^{T(r+3)}(u,{\bar{u}}),\Vert u\Vert _p^2 \}_{{\tilde{w}}^0}| \prec R^{r+1}. \end{aligned}$$

  Using the same method, the long time stability result of (7.32) under non-resonant assumption is obtained.

Final, similar to Lemma 6.1, there exists a almost full measure subset $\widetilde{\Theta }_m^0 \subset {\Theta }_m^0$, for any $V\in \widetilde{\Theta }_m^0$, the frequencies $\omega (V)$ satisfies the non resonant condition (7.39).

1.3 Proof of Theorem 4

Prior to giving the proof of Theorem 4, we list some results.

Consider an equation

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_t = -\omega _{\varepsilon } u+ \mathbf{i} \sigma \partial _{{\bar{u}}}P(u,{\bar{u}}), \\ u(0)=u_0 \end{array}\right. \end{aligned}$$

(7.41)

Let $\omega _{\varepsilon }=\text{ diag }\{ \big ((\mathbf{i}+\varepsilon ) \omega _j\big )_{j\in \mathbb {Z}^*}\}$, be an operator with domain $D(\omega _{\varepsilon })=D(\omega )$, where $D(\omega )=\ell ^2_{2}(\mathbb {Z}^*,\mathbb {C})$. For any $\alpha >0$, we define

$$\begin{aligned} \omega _{\varepsilon }^{\alpha }u=\text{ diag }\{(PV[((i+\varepsilon )j^2)^{\alpha }]u_j)_{j\in \mathbb {Z}^*}\}, \end{aligned}$$

where $PV[z^{\alpha }]$ denote the principal value of $z^{\alpha }$.

Theorem 6

Let $1/2\le \alpha <1$ and $u_0\in D(\omega _{\varepsilon }^{\alpha })=\ell ^2_{2\alpha }(\mathbb {Z}^*,\mathbb {C})$. Suppose that there exist two non-negative and non-decreasing functions g, f defined on $[0,\infty )$ such that

$$\begin{aligned} \Vert \partial _{{\bar{u}}}P(u,{\bar{u}})\Vert _{\ell ^2}\le f(\Vert \omega _{\varepsilon }^{\alpha } u \Vert _{\ell ^2}) \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _{{\bar{u}}}P(u,{\bar{u}})-\partial _{{\bar{u}}}P(v,{\bar{v}})\Vert _{\ell ^2}\le g(\Vert \omega _{\varepsilon }^{\alpha } u \Vert _{\ell ^2}+\Vert \omega _{\varepsilon }^{\alpha } v \Vert _{\ell ^2})(\Vert \omega _{\varepsilon }^{\alpha }(u-v) \Vert _{\ell ^2}),\quad \text{ for } \ u\in D(\omega _{\varepsilon }^{\alpha }). \end{aligned}$$

Then for any $\varepsilon >0$ there exists $T_0>0$ such that Eq. (7.41) has a unique local strict solution.

$$\begin{aligned} u\in C(0,T_0, D(\omega _{\varepsilon }^{\alpha }))\cap C^1(0,T_0,\ell ^2) \cap C(0,T_0, D(\omega _{\varepsilon })), \end{aligned}$$

where $T_0$ is a constant depending only on $\Vert u_0\Vert _{2\alpha }$.

The proof of Theorem 6 is found in Theorem 2.1 in [1].

Remark 7.1

We call u(t) be a strict solution to (7.41) on [0, T] if

• 1. u(t) is strongly continuous on [0, T] and strongly continuously differentiable on (0, T],

• 2. for each $t\in (0,T]$u(t) belongs to $D(\omega _{\varepsilon })$ and $\omega _{\varepsilon }u(t)$ is strongly continuous in (0, T],

• 3. u(t) satisfies (7.41).

Theorem 7

If $u_0\in D(\omega _{\varepsilon }^{\infty })=\cap _{\alpha >0} D(\omega _{\varepsilon }^{\alpha })$, then the solution u to Eq. (7.41) belongs to $C^{\infty }(0,T_0,D(\omega ^{\infty })).$

In order to prove Theorem 7, we need the following Lemma.

Lemma 7.1

Let $y(t)\in D(\omega _{\varepsilon }^{\alpha })$ and $\omega _{\varepsilon }^{\alpha } y(t)$ be continuous in [0, T] for some $\alpha >0$. Put

$$\begin{aligned} v(t)=\int _0^t e^{-(t-\tau )\omega _{\varepsilon } }y(\tau )d\tau . \end{aligned}$$

Then for any $\beta$ with $0<\beta <1$, $v(t)\in D(\omega _{\varepsilon }^{\alpha +\beta })$ for $t\in [0,T]$ and

$$\begin{aligned} \omega _{\varepsilon }^{\alpha +\beta } v(t)=\int _0^t \omega ^{\beta }_{\varepsilon }e^{-(t-\tau )\omega ^{\alpha }_{\varepsilon } } y(\tau )d\tau . \end{aligned}$$

Moreover, $\omega _{\varepsilon }^{\alpha +\beta } v(t)$ is Hölder continuous in [0, T] with exponent $(1-\alpha )$.

The proof of Lemma 7.1 see Lemma 2.5 in [1]. Now we prove the Theorem 7.

Proof

From (7.41), we have

$$\begin{aligned} u(t)=e^{-\omega _{\varepsilon } t}u_0+\int _0^t e^{-(t-\tau )\omega _{\varepsilon }} \mathbf{i} \sigma \partial _{{\bar{u}}}P(u(\tau ),{\bar{u}}(\tau ))d \tau . \end{aligned}$$

(7.42)

Assumptions on $u_0$ imply that the first term belongs to $C^{\infty }(0,T_0, D(\omega _{\varepsilon }^{\infty }))$. Because

$$\begin{aligned} \partial _{{\bar{u}}}P(u,{\bar{u}})-\partial _{{\bar{u}}}P(v,{\bar{v}})= \partial _{uu}^2 P(u_{\theta },{\bar{u}}_{\theta })\cdot (u-v) + \partial _{u{\bar{u}}}^2 P(u_{\theta '},{\bar{u}}_{\theta '})\cdot ({\bar{u}}-{\bar{v}}), \end{aligned}$$

where $u_{\theta }=u+\theta (v-u)$, $u_{\theta '}=u+\theta ' (v-u)$ with $\theta ,\ \theta '\in (0,1)$. When $u,v\in D(\omega _{\varepsilon }^{\alpha })$, it follows

$$\begin{aligned} \Vert u_{\theta }\Vert _{2\alpha }\le \Vert u\Vert _{2\alpha }+\Vert v\Vert _{2\alpha },\quad \Vert u_{\theta '}\Vert _{2\alpha }\le \Vert u\Vert _{2\alpha }+\Vert v\Vert _{2\alpha }. \end{aligned}$$

Using Lemma 3.2, we get that

$$\begin{aligned} \Vert \partial _{{\bar{u}}}P(u,{\bar{u}})\Vert _{\ell ^2}\le f(\Vert \omega _{\varepsilon }^{\alpha } u\Vert _{\ell ^2})=f(\Vert u\Vert _{2 \alpha }) \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _{{\bar{u}}}P(u,{\bar{u}})-\partial _{{\bar{u}}}P(v,{\bar{v}})\Vert _{\ell ^2}\le g(\Vert u\Vert _{2\alpha }+\Vert v\Vert _{2\alpha })\cdot \Vert u-v\Vert _1,\quad 1>\alpha >1/2, \end{aligned}$$

where g and f are functions defined on $[0,\infty )$ which are non-negative and non-decreasing. Form Theorem 6, note that $u\in C(0,T_0, D(\omega _{\varepsilon }^{\alpha }))$ for any $\alpha$ with $1/2< \alpha <1$.

In much the same way as above we see that

$$\begin{aligned} \partial _{{\bar{u}}}P(u,{\bar{u}})(t)\in D(\omega _{\varepsilon }^{\alpha +\beta -\frac{1}{2}})\ \text{ and } \ \omega _{\varepsilon }^{\alpha +\beta -\frac{1}{2}}\partial _{{\bar{u}}}P(u,{\bar{u}})(t)\ \text{ is } \text{ continuous } \text{ on } [0,T_0]. \end{aligned}$$

Then applying Lemma 7.1, for any $\beta '\in (0,1)$, we have $u\in C(0,T_0, D(\omega ^{\alpha +\beta +\beta '-1}))$. Repeatly, this argument yields

$$\begin{aligned} u\in C(0,T_0, D(\omega _{\varepsilon }^{\infty })). \end{aligned}$$

(7.43)

Then in virtue of (7.41) we obtain

$$\begin{aligned} u_t\in C(0,T_0, D(\omega _{\varepsilon }^{\infty })). \end{aligned}$$

(7.44)

Further regularities with respect to t follows immediately from (7.43) and (7.44). $\square$

Let us prove Theorem 4.

Proof

Since $D(\omega ^{\infty })=D(\omega _{\varepsilon }^{\infty })$ is dense in $D(\omega ^{p/2})$, we find sequence $\{u_{0\varepsilon }\} \subset D(\omega _{\varepsilon }^{\infty })$ such that

$$\begin{aligned} u_{0\varepsilon } \rightarrow u_0 \quad \text{ strongly } \text{ in } \ D(\omega ^{p/2}). \end{aligned}$$

Let $u_{\varepsilon }(t)$ be a solution of initial value problem.

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} (u_{\varepsilon })_t = -\omega _{\varepsilon } u_{\varepsilon }+\mathbf{i} \sigma \partial _{{\bar{u}}}P (u_{\varepsilon },{\bar{u}}_{\varepsilon }), \\ u_{\varepsilon }(0)=u_{0\varepsilon }. \end{array}\right. \end{aligned}$$

(7.45)

Then using Theorem 7, there exists $u_{\varepsilon }(t)\in C^{\infty }(0,T_0; D(\omega ^{\infty }))$. Applying ${\tilde{\omega }}^{p/2}$ (${\tilde{\omega }}:={\textit{diag}}\{(j^2)_{j\in \mathbb {Z}^*}\}$ commuted with $\omega$) to the both sides of Eq. (7.45), and taking the inner product of the resultant equation with ${\tilde{\omega }}^{p/2}{\bar{u}}_{\varepsilon }$, we have

$$\begin{aligned} \frac{d}{dt}\Vert {\tilde{\omega }}^{p/2}u_{\varepsilon } \Vert ^2_{\ell ^2}+2\varepsilon \Vert {\tilde{\omega }}^{p/2} \omega ^{1/2} u_{\varepsilon }\Vert ^2_{\ell ^2}=\{P(u_{\varepsilon },{\bar{u}}_{\varepsilon }), \Vert {\tilde{\omega }}^{p/2}u_{\varepsilon }\Vert ^2_{\ell ^2}\}. \end{aligned}$$

(7.46)

If $p>3$,

$$\begin{aligned} \Vert {\tilde{\omega }}^{p/2}u_{\varepsilon }\Vert ^2_{\ell ^2}=\Vert u_{\varepsilon }\Vert _p^2 \end{aligned}$$

and using Lemma 3.2 we get

$$\begin{aligned} \frac{d}{dt}\Vert {\tilde{\omega }}^{p/2}u_{\varepsilon } \Vert ^2_{\ell ^2}+2\varepsilon \Vert {\tilde{\omega }}^{p/2} \omega ^{1/2} u_{\varepsilon }\Vert ^2_{\ell ^2}\le C\sum _{r\ge 3} \Vert u_{\varepsilon } \Vert ^2_{p} \Vert {u}_{\varepsilon }\Vert _{2}^{r-2}, \end{aligned}$$

(7.47)

Integrating both sides of (7.47) from 0 to t, we get

$$\begin{aligned}&\Vert u_{\varepsilon }(t)\Vert _p^2-\Vert u_{\varepsilon }(0)\Vert _p^2+2\varepsilon \int _0^t \Vert {\tilde{\omega }}^{p/2} \omega ^{1/2} u_{\varepsilon }(\tau )\Vert ^2_{\ell ^2} d\tau \nonumber \\&\quad \le C \int _{0}^t \sum _{r\ge 3} \Vert u_{\varepsilon }(\tau ) \Vert ^2_{p}\cdot \Vert {u}_{\varepsilon }(\tau )\Vert _{2}^{r-2}d\tau . \end{aligned}$$

(7.48)

The solution $\Vert u_{\varepsilon }(t)\Vert _p$ of (7.48) can be majorized by the solution of

$$\begin{aligned} y(t)=\eta +c \int _0^t h(y(\tau )) d\tau \end{aligned}$$

(7.49)

with $\eta =\Vert u_{\varepsilon }(0)\Vert _p^2$, where h is a non-negative and non-decreasing function. The solution of (7.49) exists only for $0\le t<T_*:=\frac{\eta }{2c\ h(2\eta )}$. Hence, $\{u_{\varepsilon }\}$ remains bounded on $\ell ^{\infty }(0,T_*,\ell ^2_{p}(\mathbb {Z}^*,\mathbb {C}))$ for any $\varepsilon \in [0,\varepsilon _0]$ provided that we choose $0<T_1< \min (T_0,T_*)$. Thus, we have

$$\begin{aligned} \Vert u_{\varepsilon }\Vert _{\ell ^{\infty }(0,T_1, \ell ^2_{p}(\mathbb {Z}^*,\mathbb {C}))}\le C_3. \end{aligned}$$

(7.50)

Integration of (7.46) with respect to t yields

$$\begin{aligned} \varepsilon ^{1/2} \Vert u_{\varepsilon }\Vert _{\ell ^{2}(0,T_1; \ell ^2_{p+1}(\mathbb {Z}^*,\mathbb {C}))}\le C_4 \end{aligned}$$

(7.51)

and

$$\begin{aligned} \varepsilon ^{1/2} \Vert \partial _{{\bar{u}}}P(u_{\varepsilon },{\bar{u}}_{\varepsilon })\Vert _{\ell ^{\infty }(0,T_1; \ell ^2_{p-1}(\mathbb {Z}^*,\mathbb {C}))}\le C_7. \end{aligned}$$

(7.52)

In view of (7.45), we get

$$\begin{aligned} \Vert u'_{\varepsilon }(t)\Vert _{\ell ^1(0,T_1; \ell ^2_{p-2}(\mathbb {Z}^*,\mathbb {C}))}\le C_8. \end{aligned}$$

(7.53)

$u_{\varepsilon }(t)$ is strongly continuous on $C(0,T_1, \ell ^2_{p-2}(\mathbb {Z}^*,\mathbb {C})).$

We show the convergence of approximate solutions $\{u_{\varepsilon }(t)\}$. Set $v(t)=u_{\varepsilon }(t)-u_{\delta }(t)$. And v(t) satisfies the following equation

$$\begin{aligned} v_t = -\omega _{\varepsilon } v+\mathbf{i} \sigma \partial _{{\bar{u}}}P(u_{\varepsilon }) - \mathbf{i}\sigma \partial _{{\bar{u}}}P(u_{\delta })+(\varepsilon -\delta ) \omega u_{\delta }. \end{aligned}$$

(7.54)

Summing the inner product of (7.54) with ${\bar{v}}$ and the inner product of the conjugate of (7.54) with v, we get

$$\begin{aligned}&\frac{d}{dt}\Vert v\Vert ^2_{\ell ^2}+2\varepsilon \Vert \omega ^{1/2}v\Vert ^2_{\ell ^2}\nonumber \\&\quad = 2\text{ Re }\ \mathbf{i} \langle \sigma \partial _{{\bar{u}}}P(u_{\varepsilon },{\bar{u}}_{\varepsilon })-\sigma \partial _{{\bar{u}}}P(u_{\delta },{\bar{u}}_{\delta }), v\rangle +2(\varepsilon -\delta )\text{ Re }\langle \omega ^{1/2}u_{\delta },\omega ^{1/2}v \rangle ,\qquad \end{aligned}$$

(7.55)

where ${\text{ Re }}(z)$ is the real part of z. Calculus the first term in the right side, we get that

$$\begin{aligned}&\text{ Re }\ \mathbf{i} \langle \sigma \partial _{{\bar{u}}}P(u_{\varepsilon },{\bar{u}}_{\varepsilon })-\sigma \partial _{{\bar{u}}}P(u_{\delta },{\bar{u}}_{\delta }), v\rangle \nonumber \\&\quad = \text{ Re }\ \mathbf{i} \langle \sigma \partial _{u{\bar{u}}}P(u_{\theta }, {\bar{u}}_{\theta })v, v) + \text{ Re }\ \mathbf{i} \langle \sigma \partial _{uu}P(u_{\theta }, {\bar{u}}_{\theta }){\bar{v}}, v) \nonumber \\&\quad = \text{ Re }\ \mathbf{i} \langle \sigma \partial _{uu}P(u_{\theta }, {\bar{u}}_{\theta }) {\bar{v}}, v), \end{aligned}$$

(7.56)

where $u_{\theta }=u_{\varepsilon }+\theta (u_{\delta }-u_{\varepsilon })$, $u_{\theta '}=u_{\varepsilon }+\theta '(u_{\delta }-u_{\varepsilon })$, $\theta , \theta '\in (0,1)$. The last equation is holding by the fact that $\overline{\partial _{u{\bar{u}}}P(u_{\theta }, {\bar{u}}_{\theta })}= \partial _{{\bar{u}}u}P(u_{\theta }, {\bar{u}}_{\theta })$, i.e., $\langle \sigma \partial _{u{\bar{u}}}P(u_{\theta }, {\bar{u}}_{\theta })v,v\rangle$ is real. So $\text{ Re }\ \mathbf{i} \langle \sigma \partial _{u{\bar{u}}}P(u_{\theta }, {\bar{u}}_{\theta }){v}, v)=0$. $\square$

Lemma 7.2

When $P(u,{\bar{u}})$ is an r-degree homogeneous polynomial with $\beta$-type symmetric coefficients semi-bounded by $C>0$, then we have that

$$\begin{aligned} | \text{ Re }\ \mathbf{i} \langle \sigma \partial _{uu}P(u_{\theta }, {\bar{u}}_{\theta }){\bar{v}}, v\rangle |\le C^{r-2} c^{r-1} \Vert v\Vert _{\ell ^2}^2\Vert u_{\theta }\Vert _3^{r-2},\quad u_{\theta }, {\bar{u}}_{\theta },v, {\bar{v}}\in \ell ^2_{p}(\mathbb {Z}^*,\mathbb {C}),\ p\ge 3. \end{aligned}$$

Proof

We write

$$\begin{aligned}&\text{ Re }{} \mathbf{i} \langle {\bar{v}} \partial _{uu}P(u_{\theta }, {\bar{u}}_{\theta }), v\rangle = A(u_{\theta },{\bar{u}}_{\theta }, v,{\bar{v}})+ B(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})\\&\quad +\,C(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})+D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}}), \end{aligned}$$

where

$$\begin{aligned} A(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}}):= & {} \text{ Re } \sum _{(l,k)\in \alpha [1]} \mathbf{i} {\tilde{P}}_{r,lk}^i \text{ sgn }(j) l_j\cdot (l-e_j)_n {\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{l_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n,\\ B(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}}):= & {} \text{ Re } \sum _{\alpha [2]} \mathbf{i} {\tilde{P}}_{r,lk}^i \text{ sgn }(j) l_j\cdot (l-e_j)_n {\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{l_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n,\\ C(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})\!:= & {} \! \text{ Re } \sum _{\alpha [3]}{} \mathbf{i} {\tilde{P}}_{r,lk}^i \text{ sgn }(j) (\sqrt{|nj|} - |j|)l_j(l\\&\quad -e_j)_n {\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{(l-e_n-e_j)_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n, \end{aligned}$$

and

$$\begin{aligned} D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}}):= \text{ Re } \sum _{\alpha [3]} \mathbf{i} {\tilde{P}}_{r,lk}^i \text{ sgn }(j)|j|\cdot l_j (l-e_j)_n {\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{(l-e_n-e_j)_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n, \end{aligned}$$

where

$$\begin{aligned} \alpha [0]:= & {} \{ (l,k)\in \mathbb {N}^{\mathbb {Z}^*}\times \mathbb {N}^{\mathbb {Z}^*} \ \big |\ |l+k|=r,\ {{\mathcal {M}}}(l,k)=i\in \mathbb {Z} \},\\ \alpha [1]:= & {} \big \{ (l,k)\in \alpha [0] \ \big |\ \text{ there } \text{ exist } j, n\ \text{ with } l_j,\ (l-e_j)_n\ne 0,\ j=n \big \},\\ \alpha [2]:= & {} \bigg \{ (l,k)\in \alpha [0] \ \bigg |\ \begin{array}{l@{\quad }l} \text{ there } \text{ exist } j\ne n\ \text{ with } l_j,\ (l-e_j)_n\ne 0, \\ \max \{ |j|, \ |n|\} >\max \{ |t|\ | \ k_t\ne 0,\ \text{ or }\ (l-e_j-e_n)_t\ne 0 \} \end{array} \bigg \} \end{aligned}$$

and

$$\begin{aligned} \alpha [3]:= \bigg \{ (l,k)\in \alpha [0]\ \bigg |\ \begin{array}{l@{\quad }l} \text{ there } \text{ exist } j\ne n\ \text{ with } l_j,\ (l-e_j)_n\ne 0, \\ \max \{ |j|, \ |n|\} \le \max \{ |t|\ | \ k_t\ne 0,\ \text{ or }\ (l-e_j-e_n)_t\ne 0 \} \end{array} \bigg \}. \end{aligned}$$

Let us consider $D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$, first. For any $(l,k)\in \alpha [3]$, it holds true that

$$\begin{aligned} \text{ sgn }(j)\ |j| +\text{ sgn }(n)\ |n|={{\mathcal {M}}}(k,l-e_n-e_j)+i, \quad \ l_j\cdot (l-e_j)_n=l_n\cdot (l-e_n)_j.\nonumber \\ \end{aligned}$$

(7.57)

Take (7.57) into $D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ and get that

$$\begin{aligned}&D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})\nonumber \\&\quad = -\text{ Re } \sum _{\alpha [3]} \mathbf{i} {\tilde{P}}_{r,lk}^i \text{ sgn }(n)|n|\cdot l_n(l-e_n)_j {\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{(l-e_n-e_j)_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n\nonumber \\&\quad \quad +\text{ Re } \sum _{\alpha [3]} \mathbf{i} {\tilde{P}}_{r,lk}^i ({{\mathcal {M}}}(k,l-e_n-e_j)\nonumber \\&\quad \quad +i)\cdot l_j (l-e_j)_n{\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{(l-e_n-e_j)_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n. \end{aligned}$$

(7.58)

The first part of the right side of Eq. (7.58) equals to $-D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$. From (7.58), we get that

$$\begin{aligned}&D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})\nonumber \\&\quad =\frac{1}{2}\text{ Re } \sum _{\alpha [3]}{} \mathbf{i} {\tilde{P}}_{r,lk}^i ({{\mathcal {M}}}(k,l-e_n-e_j)\nonumber \\&\quad \quad +i)\cdot l_j (l-e_j)_n {\bar{u}}^k u^{l-e_j-e_n} \prod _{t} |t|^{\frac{(l-e_n-e_j)_t+k_t}{2}} {\bar{v}}_j{\bar{v}}_n. \end{aligned}$$

(7.59)

In the following we will estimate $A(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$, $B(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$, $C(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ and $D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ in (7.59). In order to do that, we should calculate the boundary of the coefficients of them. The coefficient of $u^{l-e_j-e_n}{\bar{u}}^{k}{\bar{v}}_j{\bar{v}}_n$ in $A(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ is bounded by

$$\begin{aligned}&|{\tilde{P}}_{r,lk}^i|\cdot |j|\prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}} = |{\tilde{P}}_{r,lk}^i|\cdot |i-{{\mathcal {M}}}(l-e_j-e_n,k)|\prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}}\nonumber \\&\quad \le \frac{C^{r-2}}{\langle i\rangle ^{\beta -1} }\cdot \left( \sum _{t\in \mathbb {Z}^*} (l-e_j-e_n)_t|t|+k_t|t|\right) \prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}}. \end{aligned}$$

(7.60)

The coefficients of the term with $(l,k)\in \alpha [2]$ of $B(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ is bounded by

$$\begin{aligned} |{\tilde{P}}_{r,lk}^i|\sqrt{|j|}\sqrt{|n|}\prod _{t} |t|^{\frac{(l-e_j-e_n)_t+k_t}{2}}\le \frac{C^{r-2}}{\langle i\rangle ^{\beta }}\prod |t|^{\frac{3((l-e_j-e_n)_t+k_t)}{2}}. \end{aligned}$$

The coefficients of $C(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ are bounded by the following fact

$$\begin{aligned}&|{\tilde{P}}_{r,lk}^i|\cdot \text{ sgn }(j) \sqrt{|j|}(\sqrt{|n|}-\sqrt{|j|})\prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}}\nonumber \\&\quad \le |{\tilde{P}}_{r,lk}^i|\cdot sgn(j)\sqrt{j} \frac{\big | |n|-|j|\big | }{\sqrt{|j|}+\sqrt{|n|}}\prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}}\nonumber \\&\quad \le |{\tilde{P}}_{r,lk}^i| \cdot \big ||n|-|j|\big |\cdot \prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}}\le |{\tilde{P}}_{r,lk}^i|\cdot |n+j|\prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}} \nonumber \\&\quad \le \frac{C^{r-2}}{\langle i\rangle ^{\beta -1}}\cdot \left( \sum _{t\in \mathbb {Z}^*} (l-e_j-e_n)_t|t|+k_t|t|\right) \prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}}. \end{aligned}$$

(7.61)

The coefficients of $D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})$ is bounded by

$$\begin{aligned} \frac{C^{r-2}}{\langle i\rangle ^{\beta -1}}\cdot \left( \sum _{t\in \mathbb {Z}^*} (l-e_j-e_n)_t|t|+k_t|t|\right) \prod |t|^{\frac{((l-e_j-e_n)_t+k_t)}{2}} \end{aligned}$$

(7.62)

From (7.60)–(7.62), using Lemma 3.2, we get that

$$\begin{aligned}&|A(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})+B(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})+C(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})\\&\quad +D(u_{\theta },{\bar{u}}_{\theta },v,{\bar{v}})|\le C c^{r-1} \Vert u_{\theta }\Vert _{3}^{r-2} \Vert v\Vert ^2_{\ell ^2}. \end{aligned}$$

$\square$

From Eq. (7.55) and (7.56), using Lemma 7.2, it follows that

$$\begin{aligned} \frac{d}{dt} \Vert v(t) \Vert _{\ell ^2}^2\le C_* \Vert v(t)\Vert _{\ell ^2}^2+(\varepsilon +\delta ) \Vert v(t)\Vert ^2_{\ell ^2},\quad \text{ for } \text{ any } t\in (0,T_1), \end{aligned}$$

(7.63)

Then

$$\begin{aligned} \Vert v(t)\Vert _{\ell ^2}\le C_*(\Vert v(0)\Vert _{\ell ^2}+\varepsilon +\delta ),\quad \text{ for } \ 0\le t\le T_1, \end{aligned}$$

(7.64)

which implies that $\{u_{\varepsilon }\}$ is a Cauchy sequence in $C(0,T_1,\ell ^2(\mathbb {Z}^*,\mathbb {C}))$. Thus, there exists a function u(x, t) such that as $\varepsilon \rightarrow 0$,

$$\begin{aligned} u_{\varepsilon } \rightarrow u \quad \text{ strongly } \text{ in } \ C(0,T_1,\ell ^2(\mathbb {Z}^*,\mathbb {C})). \end{aligned}$$

From the estimate of (7.50)–(7.52),

$$\begin{aligned}&u_{\varepsilon }\rightarrow u \quad \text{ weakly } \text{ star } \text{ in } \ell ^{\infty }(0,T_1,\ell ^2_{2\alpha }(\mathbb {Z}^*,\mathbb {C})),\\&\omega _{\varepsilon } u_{\varepsilon }\rightarrow \mathbf{i}\omega u \quad \text{ weakly } \text{ star } \text{ in } \ell ^{\infty }(0,T_1,\ell ^2_{2\alpha -2}(\mathbb {Z}^*,\mathbb {C})),\\&\partial _{{\bar{u}}}P(u_{\varepsilon },{\bar{u}}_{\varepsilon })\rightarrow \partial _{{\bar{u}}}P(u,{\bar{u}}) \quad \text{ weakly } \text{ star } \text{ in } \ell ^{\infty }(0,T_1,\ell ^2_{2\alpha -1}(\mathbb {Z}^*,\mathbb {C})). \end{aligned}$$

Taking the inner product of (7.45) with $v\in D(\omega _{\varepsilon }^*)=D(\omega )=\ell ^2_{2}(\mathbb {Z}^*,\mathbb {C})$ and integrating on (0, t), we find

$$\begin{aligned} \langle u_{\varepsilon }(t),v\rangle =\langle u_{\varepsilon }(0),v\rangle -\int _0^t u_{\varepsilon }(\tau ) \omega _{\varepsilon }^* v d \tau +\int _0^t\langle \mathbf{i}\sigma \partial _{{\bar{u}}}P(u_{\varepsilon },{\bar{u}}_{\varepsilon }),v\rangle d\tau . \end{aligned}$$

(7.65)

Letting $\varepsilon$ tend to zero, we get

$$\begin{aligned} \langle u(t),v\rangle =\left\langle u(0)+\int _0^t(-\mathbf{i})\bar{\omega }u(\tau ) +\mathbf{i}\sigma \partial _{{\bar{u}}}P(u,{\bar{u}}) (\tau ) d\tau ,v\right\rangle , \end{aligned}$$

for any $v\in D(\omega )$, where $\langle \cdot , \cdot \rangle$ denote the duality between $D(\omega )'=\ell ^2_{-2}(\mathbb {Z}^*,\mathbb {C})$ and $D(\omega )=\ell ^2_{2}(\mathbb {Z}^*,\mathbb {C})$ and $\bar{\omega }$ is an unique continuous linear extension of $\omega$ from $\ell ^2$ to $D(\omega )$.

Hence we obtain

$$\begin{aligned} u(t)=u_0+\int _0^t (-\mathbf{i}\bar{\omega } u)(\tau ) +\mathbf{i} \sigma \partial _{{\bar{u}}}P(u,{\bar{u}})(\tau )d\tau ,\quad t\in (0,T_1), \end{aligned}$$

which implies that u(t) is strongly absolutely continuous in $D(\omega ^{\alpha -1})$ at almost all $t\in (0,T_1)$

$$\begin{aligned} \frac{du}{dt}={-\mathbf{i}}\omega u,+\mathbf{i } \sigma \partial _{{\bar{u}}}P(u,{\bar{u}})\in \ell ^2(0,T_1,D(\omega ^{\alpha -1})). \end{aligned}$$

(7.66)

$\square$