Averaging of nonlinear Schrödinger equations with time-oscillatory coefficients

Abstract

In this paper, the limit behavior of solutions for the nonlinear Schrödinger equation $i \partial _{t} u + \gamma (\omega t) \Delta u + \theta (\omega t) |u|^{\alpha } u = 0$ in ${\mathbb {R}}^{N}$ $(N = 1,2,3)$ is studied. Here $\alpha$ is an $H^{1}$- subcritical exponent and the coefficients $\gamma$, $\theta$ are periodic functions. The coefficient $\gamma$ is further assumed to be one sign, bounded, and bounded away from zero. We prove local and global well-posedness results in $H^1$ and that the solution $u_{\omega }$ converges as $|\omega | \rightarrow \infty$ to the solution of the limiting equation with the same initial condition. Furthermore, we also prove that if the limiting solution is global and has a certain decay property, then $u_{\omega }$ is also global for $|\omega |$ sufficiently large.

1 Introduction

The interest in nonlinear Schrödinger equations with variable coefficients is found in a large number of physical models and their descriptions, for example, see [5, 10, 12, 13] and the references therein. In the paper, we consider the nonlinear Schrödinger equation with time periodic coefficients

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} i \partial _{t} u + \gamma (\omega t) \Delta u + \theta (\omega t) |u|^{\alpha } u = 0, \\ u(0) = \varphi , \end{array} \right. \end{aligned} \end{aligned}$$

(1)

in ${\mathbb {R}}^{N}$, $N = 1, 2, 3$, where

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} 0< \alpha< \infty \quad \quad &{} N = 1, 2, \\ 0< \alpha < 4 \quad \quad &{} N = 3, \end{array} \right. \end{aligned} \end{aligned}$$

(2)

$\omega \in {\mathbb {R}}$ and $\gamma , \theta$ are $\tau$–periodic functions for some $\tau > 0$. Moreover, we assume that $\theta \in C^{1}({\mathbb {R}})$ and the function $\gamma$ is one sign, bounded and bounded away from zero on $[0, \tau ]$.

As usual, we consider the integral form via Duhamel’s formula:

$$\begin{aligned} \begin{aligned} u (t) = e^{i \Gamma _{\omega } (t, 0) \Delta } \varphi + i \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } \theta (\omega s) | u (s) |^{\alpha } u (s) \, ds, \end{aligned} \end{aligned}$$

(3)

where $e^{i \Gamma _{\omega } (t, s) \Delta }$ is the unitary group determined by the associated linear Schrödinger equation, i.e., when $\theta = 0$; see Sect. 2.1 for more details.

It is well known that the Cauchy problem (1) when $\gamma = 1$ and $\theta \in L^{\infty }({\mathbb {R}})$ is well-posed in $H^{1}$, see [3] for the subcritical and [6] for the critical cases. The standard techniques they used also give us the following fundamental result for our case.

Proposition 1

Given any $\varphi \in H^{1}({\mathbb {R}}^{N})$ and $\omega \in {\mathbb {R}}$, there exists a unique $H^{1}$–solution u of (3) defined on the maximal interval $[0, T_{\max })$ with $0< T_{\max } \le \infty$. Moreover, the following properties hold:

• (i) $u \in C( [0, T_{\mathrm{max}}), H^{1} ({\mathbb {R}}^{N}) ) \cap L^{q}_{\mathrm{loc}} ( (0, T_{\max }), W^{1, r}({\mathbb {R}}^{N}) )$ for all admissible pair (q, r).

• (ii) (Blow-up alternative) If $T_\mathrm{max} < \infty$, then $\Vert u(t)\Vert _{H^{1} ({\mathbb {R}}^{N})} \rightarrow \infty$ as $t \uparrow T_\mathrm{max}$.

• (iii) If $\alpha < 4/N$, then the solution u is global, i.e., $T_\mathrm{max} = \infty$.

The main purpose is to study the behavior of solutions $u_{\omega }$ for (1) as $|\omega | \rightarrow \infty$. Since $\gamma$ and $\theta$ are periodic, we expect it to be close to the solution of the limiting equation

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} i \partial _{t} U + I(\gamma ) \Delta U + I(\theta ) |U|^{\alpha } U = 0, \\ U(0) = \varphi , \end{array} \right. \end{aligned} \end{aligned}$$

(4)

or its equivalent integral form

$$\begin{aligned} \begin{aligned} U (t) = e^{i I(\gamma )t \Delta } \varphi + i \int _{0}^{t} e^{i I(\gamma ) (t - s) \Delta } I(\theta ) | U (s) |^{\alpha } U (s) \, ds, \end{aligned} \end{aligned}$$

(5)

where $I(\gamma )$ and $I(\theta )$ are averages of $\gamma$ and $\theta$, respectively, i.e.,

$$\begin{aligned} \begin{aligned} I(\gamma ) = \frac{1}{\tau } \int _{0}^{\tau } \gamma (s) \, ds \quad \text{ and } \quad I(\theta ) = \frac{1}{\tau } \int _{0}^{\tau } \theta (s) \, ds. \end{aligned} \end{aligned}$$

(6)

The existence of the maximal solution U for the Cauchy problem (4) or (5) has been extensively studied, e.g., [2]. So we investigate that our expectation is true on the maximal interval in which solution U exists. In the following theorem, we state our main consequences.

Theorem 1

Fix an initial value $\varphi \in H^{1}({\mathbb {R}}^{N})$. Given $\omega \in {\mathbb {R}}$, denote by $u_{\omega }$ the maximal solution of (3). Let U be the solution of (5) defined on the maximal interval $[0, S_\mathrm{max})$.

• (i) For each $0< S < S_\mathrm{max}$, the solution $u_{\omega }$ exists on [0, S] provided that $|\omega |$ is sufficiently large.

• (ii) $u_\omega$ converges to U in $L^{\infty }((0, S), H^{1}({\mathbb {R}}^{N}))$ as $|\omega | \rightarrow \infty$.

Remark 1

The averaging theorem of NLS has widely been studied considering various forms of the time-dependent coefficients. In [1], the authors consider in the case of $\theta = 1$ and the fast dispersion management $\gamma$ of the form $\gamma (t /\varepsilon )$, where $\gamma$ is given by 2–periodic and piecewise constant, a typical example being $\gamma = 1$ on the interval [0, 1) and $\gamma = -1$ on the interval [1, 2). Moreover, they proved the scaling limit of fast dispersion management and the convergence in $H^2$ to an effective model with averaged dispersion. In [5, 13] an Eq. (1) with the strong dispersion management $\gamma$ of the form $\varepsilon ^{-1} \gamma (t /\varepsilon )$ and lumped amplification was studied in dimension $N = 1$, which is closely related to a physical phenomenon. In contrast, the averaging theorem for $\gamma = 1$ were obtained by Cazenave and Scialom [3] .

If $\alpha \ge 4/N$ and $S_{\max } = \infty$, one may question whether $u_{\omega }$ is also global for $|\omega |$ sufficiently large. The following theorem gives us an affirmative answer under the condition that U has suitable decay as $t \rightarrow \infty$. Moreover, the convergence holds globally in time.

Theorem 2

Assume (2) and further that $\alpha \ge 4/N$. Let r and a be defined by

$$\begin{aligned} \begin{aligned} r = \alpha + 2 \quad \text{ and } \quad a = \frac{2 \alpha (\alpha + 2)}{4 - (N-2) \alpha }. \end{aligned} \end{aligned}$$

Fix an initial value $\varphi \in H^{1}({\mathbb {R}}^{N})$. Given $\omega \in {\mathbb {R}}$, denote by $u_{\omega }$ the maximal solution of (3). Let U be the solution of (5) defined on the maximal interval $[0, S_{\max })$. Suppose that

$$\begin{aligned} \begin{aligned} S_{\max } = \infty \quad \text{ and } \quad U \in L^{a}((0, \infty ), L^{r}({\mathbb {R}}^{N})). \end{aligned} \end{aligned}$$

(7)

Then $u_{\omega }$ is global for $|\omega |$ sufficiently large. Moreover, $u_{\omega }$ converges to U in $L^{\infty }((0, \infty ), H^{1}({\mathbb {R}}^{N}))$ as $|\omega | \rightarrow \infty$.

The existence of solutions satisfying (7) is guaranteed by the scattering theory (the details can be referred in [2, 7, 11]). Thus by applying Theorem 2, we obtain the following.

Corollary 1

Assume (2). Fix an initial value $\varphi \in H^{1}({\mathbb {R}}^{N})$, let U be the maximal solution of (5). Given $\omega \in {\mathbb {R}}$, denote by $u_{\omega }$ the maximal solution of (3). If one of the following conditions is satisfied,

• (i) $I(\gamma ) I(\theta ) < 0$ and $\alpha > 4/N$

• (ii) $I(\theta ) = 0$ and $\alpha \ge 4/N$

• (iii) $I(\gamma ) I(\theta ) > 0$, $\alpha \ge 4/N$ and $\Vert \varphi \Vert _{{\dot{H}}^{s}}$ is sufficiently small, where $s = (N \alpha - 4)/2 \alpha \in [0, 1)$,

then it follows that the solution $u_{\omega }$ of (3) is global for $|\omega |$ sufficiently large. Moreover, $u_{\omega }$ converges to U in $L^{\infty }((0, \infty ), H^{1}({\mathbb {R}}^{N}))$ as $|\omega | \rightarrow \infty$.

Note that in case $I(\theta ) = 0$, i.e., linear equation, $U(t) = e^{i I(\gamma )t \Delta } \varphi$. Using the change of variables $V(t, x) = U(t/I(\gamma ), x)$, V solves

$$\begin{aligned} \begin{aligned} i \partial _{t} V + \Delta V + \frac{I(\theta )}{I(\gamma )} |V|^{\alpha } V = 0 \end{aligned} \end{aligned}$$

(8)

with the initial value $V(0) = \varphi$. The behavior of (8) is focusing or defocusing which depend only the sign of $I(\theta )/I(\gamma )$. Thus, we refer to defocusing equation when $I(\gamma )I(\theta ) < 0$, otherwise we refer to focusing equation.

Notation

We use $C > 0$ to denote various constants. For $1 \le r, q \le \infty$, the norm of mixed space $L^{r}(I, L^{q}({\mathbb {R}}^{N}))$ is denoted by $\Vert \cdot \Vert _{L^{r}(I, L^{q})}$.

The paper is organized as follows: In Sect. 2, we establish some preliminaries and lemmas and derive the well-posedness results. In Sect. 3, we give the proof of Theorem 1. Finally, the proof of Theorem 2 is devoted to Sect. 4.

2 Preliminaries and well-posedness results

2.1 The linear propagator

Before proving Proposition 1, we collect some properties for the propagator associated with the linear Schrödinger equation

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} i \partial _{t} u_\mathrm{lin} + \gamma ( \omega t ) \Delta u_\mathrm{lin} = 0, \\ u_\mathrm{lin}( 0 ) = f, \end{array} \right. \end{aligned} \end{aligned}$$

(9)

for all $\omega \in {\mathbb {R}}$, where the $\tau$–periodic function $\gamma$ satisfies our assumptions. Here and below, we denote by

$$\begin{aligned} \begin{aligned} \Gamma _{\omega } (t, s):= \int _{s}^{t} \gamma ( \omega t') \, d t' = \frac{1}{\omega } \int _{\omega s}^{\omega t} \gamma (t') \, dt' \end{aligned} \end{aligned}$$

(10)

for all $s,t \in {\mathbb {R}}$. One can express the associated propagator $e^{i \Gamma _{\omega } (t, 0) \Delta }$ that describes the solution $u_\mathrm{lin}(x, t)$ for (9) as

$$\begin{aligned} \begin{aligned} e^{i \Gamma _{\omega } (t, 0) \Delta } f(x) = \frac{1}{(2 \pi )^{N/2}} \int _{{\mathbb {R}}^{N}} e^{ - i | \xi |^{2} \Gamma _{\omega } (t, 0)}e^{i x \cdot \xi } {\widehat{f}}(\xi ) \, d\xi \end{aligned} \end{aligned}$$

(11)

for $f \in L^{2}({\mathbb {R}}^{N})$, where ${\widehat{f}}$ denotes the Fourier transform of $f \in L^{2}({\mathbb {R}}^{N})$. We now define the operator $e^{ i \Gamma _{\omega } (t, s) \Delta }$ by

$$\begin{aligned} \begin{aligned} e^{i \Gamma _{\omega } (t, s) \Delta } := e^{i \Gamma _{\omega } (t, 0)\Delta } \, e^{-i \Gamma _{\omega }(s, 0) \Delta } \end{aligned} \end{aligned}$$

on $L^{2}({\mathbb {R}}^{N})$. Then, fixed $s \in {\mathbb {R}}$, it is a unitary operator on $L^{2} ({\mathbb {R}}^{N})$ also on $H^{1} ({\mathbb {R}}^{N})$ satisfying

$$\begin{aligned} \begin{aligned} \Vert e^{i \Gamma _{\omega } (t, s) \Delta } f \Vert _{L^{2}} = \Vert f \Vert _{L^{2}} \quad \text{ and } \quad \Vert e^{i \Gamma _{\omega } (t, s) \Delta } f \Vert _{H^{1}} = \Vert f \Vert _{H^{1}} \end{aligned} \end{aligned}$$

for every $\omega \in {\mathbb {R}}$. Moreover, fixed $s \in {\mathbb {R}}$, it follows from (11) that the mapping $t \mapsto e^{i \Gamma _{\omega }( t, s ) \Delta } f$ is continuous for every $f \in L^{2} ({\mathbb {R}}^{N})$.

From our assumption of $\gamma$, it follows that for any $s, t \in {\mathbb {R}}$, there exists $C > 0$ such that

$$\begin{aligned} \begin{aligned} \left| \int _{s}^{t} \gamma (\tau ) \, d\tau \right| \ge C |t - s|, \end{aligned} \end{aligned}$$

which allows us to obtain the following result.

Lemma 1

Let $\omega \in {\mathbb {R}}$. There exists a constant C independent of $\omega$ such that if $s \ne t$, then

$$\begin{aligned} \begin{aligned} \Vert e^{i \Gamma _{\omega }(t,s) \Delta } f\Vert _{L^{\infty }} \le \frac{C}{| t - s |^{N/2}} \Vert f \Vert _{L^{1}} \end{aligned} \end{aligned}$$

for any $f \in L^{1}({\mathbb {R}}^{N})$.

Proof

Using the explicit form of the solution operator for the free Schrödinger equation

$$\begin{aligned} \begin{aligned} e^{i t \Delta } f (x) = \frac{1}{(4i \pi t)^{N/2}} \int _{{\mathbb {R}}^{N}} e^{i \frac{| x - y |^{2}}{4t}} f (y) \, dy, \quad t \ne 0, \end{aligned} \end{aligned}$$

we obtain that

$$\begin{aligned} \begin{aligned} \Vert e^{i \Gamma _{\omega } (t, s) \Delta } f \Vert _{L^{\infty }} \le \frac{1}{(4\pi |\Gamma _{\omega } (t, s)|)^{N/2}} \Vert f \Vert _{L^{1}}. \end{aligned} \end{aligned}$$

(12)

Note that since $\gamma$ is one sign and bounded away from zero, we have

$$\begin{aligned} \begin{aligned} |\Gamma _{\omega }(t, s)| = \left| \frac{1}{\omega } \int _{\omega s}^{\omega t} \gamma (t') \, d t' \right| \ge C|t - s|. \end{aligned} \end{aligned}$$

This together with (12) completes the proof of Lemma 1. $\square$

Observe that the usual Strichartz estimates hold for the semigroup $e^{i \Gamma _{\omega } (t, 0) \Delta }$. To this end, for any $1 \le p \le \infty$, let $p'$ be the Hölder conjugate, that is, $1/p + 1/p'= 1$, and a pair of exponents (q, r) is said to be admissible if

$$\begin{aligned} \begin{aligned} \frac{2}{q} =\frac{N}{2} - \frac{N}{r} \quad \text{ and } \quad \left\{ \begin{array}{ll} 2 \le r \le \infty \quad \quad &{} N = 1, \\ 2 \le r < \infty \quad \quad &{} N = 2, \\ 2 \le r \le 6 \quad \quad &{} N = 3. \end{array} \right. \end{aligned} \end{aligned}$$

Using Lemma 1, we can show the following standard Strichartz estimates with an argument similar to that of, for example, [2] and [9]. So we omit the details of the proof.

Lemma 2

(Strichartz’s estimates) Let (q, r) and $(q_{0}, r_{0})$ be admissible pairs. For any $\omega \in {\mathbb {R}}$, the following properties hold:

• (i) For every $f \in L^{2}({\mathbb {R}}^{N})$, the map $t \mapsto e^{i \Gamma _{\omega } (t, 0) \Delta } f$ belongs to $L^{q} ({\mathbb {R}}, L^{r} ({\mathbb {R}}^{N})) \cap C({\mathbb {R}}, L^{2}({\mathbb {R}}^{N}))$. Furthermore, there exists a constant C independent of $\omega$ such that

  $$\begin{aligned} \begin{aligned} \Vert e^{i \Gamma _{\omega } (\cdot , 0) \Delta } f \Vert _{L^{q} ({\mathbb {R}}, L^{r})} \le C \Vert f \Vert _{L^{2}}. \end{aligned} \end{aligned}$$

• (ii) Let I be an interval of ${\mathbb {R}}$. For every $F \in L^{q'_{0}} (I, L^{r'_{0}}({\mathbb {R}}^{N}))$, the map

  $$\begin{aligned} \begin{aligned} t \mapsto \int _{I} e^{i \Gamma _{\omega } (t, \tau ) \Delta } F(\cdot , \tau ) \, d \tau \quad \text{ for } \quad t \in I, \end{aligned} \end{aligned}$$

  belongs to $L^{q}(I, L^{r}({\mathbb {R}}^{N})) \cap C({\overline{I}}, L^{2}({\mathbb {R}}^{N}))$. Furthermore, there exists a constant C independent of $\omega$ such that

  $$\begin{aligned} \begin{aligned} \left\| \int _{I} e^{i \Gamma _{\omega } (\cdot , \tau ) \Delta } F(\cdot , \tau ) \, d \tau \right\| _{L^{q} (I, L^{r})} \le C \Vert F \Vert _{L^{q'_{0}} (I, L^{r'_{0}})}. \end{aligned} \end{aligned}$$

2.2 Well-posedness results

This subsection concentrates on proving the existence and uniqueness of solutions for (1), i.e., Proposition 1. For any $\omega \in {\mathbb {R}}$, we consider the integral equation

$$\begin{aligned} \begin{aligned} u_{\omega }(t) = e^{i\Gamma _{\omega }(t, 0) \Delta } \varphi + i \int _{0}^{t} e^{i\Gamma _{\omega } (t, s) \Delta }\theta (\omega s) |u_{\omega }(s)|^{\alpha } u_{\omega }(s) \, ds. \end{aligned} \end{aligned}$$

(13)

Recall that $\theta \in C^{1}({\mathbb {R}})$ and $\Gamma _{\omega } (t, s)$ is given by (10). For this subsection, we only need to assume $\theta \in L^{\infty }({\mathbb {R}})$ which is slightly more general that (3).

We start with the local well-posedness of (13). Based on Strichartz’s estimate mentioned in (2), the well-posedness results are quite standard, see, for example, [2, 8]. In fact, the proof of in the case $\gamma = 1$ can be found in [3]. For brevity we only state the results without detailed proofs.

Proposition 2

Assume (2).

• (i) Given $A, M > 0$, there exists $T = T(A, M) > 0$ such that if $\Vert \theta \Vert _{L^{\infty }} \le A$ and if $\varphi \in H^{1}({\mathbb {R}}^{N})$ satisfying $\Vert \varphi \Vert _{H^{1}} \le M$, then for any $\omega \in {\mathbb {R}}$, there exists a unique local solution $u_{\omega } \in C([0, T], H^{1}({\mathbb {R}}^{N}))$ of (13). In addition,

  $$\begin{aligned} \begin{aligned} \Vert u_{\omega } \Vert _{L^{q}((0, T), W^{1, r})} \le 2 C\Vert \varphi \Vert _{H^{1}} \end{aligned} \end{aligned}$$

  for all admissible pair (q, r).

• (ii) Assume further that $\alpha < 4/N$. Given $A, M' > 0$, there exists $T' = T'(A, M') > 0$ such that if $\Vert \theta \Vert _{L^{\infty }} \le A$ and if $\varphi \in L^{2}({\mathbb {R}}^{N})$ satisfying $\Vert \varphi \Vert _{L^{2}} \le M'$, then for any $\omega \in {\mathbb {R}}$, there exists a unique local solution $u_{\omega }\in C([0, T'], L^{2}({\mathbb {R}}^{N}))$ of (13).

Remark 2

• (i) Fix an initial value $\varphi \in H^{1}({\mathbb {R}}^{N})$. Given $\omega \in {\mathbb {R}}$, the solution $u_{\omega }$ of (13) obtained in Proposition 2 can be extended to a maximal interval $[0, T_{\max }(\omega ))$. Moreover, we have the blowup alternative holds: If $T_{\max }(\omega ) < \infty$, then

  $$\begin{aligned} \begin{aligned} \lim _{t \rightarrow T_{\max }(\omega )} \Vert u_\omega (t)\Vert _{H^{1}} = \infty . \end{aligned} \end{aligned}$$

• (ii) Arguing as in the case of constant coefficients, one can show that the mass is conserved, that is,

  $$\begin{aligned} \begin{aligned} \Vert u_{\omega } (t)\Vert _{L^{2}} = \Vert \varphi \Vert _{L^{2}} \end{aligned} \end{aligned}$$

  for all $0 \le t < T_{\max }(\omega )$. However, in our case, the energy is neither conserved nor decreasing.

• (iii) Suppose $\alpha < 4/N$. From Proposition 2 (ii), we know that the local existence time $T'$ depends on the $L^{2}$ norm of the initial value. It follows from the conservation of mass that the $L^{2}$–solution $u_{\omega }$ is globally defined for each $\omega \in {\mathbb {R}}$.

Proof of Proposition 1

The existence and uniqueness of the local $H^{1}$–solution of (13) follow from Proposition 2 (i). The maximal existence time and the blowup alternative are a consequence of Remark 2 (i), moreover u is in $L^{q}_\mathrm{loc}((0, T_{\max }), W^{1, r}({\mathbb {R}}^{N}))$ for all admissible pair. If $\alpha < 4/N$, then we can establish $H^{1}$ regularity of the global $L^{2}$–solution, see Theorem 5.2.2 in [2] for details. Thus, we obtain $u \in C([0, \infty ), H^{1}({\mathbb {R}}^{N}))$.

$\square$

We have the following results, which are the same as [3, Proposition 2.3] and [3, Corollary 2.4]. For proofs, the reader can consult, for example, [3, Proposition 2.3 and Corollary 2.4] and [4, Propositions 2.3 and 2.4].

Proposition 3

Assume (2) and suppose further that $\alpha \ge 4/N$. Let r, q, and a be defined by

$$\begin{aligned} \begin{aligned} r = \alpha + 2, \quad q = \frac{4(\alpha +2)}{N\alpha }, \quad a = \frac{2\alpha (\alpha +2)}{4-(N-2)\alpha }. \end{aligned} \end{aligned}$$

(14)

Given any $A > 0$, there exists $\varepsilon = \varepsilon (A)$ and $\Lambda$ such that for any $\omega \in {\mathbb {R}}$, if $\Vert \theta \Vert _{L^{\infty }} < A$ and if $\varphi \in H^{1}({\mathbb {R}}^{N})$ satisfies

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(\cdot , 0)\Delta } \varphi \Vert _{L^{a}((0, \infty ), L^{r})} \le \varepsilon , \end{aligned} \end{aligned}$$

then the corresponding solution $u_{\omega }$ of (13) is global and satisfies

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }\Vert _{L^{a}((0, \infty ), L^{r})} \le 2 \Vert e^{i\Gamma _{\omega }(\cdot , 0)\Delta } \varphi \Vert _{L^{a}((0, \infty ), L^{r})} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }\Vert _{L^{q}((0, \infty ), W^{1, r})} + \Vert u_{\omega }\Vert _{L^{\infty }((0, \infty ),H^{1})} \le \Lambda \Vert \varphi \Vert _{H^{1}}. \end{aligned} \end{aligned}$$

Conversely, if the solution $u_{\omega }$ of (13) is global and satisfies

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }\Vert _{L^{a}((0, \infty ), L^{r})} \le \varepsilon , \end{aligned} \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(\cdot , 0)\Delta } \varphi \Vert _{L^{a}((0, \infty ), L^{r})} \le 2 \Vert u_{\omega }\Vert _{L^{a}((0, \infty ), L^{r})}. \end{aligned} \end{aligned}$$

Corollary 2

Assume (2) and $\alpha \ge 4/N$. Let r, q, and a be defined by (14). Let $A > 0$ and consider $\varepsilon = \varepsilon (A)$ and $\Lambda$ as in Proposition 3. Given $\varphi \in H^{1}({\mathbb {R}}^{N})$ and $\Vert \theta \Vert _{L^{\infty }} \le A$, let $u_{\omega }$ be the corresponding solution of (13) defined on the maximal interval $[0, T_{\max })$. If there exists $0< T < T_{\max }$ such that

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(0, \cdot )\Delta } u_{\omega } (T)\Vert _{L^{a}((0, \infty ), L^{r})} \le \varepsilon , \end{aligned} \end{aligned}$$

then the solution $u_{\omega }$ is global, i.e., $T_{\max } = \infty$. Moreover,

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }\Vert _{L^{a}((T, \infty ), L^{r})} \le 2 \varepsilon \quad \text{ and } \quad \Vert u_{\omega }\Vert _{L^{q}((T, \infty ),W^{1, r})} \le \Lambda \Vert \varphi \Vert _{H^{1}}. \end{aligned} \end{aligned}$$

3 Proof of Theorem 1

The following lemmas below play a key role in our proof of the convergence result stated in Theorem 1. Similar results are considered also in [1, 3].

Lemma 3

If $g \in L^{1}((0, L), H^{1}({\mathbb {R}}^{N}))$ for some $0 < L \le \infty$, then

$$\begin{aligned} \begin{aligned} \int _{0}^{t} \theta ( \omega s ) e^{i \Gamma _{\omega } (t, s) \Delta } g(s) \, ds \underset{|\omega | \rightarrow \infty }{\longrightarrow } I(\theta ) \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } g(s) \, ds \end{aligned} \end{aligned}$$

(15)

in $L^{\infty }( (0, L), H^{1}({\mathbb {R}}^{N}))$.

Proof

Set

$$\begin{aligned} \begin{aligned} \psi (t) = \theta (t) - I (\theta ) \quad \text{ and } \quad \Psi (t) = \int _{0}^{t} \psi (t') \, dt'. \end{aligned} \end{aligned}$$

Since $\theta$ is $\tau$–periodic, $\Psi$ is also $\tau$–periodic, therefore, $\Vert \Psi \Vert _{L^{\infty }} < \infty$. Using Minkowski’s inequality and the fact that the operator $e^{i \Gamma _{\omega } (\cdot , \cdot ) \Delta }$ is unitary, it follows that

$$\begin{aligned} \begin{aligned} \left\| \int _{0}^{\cdot } \psi (\omega s) e^{i \Gamma _{\omega }(\cdot , s)\Delta } f(s) \, ds \right\| _{L^{\infty }((0, L), H^{1})} \le C \Vert \psi \Vert _{L^{\infty }} \Vert g \Vert _{L^{1}((0, L), H^{1})} \end{aligned} \end{aligned}$$

for every $g \in L^{1}((0, L), H^{1}({\mathbb {R}}^{N}))$. Therefore, by density, we only need to prove (15) for $g \in C_{c}^{1}((0, L), {\mathcal {S}}({\mathbb {R}}^{N}))$. Since $\frac{d}{ds} \Psi (\omega s) = \omega \psi (\omega s)$, an integration by parts shows that

$$\begin{aligned} \begin{aligned}&\int _{0}^{t} \psi (\omega s) e^{i \Gamma _{\omega } (t, s) \Delta } g (s) \, ds = \frac{1}{\omega } \Psi (\omega t) g (t)\\&\quad -\frac{1}{\omega } \int _{0}^{t} \Psi (\omega s) e^{i \Gamma _{\omega } (t, s) \Delta } \Big [ g_{t} (s) - i \gamma (\omega s) \Delta g(s) \Big ] \, ds. \end{aligned} \end{aligned}$$

Since $\gamma$ is bounded, we see that

$$\begin{aligned} \begin{aligned}&\left\| \frac{1}{\omega } \int _{0}^{\cdot } \Psi (\omega s) e^{i \Gamma _{\omega } (\cdot , s) \Delta } \Big [ g_{t} (s) - i \gamma (\omega s) \Delta g (s) \Big ] \, ds \right\| _{L^\infty ((0,L), H^{1})} \\&\qquad \qquad \le \frac{1}{|\omega |} \Vert \Psi \Vert _{L^{\infty }} \Vert g_{t} (s) - i \gamma (\omega s) \Delta g (s) \Vert _{L^1((0,L), H^{1})} \, \\&\qquad \qquad \le \frac{C}{|\omega |} \Vert \Psi \Vert _{L^{\infty }} \big ( \Vert g_{t} \Vert _{L^{1}((0, L), H^{1})} + \Vert \Delta g \Vert _{L^{1}((0, L), H^{1})} \big ), \end{aligned} \end{aligned}$$

where the constant C is independent of $\omega$. This yields

$$\begin{aligned} \begin{aligned}&\left\| \int _{0}^{\cdot } \psi (\omega s) e^{i \Gamma _{\omega } (\cdot , s) \Delta } g (s) \, ds \right\| _{L^{\infty }((0, L), H^{1})} \\&\qquad \le \frac{C}{|\omega |} \Vert \Psi \Vert _{L^{\infty }} \left( \sup _{t \in (0, L)} \Vert g (t) \Vert _{H^{1}} + \Vert g_{t} \Vert _{L^{1}((0, L), H^{1})} + \Vert \Delta g \Vert _{L^{1}((0, L), H^{1})} \right) . \end{aligned} \end{aligned}$$

Letting $|\omega | \rightarrow \infty$, we obtain the desired convergence, which completes the proof of Lemma 3. $\square$

Lemma 4

If $f \in H^{1}({\mathbb {R}}^{N})$ for some $0 < L \le \infty$, then for a fixed $s \in [0, L)$, we have

$$\begin{aligned} \begin{aligned} \sup _{t \in (0, L)}\left\| \left( e^{i \Gamma _{\omega } (t, s) \Delta } - e^{iI(\gamma ) (t-s) \Delta } \right) f\right\| _{H^{1}} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

(16)

Proof

Since $\gamma$ is $\tau$–periodic, we can decompose $\Gamma _{\omega }$ as

$$\begin{aligned} \begin{aligned} \Gamma _{\omega } (t, s) = I (\gamma ) (t - s) + \frac{1}{\omega } \int _{\omega s}^{\omega t} \gamma _{0} (t') \, d t' \end{aligned} \end{aligned}$$

for every $s, t \in {\mathbb {R}}$, where $I(\gamma ) \in {\mathbb {R}} - \{0\}$ denotes the average defined by (6) and $\gamma _{0}$ is a $\tau$–periodic function with mean zero. Denote by

$$\begin{aligned} \begin{aligned} \vartheta _{\omega } (t ,s) = \int _{\omega s}^{\omega t} \gamma _{0} (t') \, d t', \end{aligned} \end{aligned}$$

since

$$\begin{aligned} \begin{aligned} \left| \int _{s}^{t} \gamma _{0} (t') \, dt' \right| \le \tau (M - I(\gamma )) \end{aligned} \end{aligned}$$

we obtain that $\vartheta _{\omega } \in L^{\infty }({\mathbb {R}}^{2})$ uniformly. Hence, using Plancherel’s identity and Minkowski’s inequality, we have

$$\begin{aligned} \begin{aligned}&\sup _{t \in (0, L)} \left\| \left( e^{i \Gamma _{\omega } (t, s) \Delta } - e^{i I(\gamma ) (t-s) \Delta } \right) f \right\| ^{2}_{H^{1}} \\[0.5ex]&\qquad = \sup _{t \in (0, L)} \int _{{\mathbb {R}}^{N}} ( 1 + | \xi |^{2} ) \left| e^{i I(\gamma ) (t - s) |\xi |^{2} } ( e^{i \frac{1}{\omega } \vartheta _{\omega } (t, s) |\xi |^{2}} - 1 )\right| ^{2} | {\widehat{f}} (\xi )|^{2} \, d\xi \\[0.5ex]&\qquad \le \int _{{\mathbb {R}}^{N}} ( 1 + | \xi |^{2} ) | {\widehat{f}} (\xi ) |^{2} \sup _{t \in (0, L)} \left| e^{i \frac{1}{\omega } \vartheta _{\omega } (t, s) |\xi |^{2}} - 1 \right| ^{2} \, d\xi \end{aligned} \end{aligned}$$

for a fixed $s \in [0, L)$. Thus (16) follows from the Lebesgue dominated convergence theorem. $\square$

Lemma 5

If $g \in L^{1}( (0, L), H^{1}({\mathbb {R}}^{N}))$ for some $0 < L \le \infty$, then

$$\begin{aligned} \begin{aligned} \sup _{t \in (0, L)}\left\| \int _{0}^{t} \bigg ( e^{i \Gamma _{\omega } (t, s) \Delta } - e^{i I(\gamma ) (t - s) \Delta } \bigg ) g(s) \, ds \right\| _{H^{1}} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

Proof

Since $g(s) \in H^{1} ({\mathbb {R}}^{N})$, it follows from Lemma 4 that

$$\begin{aligned} \begin{aligned} h_{\omega } (s) := \sup _{t \in (0, L)} \left\| \left( e^{i \Gamma _{\omega } (t, s) \Delta } - e^{I(\gamma ) (t-s) \Delta } \right) g(s) \right\| _{H^{1}} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

Using Minkowski’s inequality, we get

$$\begin{aligned} \begin{aligned} \sup _{t \in (0, L)} \left\| \int _{0}^{t} \bigg ( e^{i \Gamma _{\omega } (t, s) \Delta } - e^{i I(\gamma ) (t - s) \Delta } \bigg ) g(s) \, ds \right\| _{H^{1}} \le \int _{0}^{L} h_{\omega } (s) \, ds \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0 \end{aligned} \end{aligned}$$

because the Lebegsue dominated convergence theorem with the fact that

$$\begin{aligned} \begin{aligned} h_{\omega } \le C \Vert g (\cdot )\Vert _{H^{1}} \in L^{1}(0, L). \end{aligned} \end{aligned}$$

Recall the following Gronwall-type estimate whose proof can be found in [3, Lemma A.1] $\square$

Lemma 6

Assume that $0< T <\infty$, $1 \le p < q \le \infty$, and $A, B \ge 0$. If $f \in L^{q} (0, T)$ satisfies

$$\begin{aligned} \begin{aligned} \Vert f \Vert _{L^{q}(0, t)} \le A + B \Vert f \Vert _{L^{p}(0, t)} \end{aligned} \end{aligned}$$

for all $0< t < T$, then there exists a constant $K = K (B, p, q, T)$ such that

$$\begin{aligned} \begin{aligned} \Vert f \Vert _{L^{q}(0, T)} \le A K. \end{aligned} \end{aligned}$$

For the proof of Theorem 1, we introduce the special admissible pairs (q, r) such that

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} q = \alpha + 4, \quad r = \displaystyle \frac{2N(\alpha + 4)}{N(\alpha + 4) - 4} &{} \qquad \hbox {if} \;\; N = 1, 2 \\ q = \displaystyle \frac{\alpha + 4}{2}, \quad r = \displaystyle \frac{6(\alpha + 4)}{3(\alpha + 4) - 8} &{} \qquad \hbox {if} \;\; N = 3. \end{array} \right. \end{aligned} \end{aligned}$$

(17)

Then since $\alpha < q$ and $N < r$, it follows from the Sobolev embedding theorem that

$$\begin{aligned} \begin{aligned} L^{q}((0, L), W^{1, r}({\mathbb {R}}^{N})) \hookrightarrow L^{q}((0, L), L^{\infty }({\mathbb {R}}^{N})). \end{aligned} \end{aligned}$$

(18)

Key for our proof of Theorem 1 is the following lemma.

Lemma 7

Assume (2). Fix an initial value $\varphi \in H^{1}({\mathbb {R}}^{N})$, and given $\omega \in {\mathbb {R}}$, denote by $u_{\omega }$ the maximal solution of (3). Let U be the maximal solution of (5) defined on the interval $[0, S_{\max })$. For $0< L < S_{\max }$, we assume that $u_{\omega }$ exists on [0, L] for $|\omega |$ sufficiently large and

$$\begin{aligned} \begin{aligned} \limsup _{|\omega | \rightarrow \infty } \Vert u_{\omega } \Vert _{L^{\infty }((0, L), H^{1})} < \infty \end{aligned} \end{aligned}$$

(19)

and

$$\begin{aligned} \begin{aligned} \limsup _{|\omega | \rightarrow \infty } \Vert u_{\omega } \Vert _{L^{q}((0, L), W^{1, r})} < \infty \end{aligned} \end{aligned}$$

where (q, r) is given by (17). Then it follows that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U \Vert _{L^{\infty }( (0, L), H^{1})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

Proof

From (3) and (5), we have

$$\begin{aligned} \begin{aligned} u_{\omega }(t) - U(t) = \bigg ( e^{i \Gamma _\omega (t, 0) \Delta } - e^{i I(\gamma )t \Delta } \bigg ) \varphi + i \bigg ( {\mathcal {I}}_1 (t) + {\mathcal {I}}_2 (t) + {\mathcal {I}}_3 (t) \bigg ), \end{aligned} \end{aligned}$$

(20)

where

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_{1} (t)&= \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } \theta (\omega s) \bigg ( | u_{ \omega }(s) |^{\alpha } u_{\omega }(s) - | U(s) |^{\alpha } U(s) \bigg ) \, ds,\\[0.5ex] {\mathcal {I}}_{2} (t)&= \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } \bigg ( \theta (\omega s) - I (\theta ) \bigg ) | U(s) |^{\alpha } U(s) \, ds,\\[0.5ex] {\mathcal {I}}_{3} (t)&= \int _{0}^{t} \bigg ( e^{i \Gamma _{\omega } (t, s) \Delta } - e^{i I(\gamma ) (t - s) \Delta } \bigg ) I(\theta ) | U(s) |^{\alpha } U(s) \, ds. \end{aligned} \end{aligned}$$

For the first term on the right hand side of (20), it follows from Lemma 4 that

$$\begin{aligned} \begin{aligned} \sup _{t\in (0, L)}\left\| \left( e^{i \Gamma _\omega (t, 0) \Delta } - e^{iI(\gamma )t \Delta }\right) \varphi \right\| _{H^{1} } \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

(21)

Observe that $| U |^{\alpha } U \in L^{1}((0, L), H^{1}({\mathbb {R}}^{N}))$. Indeed, using Hölder’s inequality and (18), we see that

$$\begin{aligned} \begin{aligned} \int _{0}^{L} \Vert | U(s) |^{\alpha } U(s) \Vert _{H^{1}} \, ds&\le \int _{0}^{L} \Vert U (s) \Vert _{L^{\infty }}^{\alpha } \Vert U(s) \Vert _{H^{1}} \, ds\\&\le \Vert U \Vert ^{\alpha }_{L^{q}((0, L), L^{\infty })} \Vert U \Vert _{L^{\frac{q}{q-\alpha }} ((0, L), H^{1})} \\&\le C \Vert U \Vert ^{\alpha }_{L^{q}((0, L), W^{1, r})} \Vert U \Vert _{L^{\infty } ((0, L), H^{1})}. \end{aligned} \end{aligned}$$

Thus Lemmas 3 and 5 imply that

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {I}}_{2} \Vert _{L^{\infty } ((0, L), H^{1})} + \left\| {\mathcal {I}}_{3} \right\| _{L^{\infty } ((0, L), H^{1})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

(22)

We now estimate ${\mathcal {I}}_{1}$ to show $L^{\infty } L^{2}$–convergence. Denote the nonlinearity by $g(u) = |u|^{\alpha } u$ for simplicity. Recall that for all $u, v \in {\mathbb {C}}$, it holds

$$\begin{aligned} \begin{aligned} | g(u) - g(v) | \le C \left( |u|^{\alpha } + |v|^{\alpha } \right) | u - v |. \end{aligned} \end{aligned}$$

Applying the Hölder inequality in both space and time together with the Sobolev embedding (18), we see that

$$\begin{aligned} \begin{aligned}&\Vert g(u_{\omega }) - g(U) \Vert _{L^{1}((0, t), L^{2})} \\[0.5ex]&\qquad \le C \left( \Vert u_{\omega } \Vert _{L^{q}((0, t), L^{\infty })}^{\alpha } + \Vert U \Vert _{L^{q}((0, t), L^{\infty })}^{\alpha } \right) \Vert u_{\omega } - U \Vert _{L^{\frac{q}{q - \alpha }}((0, t), L^{2})}\\&\qquad \le C \left( \Vert u_{\omega } \Vert _{L^{q}((0, t), W^{1,r})}^{\alpha } + \Vert U \Vert _{L^{q}((0, t), W^{1, r})}^{\alpha }\right) \Vert u_{\omega } - U \Vert _{L^{\frac{q}{q - \alpha }}((0, t), L^{2})} \end{aligned} \end{aligned}$$

for all $0 < t \le L$. With this we can estimate ${\mathcal {I}}_{1}$, using Strichartz’s estimate, via

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {I}}_{1} \Vert _{L^{\infty }((0, t), L^{2})}&\le C \Vert g(u_{\omega }) - g(U) \Vert _{L^{1}((0, t), L^{2})}\\&\le C \big ( \Vert u_{\omega } \Vert _{L^{q}((0, t), W^{1,r})}^{\alpha } + \Vert U \Vert _{L^{q}((0, t), W^{1, r})}^{\alpha } \big ) \Vert u_{\omega }\\ {}&\qquad - U \Vert _{L^{\frac{q}{q - \alpha }}((0, t), L^{2})} \end{aligned} \end{aligned}$$

(23)

for all $0 < t \le L$. From (21), (23), and (22) there exists a $\varepsilon _{\omega }>0$ and a constant $C > 0$ independent of $\omega$ such that we have

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U \Vert _{L^{\infty }( (0, t), L^{2} )} \le \varepsilon _{\omega } + C \Vert u_{\omega } - U \Vert _{L^{\frac{q}{q - \alpha }}((0, t), L^{2})} \end{aligned} \end{aligned}$$

for all $0 < t \le L$, which implies from Lemma 6 that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U \Vert _{L^{\infty }( (0, L), L^{2} )} \le C \varepsilon _{\omega } \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

(24)

We next prove convergence in $L^{\infty }((0, L), H^{1}({\mathbb {R}}^{N}))$. For this, we use an argument of Kato [8]. Observe that by (20)

$$\begin{aligned} \begin{aligned} \nabla u_{\omega }(t) - \nabla U(t) = \bigg ( e^{i \Gamma _\omega (t, 0) \Delta } - e^{iI(\gamma )t \Delta } \bigg ) \nabla \varphi + i \bigg ( \nabla {\mathcal {I}}_1 (t) + \nabla {\mathcal {I}}_2 (t) + \nabla {\mathcal {I}}_3 (t) \bigg ). \end{aligned} \end{aligned}$$

Here $\nabla {\mathcal {I}}_1 (t)$ can be rewritten as

$$\begin{aligned} \begin{aligned} \nabla {\mathcal {I}}_{1} (t) = {\mathcal {J}}_{1} (t) + {\mathcal {J}}_{2} (t), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {\mathcal {J}}_{1} (t)&= \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } \theta (\omega s) g'(u_{\omega }(s)) \cdot \bigg ( D u_{ \omega } (s) - D U(s) \bigg ) \, ds,\\ {\mathcal {J}}_{2} (t)&= \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } \theta (\omega s) \bigg ( g' ( u_{ \omega }(s) ) - g'( U(s)) \bigg ) \cdot D U(s) \, ds, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} g'(u) = \left( \begin{array}{c} \frac{\alpha + 2}{2} |u|^{\alpha } \\ \frac{\alpha }{2} |u|^{\alpha - 2} u^{2} \\ \end{array} \right) \quad \text{ and } \quad Du = \left( \begin{array}{c} \nabla u \\ \nabla {\overline{u}} \\ \end{array} \right) . \end{aligned} \end{aligned}$$

Since $|g'(u_{\omega })| \le C |u_{\omega }|^{\alpha }$, using Strichartz’s estimate, Hölder’s inequality in time and (18), we obtain

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {J}}_{1} \Vert _{L^{\infty }((0, L), L^{2})}&\le C \Vert g'(u_{\omega }) \cdot \left( D u_{\omega } - D U \right) \Vert _{L^{1}((0, L), L^{2})}\\&\le C \Vert u_{\omega } \Vert _{L^{q}((0, L), W^{1,r})}^{\alpha } \Vert \nabla u_{\omega } - \nabla U \Vert _{L^{\frac{q}{q - \alpha }}((0, L), L^{2})}\\&\le C \Vert \nabla u_{\omega } - \nabla U \Vert _{L^{\frac{q}{q - \alpha }}((0, L), L^{2})}. \end{aligned} \end{aligned}$$

(25)

Again, applying Strichartz’s estimate and Hölder’s inequality, we see that

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {J}}_{2} \Vert _{L^{\infty }((0, L), L^{2})}&\le C \Vert ( g'(u_{\omega }) - g'(U) ) \cdot D U \Vert _{L^{\rho '}((0,L), L^{(\alpha + 2)'})} \\&\le C \Vert \nabla U \Vert _{L^{\rho }((0, L), L^{\alpha + 2})} \Vert g' (u_{\omega }) - g'(U) \Vert _{L^{\frac{\rho }{\rho - 2}}((0,L), L^{\frac{\alpha + 2}{\alpha }})}, \end{aligned} \end{aligned}$$

where $(\rho , \alpha + 2)$ is an admissible pair, i.e., $\rho = 4(\alpha +2)/N\alpha$.

If we assume

$$\begin{aligned} \begin{aligned} \Vert g'(u_{\omega }) - g'(U) \Vert _{L^{\infty }((0,L), L^{\frac{\alpha +2}{\alpha }})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0, \end{aligned} \end{aligned}$$

(26)

we can obtain

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {J}}_{2} \Vert _{L^{\infty }((0, L), L^{2})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0, \end{aligned} \end{aligned}$$

(27)

which, by (21), (22), (25), and (27), and virtue of Lemma 6, implies that

$$\begin{aligned} \begin{aligned} \Vert \nabla u_{\omega } - \nabla U \Vert _{L^{\infty } ((0, L), L^{2}) } \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

Hence to completes the proof, it suffices to show (26). It follows from (19) and (24) that $u_{\omega } \rightarrow U$ in $C([0, L], H^{s}({\mathbb {R}}^{N}))$ as $|\omega | \rightarrow \infty$ for all $0 \le s < 1$. Choosing $s < 1$ sufficiently close to 1 so that $H^{s}({\mathbb {R}}^{N}) \hookrightarrow L^{\alpha + 2}({\mathbb {R}}^{N})$, we deduce that $u_{\omega } \rightarrow U$ in $C([0, L], L^{\alpha + 2}({\mathbb {R}}^{N}))$. From the well-known fact

$$\begin{aligned} \begin{aligned} | g'(u) - g'(v) | \le \left\{ \begin{array}{ll} C | u - v |^{\alpha } &{} \hbox {if} \;\; 0 < \alpha \le 1 \\ C ( |u|^{\alpha - 1} + |v|^{\alpha - 1} ) | u-v | &{} \hbox {if} \;\; \alpha > 1, \end{array} \right. \end{aligned} \end{aligned}$$

we obtain the mapping $u \mapsto g'(u)$ is continuous $L^{\alpha + 2}({\mathbb {R}}^{N}) \rightarrow L^{(\alpha +2)/\alpha }({\mathbb {R}}^{N})$, which yields (26). This completes the proof of Lemma 7. $\square$

Now, we are ready to complete the proof of Theorem 1.

Proof of Theorem 1

From Lemma 7, we only show that the conditions of Lemma 7 hold under the assumptions of Theorem 1. Fix $0< S < S_{\max }$ and set

$$\begin{aligned} \begin{aligned} M = 2 \sup _{0 \le t \le S} \Vert U(t)\Vert _{H^{1}}. \end{aligned} \end{aligned}$$

It follows from Proposition 2 that for $\Vert \varphi \Vert _{H^{1}} \le M$ there exists $T = T(A, M) > 0$, where $A = \Vert \theta \Vert _{L^{\infty }}$, such that $u_{\omega }$ exists on [0, T] for all $\omega \in {\mathbb {R}}$, moreover,

$$\begin{aligned} \begin{aligned} \sup _{\omega \in {\mathbb {R}}} \Vert u_{\omega }\Vert _{L^{\infty }((0, T), H^{1})} \le C \Vert \varphi \Vert _{H^{1}} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \sup _{\omega \in {\mathbb {R}}} \Vert u_{\omega }\Vert _{L^{q}((0, T), W^{1, r})} \le C \Vert \varphi \Vert _{H^{1}} \end{aligned} \end{aligned}$$

where (q, r) is given by (17). Next, let $0 < L \le S$ be such that $u_{\omega }$ exists on [0, L] for $|\omega |$ sufficiently large,

$$\begin{aligned} \begin{aligned} \limsup _{|\omega | \rightarrow \infty } \Vert u_{\omega }\Vert _{L^{\infty }((0, L), H^{1})} < \infty , \end{aligned} \end{aligned}$$

(28)

and

$$\begin{aligned} \begin{aligned} \limsup _{|\omega | \rightarrow \infty } \Vert u_{\omega }\Vert _{L^{q}((0, L), W^{1, r})} < \infty . \end{aligned} \end{aligned}$$

(29)

Note that $L = T$ is always a possible choice. Then by Lemma 7, we have that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U\Vert _{L^{\infty }((0, L), H^{1})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0 \end{aligned} \end{aligned}$$

and, since $u_{\omega } - U \in C([0, L], H^{1}({\mathbb {R}}^{N}))$, it follows that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }(L) - U(L) \Vert _{H^{1}} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

Hence $\Vert u_{\omega } (L)\Vert _{H^{1}} \le M$ for $|\omega |$ sufficiently large. Applying Proposition 2 to (3) translated by L, we deduce that for $|\omega |$ sufficiently large, $u_{\omega }$ exists on $[0, L + T]$, moreover, applying (28) and (29) yields

$$\begin{aligned} \begin{aligned} \limsup _{|\omega | \rightarrow \infty } \Vert u_{\omega }\Vert _{L^{\infty }((0, L + T), H^{1})} < \infty , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \limsup _{|\omega | \rightarrow \infty } \Vert u_{\omega } \Vert _{L^{q}((0, L + T), W^{1, r})} < \infty . \end{aligned} \end{aligned}$$

This means that the estimates (28) and (29) hold with L replaced by $L+T$, provided $L+T \le S$. Iterating this argument, we see that the estimates (28) and (29) hold L replaced by S, which proves Theorem 1. $\square$

4 Proof of Theorem 2

We give the proof of Theorem 2 at the end of this section after some lemmas.

Lemma 8

Assume (2) and $\alpha \ge 4/N$. Let r and a be defined by (14). Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(\cdot , 0)\Delta }f \Vert _{L^{a}({\mathbb {R}}, L^{r})} \le C \Vert \nabla f \Vert _{L^{2}}^{\frac{N\alpha -4}{2\alpha }} \Vert f \Vert _{L^{2}}^{\frac{4-(N-2)\alpha }{2\alpha }} \end{aligned} \end{aligned}$$

for all $f \in H^{1}({\mathbb {R}}^{N})$.

Proof

Using the Strichartz estimates in Lemma 2, the proof is virtually identical to the proof of [3, Lemma 3.2]. $\square$

Lemma 9

Assume (2) and $\alpha \ge 4/N$. Let r and a be defined by (14). If $f \in H^{1}({\mathbb {R}}^{N})$, then we have

$$\begin{aligned} \begin{aligned} \left\| \left( e^{i \Gamma _{\omega }(\cdot , 0) \Delta }-e^{i I(\gamma ) \cdot \Delta } \right) f \right\| _{L^{a}((0, \infty ), L^{r})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

(30)

Proof

In the following, we denote the operator by $A(t):=e^{i\Gamma _{\omega } (t,0) \Delta }-e^{i I(\gamma ) t\Delta }$ for simplicity. First, we consider the case $\alpha > 4/N$. Then we have $a > q$, where q is given by (14). The Hölder inequality yields

$$\begin{aligned} \begin{aligned} \Vert A(\cdot ) f\Vert _{L^{a}((0, \infty ), L^{r})} \le \Vert A(\cdot ) f\Vert _{L^{\infty }((0, \infty ), L^{r})}^{\frac{(\alpha +2)(N\alpha -4)}{N\alpha ^{2}}} \Vert A(\cdot )f\Vert _{L^{q}((0, \infty ), L^{r})}^{\frac{8-2(N-2)\alpha }{N\alpha ^{2}}}. \end{aligned} \end{aligned}$$

Since (q, r) is an admissible pair, we use the triangle inequality and the Strichartz estimate to see that there exists a constant $C > 0$, independent of $\omega$, such that

$$\begin{aligned} \Vert A(\cdot ) f\Vert _{L^{q}((0, \infty ), L^{r})}\le & {} C \left( \Vert e^{i\Gamma _\omega (\cdot , 0)\Delta } f\Vert _{L^{q}((0, \infty ), L^{r})} + \Vert e^{i I(\gamma ) \cdot \Delta } f \Vert _{L^{q}((0, \infty ), L^{r})}\right) \nonumber \\\le & {} C \Vert f \Vert _{L^{2}}. \end{aligned}$$

(31)

From Gagliardo–Nirenberg’s inequality, we also obtain

$$\begin{aligned} \begin{aligned} \Vert A(t) f\Vert _{L^{r}}&\le C \Vert \nabla (A(t) f) \Vert _{L^{2}}^{\frac{N\alpha }{2(\alpha +2)}} \Vert A(t)f \Vert _{L^{2}}^{\frac{4-(N-2)\alpha }{2(\alpha +2)}}\\&\le C \Vert A(t)f \Vert _{H^{1}}^{\frac{N\alpha }{2(\alpha +2)}} \left( \Vert e^{i\Gamma _\omega (t,0)\Delta } f\Vert _{L^{2}}+\Vert e^{i I(\gamma ) t\Delta }f\Vert _{L^{2}}\right) ^{\frac{4-(N-2)\alpha }{2(\alpha +2)}}\\&\le C \Vert f \Vert _{L^{2}}^{\frac{4-(N-2)\alpha }{2(\alpha +2)}}\Vert A(t)f\Vert _{H^{1}}^{\frac{N\alpha }{2(\alpha +2)}}, \end{aligned} \end{aligned}$$

(32)

where we used the fact that $e^{i\Gamma _\omega (t,0)\Delta }$ and $e^{i I(\gamma ) t \Delta }$ are unitary operators in $L^{2}({\mathbb {R}}^{N})$. Collecting (31) and (32), if follows that

$$\begin{aligned} \begin{aligned} \Vert A(\cdot ) f \Vert _{L^{a}((0, \infty ), L^{r})} \le C \Vert f \Vert _{L^{2}}^{\frac{4-(N-2)\alpha }{2\alpha }} \Vert A(\cdot ) f \Vert _{L^{\infty }((0, \infty ),H^{1})}^{\frac{N\alpha -4}{2\alpha }}. \end{aligned} \end{aligned}$$

Applying Lemma 4 to the second factor of the right-hand side above, we conclude (30).

Next, in the case of $\alpha = 4/N$, since $(a, r)=(\alpha +2, \alpha +2)$ is an admissible pair, it follows from Strichartz’s estimate that

$$\begin{aligned} \begin{aligned} \left\| A(\cdot ) f \right\| _{L^{a}((0, \infty ), L^{r})}&\le C \left( \Vert e^{i\Gamma _\omega (\cdot , 0)\Delta }f \Vert _{L^{a}((0, \infty ), L^{r})} +\Vert e^{i \cdot I(\gamma ) \Delta }f\Vert _{L^{a}((0, \infty ), L^{r})}\right) \\&\le C \Vert f \Vert _{L^{2}}. \end{aligned} \end{aligned}$$

Given any $\varepsilon > 0$, therefore, we can choose $0< {\widetilde{T}} = {\widetilde{T}}(\varepsilon ) < \infty$ such that

$$\begin{aligned} \begin{aligned} \left\| A(\cdot ) f\right\| _{L^{a}(({\widetilde{T}}, \infty ), L^{r})} \le \frac{\varepsilon }{2} \end{aligned} \end{aligned}$$

(33)

for every $\omega \in {\mathbb {R}}$. Note from the embedding $H^{1}({\mathbb {R}}^{N}) \hookrightarrow L^{r}({\mathbb {R}}^{N})$ that

$$\begin{aligned} \begin{aligned} \left\| A(\cdot ) f \right\| _{L^{a}((0,{\widetilde{T}}), L^{r})}&\le \left\| A(\cdot ) f \right\| _{L^{a}((0,{\widetilde{T}}), H^{1})} \le {\widetilde{T}}^{1 /{r}} \left\| A(\cdot ) f \right\| _{L^{\infty }((0,{\widetilde{T}}), H^{1})}. \end{aligned} \end{aligned}$$

Thus, applying Lemma 4 together with (33), we have

$$\begin{aligned} \begin{aligned} \Vert A(\cdot ) f\Vert _{L^{a}((0, \infty ), L^{r})} \le \varepsilon \end{aligned} \end{aligned}$$

(34)

for $|\omega |$ sufficiently large, which finishes the proof of Lemma 9. $\square$

Now we are ready to give

Proof of Theorem 2

By Theorem 1, we know that the existence time S of $u_{\omega }$ goes to infinity as $|\omega | \rightarrow \infty$ and that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U\Vert _{L^{\infty }((0,S), H^{1})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0 \end{aligned} \end{aligned}$$

for all $S < \infty$. In particular,

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }(S) - U(S)\Vert _{H^{1}} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

(35)

To prove the global existence of $u_{\omega }$ for $|\omega |$ sufficiently large, let $\varepsilon > 0$ such that $\varepsilon \le \varepsilon (A)$, where $A = \Vert \theta \Vert _{L^{\infty }}$ and $\varepsilon (A)$ is defined in Proposition 3. Since $U \in L^{a}((0, \infty ), L^{r}({\mathbb {R}}^{N}))$, we can choose S sufficiently large so that

$$\begin{aligned} \begin{aligned} \Vert U \Vert _{L^{a}((S, \infty ), L^{r})} \le \frac{\varepsilon }{6}. \end{aligned} \end{aligned}$$

Moreover, it follows from Proposition 3 with $\Gamma _\omega (t,0)$ replaced by $I(\gamma )t$, see also [3, Proposition 2.4] or [4], that

$$\begin{aligned} \begin{aligned} \Vert e^{i I(\gamma ) \cdot \Delta } U(S)\Vert _{L^{a}((0, \infty ), L^{r})} \le 2 \Vert U \Vert _{L^{a}((S, \infty ), L^{r})} \le \frac{\varepsilon }{3}. \end{aligned} \end{aligned}$$

(36)

Notice that

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(\cdot , 0 ) \Delta } u_{\omega } (S)\Vert _{L^{a}((0, \infty ), L^{r})}&\le \Vert e^{i\Gamma _{\omega }(\cdot , 0)\Delta } (u_{\omega }(S) - U(S))\Vert _{L^{a}((0, \infty ), L^{r})}\\&\qquad +\Vert \left( e^{i\Gamma _{\omega }(\cdot , 0)\Delta }-e^{i I(\gamma ) \cdot \Delta } \right) U(S)\Vert _{L^{a}((0, \infty ), L^{r})} \\&\qquad + \Vert e^{i I(\gamma ) \cdot \Delta } U(S)\Vert _{L^{a}((0, \infty ), L^{r})}. \end{aligned} \end{aligned}$$

By Lemma 8 and (35), we infer

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(\cdot , 0) \Delta } (u_\omega (S)-U(S))\Vert _{L^{a}((0, \infty ), L^{r})} \le C\Vert u_{\omega }(S)-U(S)\Vert _{H^{1}} \le \frac{\varepsilon }{3}. \end{aligned} \end{aligned}$$

(37)

Combining (36), (37), and Lemma 9, we conclude

$$\begin{aligned} \begin{aligned} \Vert e^{i\Gamma _{\omega }(\cdot , 0)\Delta }u_{\omega }(S)\Vert _{L^{a}((0, \infty ), L^{r})} \le \varepsilon . \end{aligned} \end{aligned}$$

for $|\omega |$ sufficiently large. By virtue of Corollary 2, we see that $u_{\omega }$ is global and that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }\Vert _{L^{a}((S, \infty ), L^{r})} \le 2 \varepsilon \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert u_{\omega }\Vert _{L^{q}((S, \infty ), W^{1, r})} + \Vert u_{\omega }\Vert _{L^{\infty }((S, \infty ), H^{1})} \le \Lambda \Vert u_{\omega }(S)\Vert _{H^{1}} \end{aligned} \end{aligned}$$

(38)

provided $|\omega |$ is sufficiently large. In the same way, we also obtain

$$\begin{aligned} \begin{aligned} \Vert U\Vert _{L^{a}((S, \infty ), L^{r})} \le 2 \varepsilon \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert U\Vert _{L^{q}((S, \infty ), W^{1, r})} + \Vert U\Vert _{L^{\infty }((S, \infty ), H^{1})} \le \Lambda \Vert U(S)\Vert _{H^{1}}. \end{aligned} \end{aligned}$$

(39)

Hence there exits a constant M such that for L sufficiently large,

$$\begin{aligned} \begin{aligned} \sup _{\omega \ge L} \sup _{t \ge 0} \Vert u_{\omega } (t) \Vert _{H^{1}} + \sup _{t \ge 0} \Vert U(t)\Vert _{H^{1}} \le M < \infty . \end{aligned} \end{aligned}$$

(40)

We now prove $u_{\omega } \rightarrow U$ in $L^{\infty }((0, \infty ), H^{1}({\mathbb {R}}^{N}))$ as $|\omega | \rightarrow \infty$. Observe that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U \Vert _{L^{\infty }((0, \infty ), H^{1})} \le \Vert u_{\omega } - U \Vert _{L^{\infty }((0, S), H^{1})}+ \Vert u_{\omega } - U \Vert _{L^{\infty }((S, \infty ), H^{1})}, \end{aligned} \end{aligned}$$

where $S > 0$ to be chosen later. Theorem 1 implies that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U \Vert _{L^{\infty }((0, S), H^{1})} \underset{|\omega | \rightarrow \infty }{\longrightarrow } 0. \end{aligned} \end{aligned}$$

We claim that for every $\eta >0$, there exists $S > 0$ such that

$$\begin{aligned} \begin{aligned} \Vert u_{\omega } - U \Vert _{L^{\infty }((S, \infty ), H^{1})} \le \eta \end{aligned} \end{aligned}$$

(41)

for $|\omega |$ sufficiently large. To prove this, note that

$$\begin{aligned} \begin{aligned} u_{\omega }(S + t) - U(S + t)&= e^{i \Gamma _{\omega }(t, 0) \Delta } (u_{\omega }(S) - U(S)) + \big ( e^{i \Gamma _{\omega }(t, 0) \Delta } - e^{iI(\gamma )t \Delta } \big ) U(S)\\&\quad + i (a(t)-b(t)), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} a(t):= \int _{0}^{t} e^{i \Gamma _{\omega } (t, s) \Delta } \theta (\omega (S+s))|u_{\omega } (S+s)|^{\alpha } u_{\omega }(S+s) \, ds \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} b(t):= \int _{0}^{t} e^{i I(\gamma ) (t - s) \Delta } I(\theta ) | U(S+s) |^{\alpha } U(S+s) \, ds. \end{aligned} \end{aligned}$$

Using Strichartz’s estimate and Hölder’s inequality in time, there exists a constant $C > 0$, independent of S, such that

$$\begin{aligned} \begin{aligned} \Vert a \Vert _{L^{\infty }((0, \infty ), H^{1})}&\le C \Vert |u_{\omega }|^{\alpha }u_{\omega }\Vert _{L^{q'}((S, \infty ), W^{1, r'})} \\&\le C \Vert u_{\omega } \Vert _{L^{a}((S, \infty ), L^{r})}^{\alpha } \Vert u_{\omega } \Vert _ {L^{q}((S, \infty ), W^{1, r})}, \end{aligned} \end{aligned}$$

and similarly,

$$\begin{aligned} \begin{aligned} \Vert b \Vert _{L^{\infty }((0, \infty ), H^{1})}&\le C \Vert |U|^{\alpha } U \Vert _{L^{q'}((S, \infty ), W^{1, r'})}\\&\le C \Vert U \Vert _{L^{a}((S, \infty ), L^{r})}^{\alpha } \Vert U \Vert _ {L^{q}((S, \infty ), W^{1, r})}. \end{aligned} \end{aligned}$$

Given now $\eta > 0$, we choose $\varepsilon > 0$ sufficiently small so that $2^{\alpha + 1} \varepsilon ^{\alpha } C M \le \eta /2$. We then fix S sufficiently large so that

$$\begin{aligned} \begin{aligned} \Vert U \Vert _{L^{a}((S, \infty ), L^{r})} \le \frac{\varepsilon }{4}. \end{aligned} \end{aligned}$$

Then it follows from (38), (39) and (40) that

$$\begin{aligned} \begin{aligned} \Vert a \Vert _{L^{\infty }((0, \infty ), H^{1})} + \Vert b \Vert _{L^{\infty }((0, \infty ), H^{1})} \le 2^{\alpha +1} \varepsilon ^{\alpha } C M \le \frac{\eta }{2} \end{aligned} \end{aligned}$$

for $|\omega |$ sufficiently large. It follows from (35) and Lemma 4 that

$$\begin{aligned} \begin{aligned} \sup _{t \ge 0} \Vert e^{i \Gamma _{\omega }(t, 0) \Delta } (u_{\omega }(S)- U(S))\Vert _{H^{1}} + \sup _{t \ge 0} \Vert \big ( e^{i \Gamma _{\omega }(t, 0) \Delta } - e^{iI(\gamma )t \Delta } \big )U(S)\Vert _{H^{1}} \le \frac{\eta }{2} \end{aligned} \end{aligned}$$

for $|\omega |$ sufficiently large, which proves (41). This completes the proof of Theorem 2.

$\square$