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.
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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
in \({\mathbb {R}}^{N}\), \(N = 1, 2, 3\), where
\(\omega \in {\mathbb {R}}\) 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:
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
or its equivalent integral form
where \(I(\gamma )\) and \(I(\theta )\) are averages of \(\gamma \) and \(\theta \), respectively, i.e.,
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
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
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
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
for all \(\omega \in {\mathbb {R}}\), where the \(\tau \)–periodic function \(\gamma \) satisfies our assumptions. Here and below, we denote by
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
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
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
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
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
for any \(f \in L^{1}({\mathbb {R}}^{N})\).
Proof
Using the explicit form of the solution operator for the free Schrödinger equation
we obtain that
Note that since \(\gamma \) is one sign and bounded away from zero, we have
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
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
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
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
then the corresponding solution \(u_{\omega }\) of (13) is global and satisfies
and
Conversely, if the solution \(u_{\omega }\) of (13) is global and satisfies
then
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
then the solution \(u_{\omega }\) is global, i.e., \(T_{\max } = \infty \). Moreover,
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
in \(L^{\infty }( (0, L), H^{1}({\mathbb {R}}^{N}))\).
Proof
Set
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
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
Since \(\gamma \) is bounded, we see that
where the constant C is independent of \(\omega \). This yields
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
Proof
Since \(\gamma \) is \(\tau \)–periodic, we can decompose \(\Gamma _{\omega }\) as
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
since
we obtain that \(\vartheta _{\omega } \in L^{\infty }({\mathbb {R}}^{2})\) uniformly. Hence, using Plancherel’s identity and Minkowski’s inequality, we have
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
Proof
Since \(g(s) \in H^{1} ({\mathbb {R}}^{N})\), it follows from Lemma 4 that
Using Minkowski’s inequality, we get
because the Lebegsue dominated convergence theorem with the fact that
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
for all \(0< t < T\), then there exists a constant \(K = K (B, p, q, T)\) such that
For the proof of Theorem 1, we introduce the special admissible pairs (q, r) such that
Then since \(\alpha < q\) and \(N < r\), it follows from the Sobolev embedding theorem that
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
and
where (q, r) is given by (17). Then it follows that
Proof
where
For the first term on the right hand side of (20), it follows from Lemma 4 that
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
Thus Lemmas 3 and 5 imply that
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
Applying the Hölder inequality in both space and time together with the Sobolev embedding (18), we see that
for all \(0 < t \le L\). With this we can estimate \({\mathcal {I}}_{1}\), using Strichartz’s estimate, via
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
for all \(0 < t \le L\), which implies from Lemma 6 that
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)
Here \(\nabla {\mathcal {I}}_1 (t)\) can be rewritten as
where
with
Since \(|g'(u_{\omega })| \le C |u_{\omega }|^{\alpha }\), using Strichartz’s estimate, Hölder’s inequality in time and (18), we obtain
Again, applying Strichartz’s estimate and Hölder’s inequality, we see that
where \((\rho , \alpha + 2)\) is an admissible pair, i.e., \(\rho = 4(\alpha +2)/N\alpha \).
If we assume
we can obtain
which, by (21), (22), (25), and (27), and virtue of Lemma 6, implies that
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
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
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,
and
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,
and
Note that \(L = T\) is always a possible choice. Then by Lemma 7, we have that
and, since \(u_{\omega } - U \in C([0, L], H^{1}({\mathbb {R}}^{N}))\), it follows that
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
and
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
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
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
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
From Gagliardo–Nirenberg’s inequality, we also obtain
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
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
Given any \(\varepsilon > 0\), therefore, we can choose \(0< {\widetilde{T}} = {\widetilde{T}}(\varepsilon ) < \infty \) such that
for every \(\omega \in {\mathbb {R}}\). Note from the embedding \(H^{1}({\mathbb {R}}^{N}) \hookrightarrow L^{r}({\mathbb {R}}^{N})\) that
Thus, applying Lemma 4 together with (33), we have
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
for all \(S < \infty \). In particular,
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
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
Notice that
Combining (36), (37), and Lemma 9, we conclude
for \(|\omega |\) sufficiently large. By virtue of Corollary 2, we see that \(u_{\omega }\) is global and that
and
provided \(|\omega |\) is sufficiently large. In the same way, we also obtain
and
Hence there exits a constant M such that for L sufficiently large,
We now prove \(u_{\omega } \rightarrow U\) in \(L^{\infty }((0, \infty ), H^{1}({\mathbb {R}}^{N}))\) as \(|\omega | \rightarrow \infty \). Observe that
where \(S > 0\) to be chosen later. Theorem 1 implies that
We claim that for every \(\eta >0\), there exists \(S > 0\) such that
for \(|\omega |\) sufficiently large. To prove this, note that
where
and
Using Strichartz’s estimate and Hölder’s inequality in time, there exists a constant \(C > 0\), independent of S, such that
and similarly,
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
Then it follows from (38), (39) and (40) that
for \(|\omega |\) sufficiently large. It follows from (35) and Lemma 4 that
for \(|\omega |\) sufficiently large, which proves (41). This completes the proof of Theorem 2.
\(\square \)
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Acknowledgements
Mi-Ran Choi has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE–2021R1I1A1A01045900) and (MSIT–2020R1A2C1A01010735). Dugyu Kim has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF–2020R1I1A1A0107237112).
Funding
National Research Foundation of Korea, Grant No. MSIP-2015R1A5A1009350; National Research Foundation of Korea, Grant No. NRF-2020R1I1A1A0107237112; National Research Foundation of Korea, Grant No. MSIT-2020R1A2C1A01010735; National Research Foundation of Korea, Grant No. MOE-2019R1I1A1A01058151.
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Choi, MR., Kim, D. Averaging of nonlinear Schrödinger equations with time-oscillatory coefficients. J. Evol. Equ. 22, 44 (2022). https://doi.org/10.1007/s00028-022-00803-9
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DOI: https://doi.org/10.1007/s00028-022-00803-9