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Almost Global Existence for the Fractional Schrödinger Equations

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Abstract

We study the time of existence of the solutions of the following nonlinear Schrödinger equation (NLS)

$$\begin{aligned} \hbox {i}u_t =(-\Delta +m)^su - |u|^2u \end{aligned}$$

on the finite x-interval \([0,\pi ]\) with Dirichlet boundary conditions

$$\begin{aligned} u(t,0)=0=u(t,\pi ),\qquad -\infty< t<+\infty , \end{aligned}$$

where \((-\Delta +m)^s\) stands for the spectrally defined fractional Laplacian with \(0<s<1/2\). We prove an almost global existence result for the above fractional Schrödinger equation, which generalizes the result in Bambusi and Sire (Dyn PDE 10(2):171–176, 2013) from \(s>1/2\) to \(0<s<1/2\).

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Acknowledgements

The first author is supported by NNSFC No. 11401041, NSFSP No. ZR2019MA062 and the second author is supported by NNSFC No. 11671066 and the Fundamental Research Funds for the Central Universities No. DUT18LK02.

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Correspondence to Hongzi Cong.

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Appendix: Proof of the Nonresonance Hypothesis

Appendix: Proof of the Nonresonance Hypothesis

In this section, we firstly give some technical lemmas to prove nonresonance condition. These lemmas can be also find in [6]. Here assume

$$\begin{aligned} m\in (0,1]={\mathcal {W}}, \end{aligned}$$

and

$$\begin{aligned} {\Omega }_{j}=\left( {j^2+m}\right) ^{s},~~~j\ge 1 \end{aligned}$$

are the frequencies.

Lemma 5.1

For any \(K\le r\), consider K indexes \(j_{1}<\cdots <j_{K}\le N\); consider the determinant

$$\begin{aligned} D:=\left| \begin{array}{cccc} {\Omega }_{j_{1}} &{} {\Omega }_{j_{2}} &{} \cdots &{} {\Omega }_{j_{K}} \\ \frac{d{\Omega }_{j_{1}}}{dm} &{} \frac{d{\Omega }_{j_{2}}}{dm} &{} \cdots &{} \frac{d{\Omega }_{j_{K}}}{dm} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \frac{d^{K-1}{\Omega }_{j_{1}}}{dm^{K-1}} &{} \frac{d^{K-1}{\Omega }_{j_{2}}}{dm^{K-1}} &{} \cdots &{} \frac{d^{K-1}{\Omega }_{j_{K}}}{dm^{K-1}} \end{array}\right| . \end{aligned}$$
(5.1)

One has

$$\begin{aligned} |D|\ge \frac{C(s,r)}{N^{K^{2}}}, \end{aligned}$$
(5.2)

where C(sr) is a positive constant depending on s and r.

Proof

First, for any \(n\ge 1\) with \(n\in {\mathbb {N}}\), we have

$$\begin{aligned} \frac{d^n{\Omega }_j}{d m^n}=s(s-1)\cdots (s-n+1){(j^2+m) ^{s-n}}. \end{aligned}$$
(5.3)

Substituting (5.3) in the left-hand side of (5.1), we get the determinant to be computed. More precisely,

$$\begin{aligned} D=\prod _{l=1}^{K}{\Omega }_{j_l}\prod _{i=1}^{K-1}s(s-1)\cdots (s-i+1) \left| \begin{array}{cccc} 1 &{} 1 &{} \cdots &{} 1 \\ x_{j_1} &{} x_{j_2} &{} \cdots &{} x_{j_K} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ x_{j_1}^{K-1} &{} x_{j_2}^{K-1} &{} \cdots &{} x_{j_K}^{K-1} \end{array}\right| , \end{aligned}$$

where \(x_{j}:=\left( j^2+m\right) ^{-1}\). The last determinant is a Vandermonde determinant whose value is given by

$$\begin{aligned} \prod _{1\le l<k\le K}(x_{j_k}-x_{j_l})= & {} \prod _{1\le l<k\le K}\frac{j_l^2-j_k^2}{(j_l^2+m)(j_k^2+m)}\\= & {} \left( \prod _{1\le l<k\le K}(j_l^2-j_k^2)\right) \prod _{l=1}^K\frac{1}{(j_l^2+m)^{K-1}}. \end{aligned}$$

Using the fact that all the eigenvalues are different, one gets

$$\begin{aligned} |D|= & {} \left| \prod _{l=1}^{K}\frac{{\Omega }_{j_l}}{(j_l^2+m)^{K-1}} \prod _{i=1}^{K-1}s(s-1)\cdots (s-i+1)\prod _{1\le l<k\le K}(j_l^2-j_k^2)\right| \\\ge & {} C_1(s,r)\prod _{l=1}^K\frac{1}{(j_l^2+m)^{K-1-s}}\\\ge & {} \frac{C_1(s,r)}{(2N^2)^{K^2-(1 +s)K}}\\\ge & {} \frac{{C}(s,r)}{N^{2K^2}}, \end{aligned}$$

where using N large enough, and \(C_1(s,r)\) and C(sr) are positive constants depending on s and r only. \(\square \)

Lemma 5.2

Let \(u^{(1)}, \ldots , u^{(K)}\) be K independent vectors with \(\Vert u^{(i)}\Vert _{\ell ^{1}}\le 1\). Let \(w\in {\mathbb {R}}^{K}\) be an arbitrary vector, then there exists \(i\in \{1,\ldots ,K\},\) such that

$$\begin{aligned} |u^{(i)}\cdot w|\ge \frac{\Vert w\Vert _{\ell ^{1}}\det \left( u^{(1)},\ldots ,u^{(K)}\right) }{K^{3/2}}, \end{aligned}$$

where \(\det (u^{(i)})\) is the determinant of the matrix formed by the components of the vectors \(u^{(i)}\).

Proof

From [8] appendix B we can learn the conclusion. \(\square \)

Combining Lemmas 5.1 and 5.2, we deduce the following lemma.

Lemma 5.3

Let \(w\in {\mathbb {Z}}^{\infty }\) be a vector with K component different from zero, namely those with index \(j_{1},\ldots ,j_{K}\); assume that \(K\le r\), and assume that \(j_{1}<\cdots < j_{K}\le N\). Then for any \(m\in {\mathcal {W}}\), there exists an index \(n\in \{0,\ldots ,K-1\}\) such that

$$\begin{aligned} \bigg |w\cdot \frac{d^{n}{\Omega }}{dm^{n}}(m)\bigg |\ge \frac{C(s,r)\Vert w\Vert _{\ell ^{1}}}{N^{2K^{2}+3/2}}, \end{aligned}$$
(5.4)

where \({\Omega }=({\Omega }_{j_{1}},{\Omega }_{j_{2}},\ldots ,{\Omega }_{j_{K}})\) is the frequency vector and C(sr) is a constant depending on s and r.

Lemma 5.4

Suppose that g(m) is r times differentiable on an interval \(J\subset {\mathbb {R}}\). Let \(J_{\gamma }:=\{m\in J: |g(m)|<\gamma \},\;\gamma >0.\) If \(\left| g^{(r)}(m)\right| \ge d>0\) on J, then \(|J_{\gamma }|\le M\gamma ^{1/r}\), where \(M:=2(2+3+\cdots +r+d^{-1}).\) Here \(|\cdot |\) denotes the Lebesgue measure of set.

Proof

The proof can be found in Lemma 5.4 of [6]. \(\square \)

Proposition 5.5

For a given positive large number N, there exists a set \({\mathcal {F}}\) satisfying \(\left| {\mathcal {W}}-{\mathcal {F}}\right| \rightarrow 0\) as \(N \rightarrow +\infty ,\) such that for any \(m\in {\mathcal {F}}\),

$$\begin{aligned} \left| \langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle + \varepsilon _1{\Omega }_{{j}_1}+\varepsilon _2{\Omega }_{j_2}\right| \ge \frac{1}{N^{\frac{17 r^{6}}{1-2s}}}, \end{aligned}$$
(5.5)

where \({\tilde{k}}\in {\mathbb {Z}}^N\) with \(|{\tilde{k}}|\le r+2,\)\(\varepsilon _1,\varepsilon _2\in \{-1,0,1\}\text { and }j_1, j_2>N, ~~ |{\tilde{k}}|+|\varepsilon _1|+|\varepsilon _2|\ne 0\), \({{\tilde{\Omega }}}^{(N)}=(\Omega _j)_{j\le N}\).

Proof

For a given positive number N, we define the resonant set \({\mathcal {R}}\) by

$$\begin{aligned} {\mathcal {R}}=\bigcup _{{\tilde{k}},j_1,j_2}{{\mathcal {R}}}_{{\tilde{k}}j_1j_2}=\left\{ m\in {\mathcal {W}}:\left| \langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle + \varepsilon _1{\Omega }_{j_1}+\varepsilon _2{\Omega }_{j_2}\right| < \frac{1}{N^{\frac{17 r^{6}}{1-2s}}}\right\} , \end{aligned}$$
(5.6)

where \(|{\tilde{k}}|\le r+2,\,\varepsilon _1,\varepsilon _2\in \{-1,0,1\}\text { and }j_1, j_2>N\) and \( |{\tilde{k}}|+|\varepsilon _1|+|\varepsilon _2|\ne 0.\)

Case 1:   \(\varepsilon _1=\varepsilon _2=0\).

Denote the resonant set

$$\begin{aligned} {\mathcal {R}}_{{\tilde{k}}}=\left\{ m\in {\mathcal {W}}:\left| \langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle \right|< \frac{1}{N^{4 r^{3}}}, ~~\hbox {for} ~~0<|{\tilde{k}}|\le r+2\right\} . \end{aligned}$$

By combining Lemmas 5.3 and 5.4, we can get

$$\begin{aligned} |{\mathcal {R}}_{{\tilde{k}}}|\le & {} 2\left( 2+3+\cdots +r+1+\left( C(s,r) \right) ^{-1}N^{2(r+2)^2+3/2}\right) \frac{1}{N^{\frac{4 r^3}{r+1}}}\nonumber \\\le & {} 2\left( r^2+\left( C(s,r)\right) ^{-1}N^{2(r+2)^2+3/2}\right) \frac{1}{N^{\frac{4 r^3}{r+1}}}\nonumber \\\le & {} \frac{1}{N^{\frac{4 r^3}{r+1}-2(r+2)^2-3}}, \end{aligned}$$
(5.7)

where assuming N is large enough. Setting

$$\begin{aligned} \widetilde{R}_{1}=\bigcup _{|{\tilde{k}}|\le r+2}{\mathcal {R}}_{{\tilde{k}}}, \end{aligned}$$

then we have

$$\begin{aligned} |\widetilde{R}_{1}|\le & {} \sum _{|{\tilde{k}}|\le r+2}|{\mathcal {R}}_{{\tilde{k}}}|\nonumber \\= & {} \sum _{|{\tilde{k}}|\le r+2}\frac{1}{N^{\frac{4 r^3}{r+1}-2(r+2)^2-3}}\nonumber \\\le & {} \frac{1}{N^{\frac{4 r^3}{r+1}-2(r+2)^2-3}}(2N+1)^{r+2} \nonumber \\< & {} \frac{1}{4N}, \end{aligned}$$
(5.8)

where assuming r is large.

Case 2:   \(\varepsilon _1=\pm 1,\,\varepsilon _2=0\) or \(\varepsilon _1=0,\,\varepsilon _2=\pm 1\) or \(\varepsilon _1\varepsilon _2=1\).

Without loss of generality we take \(\varepsilon _1=1,\,\varepsilon _2=0\) to prove. Denote the resonant set

$$\begin{aligned} {\mathcal {R}}_{{\tilde{k}}j_1}=\Big \{m\in {\mathcal {W}}:\left| \langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle +{\Omega }_{j_1}\right| < \frac{1}{N^{4 r^{3}}},\ ~~\hbox {for }~~ |{\tilde{k}}|\le r+1\Big \}. \end{aligned}$$

Due to \({\Omega }_{j_1}=\left( {j_1^2+m}\right) ^{s},\) one has

$$\begin{aligned}&\left| \langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle +{\Omega }_{j_1}\right| \\&\quad \ge {\Omega }_{j_1}-\left| \langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle \right| \\&\quad \ge \left( {j_1^2+m}\right) ^{s}-(r+1)\left( N^2+m\right) ^s\\&\quad > 1 \end{aligned}$$

if

$$\begin{aligned} j_1>2^{\frac{1}{2s}}(r+1)^{\frac{1}{2s}}N. \end{aligned}$$

Then the resonant set \({\mathcal {R}}_{{\tilde{k}}j_1}\) is empty. So it is sufficient to consider

$$\begin{aligned} j_1\le 2^{\frac{1}{2s}}(r+1)^{\frac{1}{2s}}N. \end{aligned}$$

Setting \(\langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle +{\Omega }_{j_1}\) in place of \(\langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle \) and \(\widetilde{N}=2^{\frac{1}{2s}}(r+1)^{\frac{1}{2s}}N\) in place of N, then according to Case 1, we have

$$\begin{aligned} |{\mathcal {R}}_{{\tilde{k}}j_1}|\le & {} C_1( s,r)N^{2(r+2)^2+2}\cdot \frac{1}{N^{\frac{4r^3}{r+1}}}, \end{aligned}$$

where \(C_1(s,r)\) is a positive constant depending on s and r. Setting

$$\begin{aligned} \widetilde{R}_{2}=\bigcup _{|{\tilde{k}}|\le r+1}\;\;\bigcup _{j_1\le \widetilde{N}}{\mathcal {R}}_{{\tilde{k}}j_1}, \end{aligned}$$

then we have

$$\begin{aligned} |\widetilde{R}_{2}|\le & {} \sum _{|{\tilde{k}}|\le r+1}\;\;\sum _{ j_1\le \widetilde{N}}|{\mathcal {R}}_{kj_1}|\nonumber \\= & {} \sum _{|{\tilde{k}}|\le r+1}\;\; \sum _{j_1\le \widetilde{N}} C_1(s,r)N^{2(r+2)^2+2}\cdot \frac{1}{N^{\frac{4r^3}{r+1}}}\nonumber \\\le & {} C_1(s,r)N^{2(r+2)^2+2}\cdot \frac{1}{N^{\frac{4r^3}{r+1}}}\cdot \widetilde{N}\cdot N^{r+1}\nonumber \\< & {} \frac{1}{4N}, \end{aligned}$$
(5.9)

where the last inequality is based on r and N large.

Case 3: \(\varepsilon _1\varepsilon _2=-1.\)

In this case, we take \(\varepsilon _1=1,\,\varepsilon _2=-1\) and \(j_1> j_2>N\) without loss of generality.

Firstly, from the zero momentum

$$\begin{aligned} \sum _{j\le N}j{\tilde{k}}_{j}+j_1-j_2=0 \end{aligned}$$
(5.10)

we obtain that

$$\begin{aligned} \left| j_1-j_2\right| =\left| \sum _{j\le N}{j}_i{\tilde{k}}_{{j}_i}\right| \le rN<N^2. \end{aligned}$$
(5.11)

Subcase 3.1 For \(j_2>N^{\frac{8r^3}{1-2s}},\) we have

$$\begin{aligned}&\left( {j_1^2+m}\right) ^s-\left( {j_2^2 +m}\right) ^s\nonumber \\&\quad =\nonumber j_1^{2s}\left( 1+\frac{sm}{j_1^{2}} +O\left( \frac{1}{j_1^{4}}\right) \right) -j_2^{2s}\left( 1+\frac{sm}{j_2^{2}} +O\left( \frac{1}{j_2^{4}}\right) \right) \\&\quad =j_1^{2s}-j_2^{2s}+O\left( {\frac{1}{j_2^{2(1-s)}}}\right) . \end{aligned}$$
(5.12)

By Taylor’s formula, one has

$$\begin{aligned} j_1^{2s}-j_2^{2s} =\frac{2s(j_1-j_2)}{j_2^{1-2s}}+O\left( \frac{(j_1-j_2)^2}{j_2^{2-2s}}\right) . \end{aligned}$$
(5.13)

In view of (5.11), (5.12) and (5.13), we have

$$\begin{aligned} \left( {j_1^2+m}\right) ^s-\left( {j_2^2+m}\right) ^s\le \frac{4s(j_1-j_2)}{j_2^{1-2s}}\le \frac{4s}{N^{8r^3-2}}\le \frac{1}{N^{3r^3}}. \end{aligned}$$
(5.14)

From (5.10) and \(j_1-j_2\ne 0\) we know that \({\tilde{k}}\ne 0\). Consider the resonant set

$$\begin{aligned} \breve{{\mathcal {R}}}_{{\tilde{k}}}=\left\{ m\in {\mathcal {W}}:~~~~~|\langle {\tilde{k}}, {\tilde{\Omega }}^{(N)}\rangle |< \frac{2}{N^{3r^3}}, 0<|{\tilde{k}}|\le r\right\} . \end{aligned}$$
(5.15)

By the same method as Case 1 we obtain

$$\begin{aligned} |\breve{{\mathcal {R}}}_{{\tilde{k}}}|\le & {} \frac{1}{N^{\frac{3r^3}{r-1}-2r^2-3}}. \end{aligned}$$
(5.16)

Setting

$$\begin{aligned} \tilde{R}_3=\bigcup _{|{\tilde{k}}|\le r} \breve{{\mathcal {R}}}_{{\tilde{k}}}, \end{aligned}$$
(5.17)

then we have

$$\begin{aligned} |\widetilde{R}_3 |\le & {} \sum _{|{\tilde{k}}|\le r}| \breve{{\mathcal {R}}}_{{\tilde{k}}}|\le \sum _{|{\tilde{k}}|\le r} \frac{1}{N^{\frac{r^3+2r^2-3r+3}{r-1}}}\nonumber \\\le & {} \frac{1}{N^{\frac{r^3+2r^2-3r+3}{r-1}}}\cdot (2N+1)^{r} \nonumber \\\le & {} \frac{1}{4N}, \end{aligned}$$
(5.18)

where the last inequality is based on r is large enough.

For \(m\in {\mathcal {W}}-\widetilde{R}_3\) and in view of (5.14), one has

$$\begin{aligned}&|\langle {\tilde{k}}, {\tilde{\Omega }}^{(N)}\rangle +{\Omega }_{j_1}-{\Omega }_{j_2}| \nonumber \\&\quad \ge |\langle {\tilde{k}}, {\tilde{\Omega }}^{(N)}\rangle |-|{\Omega }_{j_1}-{\Omega }_{j_2}| \end{aligned}$$
(5.19)
$$\begin{aligned}&\quad \ge \frac{2}{N^{3r^3}}-\frac{1}{N^{3r^3}}\nonumber \\&\quad =\frac{1}{N^{3r^3}}. \end{aligned}$$
(5.20)

Subcase3.2 When \(j_2\le N^{\frac{8r^3}{1-2s}},\) from (5.11), we have

$$\begin{aligned} j_1\le N^{\frac{8r^3}{1-2s}}+rN<2N^{\frac{8r^3}{1-2s}}. \end{aligned}$$

Denote the resonant set

$$\begin{aligned} {\mathcal {R}}_{{\tilde{k}}j_1j_2}=\Big \{m\in {\mathcal {W}}:\left| \langle {\tilde{k}},{\tilde{\Omega }}\rangle +{\Omega }_{j_1}-{\Omega }_{j_2}\right|< \frac{1}{N^{\frac{17 r^{6}}{1-2s}}}, ~~\hbox {for}~~ 0<|{\tilde{k}}|\le r\Big \}. \end{aligned}$$

Setting \(\langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle +{\Omega }_{j_1}-{\Omega }_{j_2}\) in place of \(\langle {\tilde{k}},{\tilde{\Omega }}^{(N)}\rangle \) and \(\widetilde{N}=2N^{\frac{8r^3}{1-2s}}\) in place of N, then according to Case 1, we have

$$\begin{aligned} |{\mathcal {R}}_{{\tilde{k}}j_1j_2}|\le & {} \frac{1}{N^{\frac{17r^6}{(r+1)(1-2s)}}} \left( 4N^{\frac{8r^3}{1-2s}}+1\right) ^{2(r+2)^2+3/2}. \end{aligned}$$

Setting

$$\begin{aligned} \widetilde{R}_{4}=\bigcup \limits _{|{\tilde{k}}|\le r}\;\;\bigcup \limits _{j_1\le 2N^{\frac{8r^3}{1-2s}}}\;\;\bigcup \limits _{j_2\le N^{\frac{8r^3}{1-2s}}}{\mathcal {R}}_{{\tilde{k}}j_1j_2}, \end{aligned}$$

then we have

$$\begin{aligned} |\widetilde{R}_{4}|\le & {} \sum _{|{\tilde{k}}|\le r}\;\;\sum _{j_1\le 2N^{\frac{8r^3}{1-2s}}}\;\; \sum _{j_2\le N^{\frac{8r^3}{1-2s}}}|{\mathcal {R}}_{\tilde{k}j_1j_2}|\nonumber \\= & {} \sum _{|{\tilde{k}}|\le r}\;\;\sum _{j_1\le 2N^{\frac{8r^3}{1-2s}}}\;\; \sum _{j_2\le N^{\frac{8r^3}{1-2s}}}\frac{1}{N^{\frac{17r^6}{(r+1)(1-2s)}}}\left( 4N^{\frac{8r^3}{1-2s}}+1\right) ^{2(r+2)^2+3} \nonumber \\\le & {} \frac{1}{N^{\frac{17r^6}{(r+1)(1-2s)}}} \left( 4N^{\frac{8r^3}{1-2s}}+1\right) ^{2(r+2)^2+3} (2N+1)^{r}\left( 2N^{\frac{8r^3}{1-2s}}\right) ^{2}\nonumber \\< & {} \frac{1}{4N}\quad (r \text { is large enough}). \end{aligned}$$
(5.21)

In view of (5.6), (5.8), (5.18) and (5.21), we obtain

$$\begin{aligned} \left| {\mathcal {R}}\right|\le & {} |\widetilde{R}_{1}|+|\widetilde{R}_{2}|+|\widetilde{R}_{3}|+|\widetilde{R}_{4}| <\frac{1}{N}. \end{aligned}$$

Let \({\mathcal {F}}={\mathcal {W}}-{\mathcal {R}}\), then the proposition is proved. \(\square \)

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Mi, L., Cong, H. Almost Global Existence for the Fractional Schrödinger Equations. J Dyn Diff Equat 32, 1553–1575 (2020). https://doi.org/10.1007/s10884-019-09783-w

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