Abstract
We study the time of existence of the solutions of the following nonlinear Schrödinger equation (NLS)
on the finite x-interval \([0,\pi ]\) with Dirichlet boundary conditions
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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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
and
are the frequencies.
Lemma 5.1
For any \(K\le r\), consider K indexes \(j_{1}<\cdots <j_{K}\le N\); consider the determinant
One has
where C(s, r) is a positive constant depending on s and r.
Proof
First, for any \(n\ge 1\) with \(n\in {\mathbb {N}}\), we have
Substituting (5.3) in the left-hand side of (5.1), we get the determinant to be computed. More precisely,
where \(x_{j}:=\left( j^2+m\right) ^{-1}\). The last determinant is a Vandermonde determinant whose value is given by
Using the fact that all the eigenvalues are different, one gets
where using N large enough, and \(C_1(s,r)\) and C(s, r) 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
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
where \({\Omega }=({\Omega }_{j_{1}},{\Omega }_{j_{2}},\ldots ,{\Omega }_{j_{K}})\) is the frequency vector and C(s, r) 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 \(m\in {\mathcal {F}}\),
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
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
By combining Lemmas 5.3 and 5.4, we can get
where assuming N is large enough. Setting
then we have
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
Due to \({\Omega }_{j_1}=\left( {j_1^2+m}\right) ^{s},\) one has
if
Then the resonant set \({\mathcal {R}}_{{\tilde{k}}j_1}\) is empty. So it is sufficient to consider
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
where \(C_1(s,r)\) is a positive constant depending on s and r. Setting
then we have
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
we obtain that
Subcase 3.1 For \(j_2>N^{\frac{8r^3}{1-2s}},\) we have
By Taylor’s formula, one has
In view of (5.11), (5.12) and (5.13), we have
From (5.10) and \(j_1-j_2\ne 0\) we know that \({\tilde{k}}\ne 0\). Consider the resonant set
By the same method as Case 1 we obtain
Setting
then we have
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
Subcase3.2 When \(j_2\le N^{\frac{8r^3}{1-2s}},\) from (5.11), we have
Denote the resonant set
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
Setting
then we have
In view of (5.6), (5.8), (5.18) and (5.21), we obtain
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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DOI: https://doi.org/10.1007/s10884-019-09783-w