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Exponential stability for the nonlinear Schrödinger equation on a star-shaped network

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Abstract

In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network \(\mathcal {R}\), where the damping is localized on one branch at the infinity and the initial data are assumed to be in \(L^{2}(\mathcal {R})\). We use the fixed point argument and Strichartz estimates on a star-shaped network to obtain results of local and global well-posedness. The proof of the exponential decay is based on smoothing properties for Schrödinger equation, on the unique continuation and on the semigroup properties.

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References

  1. Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Variational properties and orbital stability of standing waves for NLS equation on a star graph. J. Differ. Equ. 257, 3738–3777 (2014)

    Article  MathSciNet  Google Scholar 

  2. Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. H. Poincaré Anal. Nonlin. 31, 1289–1310 (2014)

    Article  MathSciNet  Google Scholar 

  3. Adami, R., Serra, E., Tilli, P.: Negative energy ground states for the \(L^{2}\)-critical NLSE on metric graphs. Commun. Math. Phys. 352, 387–406 (2017)

    Article  Google Scholar 

  4. Ali Mehmeti, F., Ammari, K., Nicaise, S.: Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks. Port. Math. 72, 309–355 (2015)

    Article  MathSciNet  Google Scholar 

  5. Ali Mehmeti, F., Ammari, K., Nicaise, S.: Dispersive effects for the Schrödinger equation on the tadpole graph. J. Math. Anal. Appl. 448, 262–280 (2017)

    Article  MathSciNet  Google Scholar 

  6. Banica, V., Ignat, L.I.: Dispersion for the Schrödinger equation on networks. J. Math. Phys. 52, 083703 (2011)

    Article  MathSciNet  Google Scholar 

  7. Cacciapuoti, C., Finco, D., Noja, D.: Ground state and orbital stability for the NLS equation on a general starlike graph with potentials. Nonlinearity 30, 3271–3303 (2017)

    Article  MathSciNet  Google Scholar 

  8. Cazenave, T.: Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10 (2003)

  9. Dovetta, S.: Existence of infinitely many stationary solutions of the \(L^{2}\)-subcritical and critical NLSE on compact metric graphs. J. Differ. Equ. 264, 4806–4821 (2018)

    Article  MathSciNet  Google Scholar 

  10. Dovetta, S., Tentarelli, L.: \(L^{2}\)-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features. Calc. Var. Partial Differ. Equ. 58(3), Paper No. 108, 26 pp (2019)

  11. Escauriaza, L., Kenig, C.E., Ponce, G., Vega, L.: On uniqueness properties of solutions of Schrödinger equations. Commun. Partial Differ. Equ. 31, 1811–1823 (2006)

    Article  Google Scholar 

  12. Grecu, A., Ignat, L. I.: The Schrödinger equation on a star-shaped graph under general coupling conditions. J. Phys. A 52(3):035202, 26 (2019)

  13. Ignat, L.I.: Strichartz estimates for the Schrödinger equation on a tree and applications. SIAM. J. Math. Anal 42, 2041–2057 (2010)

    Article  MathSciNet  Google Scholar 

  14. Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 46, 113–129 (1987)

    MATH  Google Scholar 

  15. Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)

    Article  MathSciNet  Google Scholar 

  16. Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations. Springer, New York (2009)

    MATH  Google Scholar 

  17. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars (1969)

    MATH  Google Scholar 

  18. Natali, F.: Exponential stabilization for the nonlinear Schrödinger equation with localized damping. J. Dyn. Control Syst. 21, 461–474 (2015)

    Article  MathSciNet  Google Scholar 

  19. Natali, F.: A note on the exponential decay for the nonlinear Schrödinger equation. Osaka J. Math. 53, 717–729 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Simon, J.: Compact sets in \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the referees for their attentive reading of the manuscript and the questions they asked.

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Correspondence to Ahmed Bchatnia.

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Appendix

Appendix

In this appendix, we present some useful results used in the paper.

First, we recall the \(H^{\frac{1}{2}}\) smoothing effect for the linear Schrödinger equation.

Theorem 4.5

[16, Theorem 4.3] Let v be a solution of the following linear Schrödinger equation:

$$\begin{aligned} \left\{ \begin{array}{lll} i \partial _t v + \partial ^2_x v = 0, &{}\quad (0,T)\times \mathbb {R},\\ v(x,0)=v_{0}, &{}\quad x\in \mathbb {R}. \end{array}\right. \end{aligned}$$
(4.33)

Then, there exists \(C>0\) such that for every \(v_{0}\in L^{2}(\mathbb {R})\),

$$\begin{aligned} \underset{x\in \mathbb {R}}{\sup }\left( \int \limits _{\mathbb {R}}\left. \vert D_{x}^{\frac{1}{2}}e^{it\partial ^2_x}v_{0}(x)\right. \vert ^{2} \mathrm{d}x \mathrm{d}t \right) ^{{1}/{2}}\le C\; \Vert v_{0}\Vert _{L^{2}}. \end{aligned}$$

The following result is a consequence of the previous theorem:

Corollary 4.6

[16, Corollary 4.2] Given R a positive real number, then there exists \(C>0\) such that

$$\begin{aligned} \left( \int \limits _{\mathbb {R}}\int \limits _{\vert x\vert \le R}\left. \vert D_{x}^{\frac{1}{2}}e^{it\partial ^2_x}v_{0}(x)\right. \vert ^{2} \mathrm{d}x \mathrm{d}t \right) ^{{1}/{2}}\le C\,R^{{1}/{2}} \Vert v_{0}\Vert _{L^{2}}, \end{aligned}$$

for every solution v of (4.33) with initial data \(v_{0}\in L^{2}(\mathbb {R})\).

The next result describes the \(C^{\infty }\) smoothing effect of nonlinear Schrödinger equation.

Theorem 4.7

[8, Theorem 5.7.3] Consider \(T>0\), \(\lambda =\pm 1\) and \(\alpha \) an odd positive number. Let v be the global solution in \( C\left( \left[ 0,T \right] ;L^{2}(\mathbb {R}) \right) \cap L^2\big ((0,T);L^{\infty }(\mathbb {R})\big )\) of

$$\begin{aligned} \left\{ \begin{array}{lll} i \partial _t v + \partial ^2_x v + \lambda |v|^{\alpha -1}v = 0, &{}\quad (0,T)\times \mathbb {R},\\ v(x,0)=v_{0}, &{}\quad x\in \mathbb {R}. \end{array}\right. \end{aligned}$$
(4.34)

Then, \(v\in C^{\infty }([0,T]\times \mathbb {R})\) for all \(v_{0}\in L^{2}(\mathbb {R})\) with a compact support.

We recall also the following unique continuation theorem for regular solutions of the nonlinear equation in (4.34). This result is more general in the sense that it deserves for nonlinear Schrödinger equation in the domain \((x, t)\in \mathbb {R}^{n}\times [0,T ], n\ge 1\), with a general nonlinearity \(F(v, \bar{v})\). In such case, we must consider \(k \in \mathbb {N}\) satisfying \(k>\frac{n}{2}+1\).

Let us define the weighted Sobolev space \( H^{1}\left( e^{\beta \vert x\vert ^{\rho }}\mathrm{d}x\right) \), as

$$\begin{aligned} H^{1}\left( e^{\beta \vert x\vert ^{\rho }}\mathrm{d}x\right) =\left\{ g;\int \limits _{\mathbb {R}}\left| g(x)\right| ^{2} e^{\beta \vert x\vert ^{\rho }}\mathrm{d}x+\int \limits _{\mathbb {R}}\left| g'(x)\right| ^{2} e^{\beta \vert x\vert ^{\rho }}\mathrm{d}x <\infty \right\} . \end{aligned}$$

Theorem 4.8

[11, Theorem 2.1] Let \(w\in C\left( [0,T];H^{k}(\mathbb {R}) \right) \), \(k\in \mathbb {N}\), \(k>{3}/{2}\) be a strong solution of the equation in (4.34) in the domain \((t,x)\in [0,T]\times \mathbb {R}\). If there is \(t_{1},t_{2}\in [0,T], t_{1}\ne t_{2}\), \(\rho >2\) and \(\beta >0\) such that

$$\begin{aligned} w(t_{1},\cdot ), w(t_{2},\cdot )\in H^{1}\left( e^{\beta \vert x\vert ^{\rho }}\mathrm{d}x\right) , \end{aligned}$$

then \(w\equiv 0\).

Now, notice the following lemmas.

Lemma 4.9

(Lions’s Lemma [17, Lemma 1.3]) Let \(\omega \) be an open bounded subset of \(\mathbb {R}\times \mathbb {R}\). Consider \(\lbrace f_{n}\rbrace _{n\in \mathbb {N}}\) a sequence in \(L^{q}(\omega )\), \(1<q<\infty \), satisfying \(\Vert f_{n}\Vert _{L^{q}(\omega )}\le C\) and \(f_{n}\longrightarrow f\) a.e. in \(\omega \). Thus, \(f_{n}\rightharpoonup f\) in \(L^{q}(\omega )\), as \(n\longrightarrow +\infty \).

Lemma 4.10

(Aubin–Lions’s Lemma [20, Corollary 4]) Let \(X_{0}, X\), and \(X_{1}\) be three Banach spaces with \(X_{0}\subset X\subset X_{1}\). Suppose that \(X_{0}\) is compactly embedded in X and that X is continuously embedded in \(X_{1}\). Suppose also that \(X_{0}\) and \(X_{1}\) are reflexive spaces. For \(1<p,q<\infty \), let \(W=\left\{ u\in L^{p}((0,T);X_{0}), \frac{\mathrm{d}u}{\mathrm{d}t}\right. \left. \in L^{q}((0,T);X_{1})\right\} \). Then, the embedding of W into \(L^{p}((0,T);X)\) is compact.

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Ammari, K., Bchatnia, A. & Mehenaoui, N. Exponential stability for the nonlinear Schrödinger equation on a star-shaped network. Z. Angew. Math. Phys. 72, 35 (2021). https://doi.org/10.1007/s00033-020-01458-7

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