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

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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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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Mathematics Subject Classification

  • 35L05
  • 34K35

Keywords

  • Exponential stability
  • Dissipative Schrödinger equation
  • Star-shaped network