Failure of scattering to standing waves for a Schrödinger equation with long-range nonlinearity on star graph

Abstract

We consider the Schrödinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, $\delta$, or $\delta '$ boundary condition, we show that the non-trivial global solution does not scatter to standing waves. Our proof is based on the argument by Murphy and Nakanishi (Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations), who treated the long-range nonlinear Schrödinger equation with a general potential in the Euclidean space, in order to consider general boundary conditions.

A Weak solution

In the appendix, we discuss that an $L^2$-solution is a weak solution. We say that a pair (q, r) is admissible if $2/q = 1/2-1/r$ and $2\le q,r \le \infty$. We define the $L^2$-solution (or the Strichartz class solution) as follows.

Definition 1

Let I be a time interval containing 0. We say that u is an $L^2$-solution (or a Strichartz class solution) to (NLS) on I if $u \in C(I:L^2({\mathcal {G}})) \cap \bigcap _{(q,r):\text {admissible}} L^q(I:L^r({\mathcal {G}}))$ and u satisfies

$$\begin{aligned} u(t)=e^{it\varDelta _{M}} u_0 + i \lambda \int _{0}^{t} e^{i(t-s)\varDelta _{M}} (|u|^p u)(s) ds \end{aligned}$$

for all $t \in I$ and almost all $x \in {\mathcal {G}}$.

A unique $L^2$-solution to (NLS) exists if $u_0 \in L^2({\mathcal {G}})$ by Grecu and Ignat [10]. They also showed $L^2$-conservation law, i.e., $\left\| u(t) \right\| _{L^2({\mathcal {G}})}=\left\| u_0 \right\| _{L^2({\mathcal {G}})}$, and thus the solution is global. The $L^2$-solution is a weak solution in the following sense.

Lemma 11

If u is an $L^2$-solution and $\varphi \in \{f \in H^2((0,\infty )) \cap C([0,\infty )) : f(0)=0\}$, we have

$$\begin{aligned}&i \left\langle u(\tau ) , \varphi \right\rangle _{j} - i \left\langle u_0 , \varphi \right\rangle _{j} + \int _{0}^{\tau } \left\langle u , \varphi '' \right\rangle _{j} dt + \int _{0}^{\tau } u_j(t,0+) \overline{\partial _x \varphi (0+)} dt\nonumber \\&\qquad =-\lambda \int _{0}^{\tau } \left\langle |u|^p u , \varphi \right\rangle _{j} dt. \end{aligned}$$

(A.1)

Proof

Let $H=-\varDelta _{M}$. We define $u_\varepsilon :=(I+\varepsilon ^2 H )^{-1}u$, where u is the $L^2$-solution. Then, from the argument in [10], it holds that $u_{\varepsilon } \in C(I:{\mathscr {D}}(H)) \cap W^{1,1,}(I:L^2({\mathcal {G}}))$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} i \partial _t u_{\varepsilon } = Hu_{\varepsilon } - (I+\varepsilon ^2 H)^{-1} F(u), \\ u_{\varepsilon }(0)=(I+\varepsilon ^2 H )^{-1}u_{0}, \end{array}\right. } \end{aligned}$$

where $F(u)=\lambda |u|^p u$ and $(I+\varepsilon ^2 H )^{-1}F(u) \in L^1(I:{\mathscr {D}}(H))$. Multiplying the equation by the complex conjugate of $\varphi \in \{f \in H^2((0,\infty )) \cap C([0,\infty )) : f(0)=0\}$ and integrating on an edge $e_j$, it follows from the integration by parts that

$$\begin{aligned} i \left\langle \partial _t u_{\varepsilon } , \varphi \right\rangle _{j} =- \left\langle u_{\varepsilon } , \varphi '' \right\rangle _{j} +u_{\varepsilon j}(t,0+) \overline{\partial _x \varphi (0+)} - \left\langle (I+\varepsilon ^2 H)^{-1} F(u) , \varphi \right\rangle _{j}. \end{aligned}$$

Integrating this on $[0,\tau )$ for $\tau \in I$, we obtain

$$\begin{aligned} i \left\langle u_{\varepsilon }(\tau ) , \varphi \right\rangle _{j} - \left\langle u_{\varepsilon }(0) , \varphi \right\rangle _{j} =&-\int _{0}^{\tau } \left\langle u_{\varepsilon } , \varphi '' \right\rangle _{j}dt +\int _{0}^{\tau } u_{\varepsilon j}(t,0+) \overline{\partial _x \varphi (0+)} dt \\&- \int _{0}^{\tau } \left\langle (I+\varepsilon ^2 H)^{-1} F(u) , \varphi \right\rangle _{j} dt. \end{aligned}$$

By the argument in [10], taking $\varepsilon \rightarrow 0$, we have (A.1). $\square$

From this lemma, we get the following lemma, which is one of keys to show Theorem 1 (see Sect. 2.3).

Lemma 12

Let u be an $L^2$-solution and $w=e^{it\varDelta _{D}}\varphi$, where $\varphi \in \{f \in H^2({\mathcal {G}}) : f(0)=0\}$. Then we have

$$\begin{aligned} i \left\langle u(\tau ) , w(\tau ) \right\rangle _{j} - i \left\langle u_0 , \varphi \right\rangle _{j}+ \int _{0}^{\tau } u_j(t,0+) \overline{\partial _x w (t,0+)} dt =-\lambda \int _{0}^{\tau } \left\langle |u|^p u , w \right\rangle _{j} dt. \end{aligned}$$

Proof

Multiplying the equation by the complex conjugate of w and integrating on an edge $e_j$, it follows from the argument in the proof of Lemma 11 that

$$\begin{aligned} i \left\langle \partial _t u_{\varepsilon } , w \right\rangle _{j} = -\left\langle u_{\varepsilon } , w'' \right\rangle _{j} +u_{\varepsilon j}(t,0+) \overline{\partial _x w (t,0+)} - \left\langle (I+\varepsilon ^2 H)^{-1} F(u) , w \right\rangle _{j}, \end{aligned}$$

where $u_{\varepsilon }$ is as in the proof and note that $w_j (t) \in \{f \in H^2((0,\infty )) \cap C([0,\infty )) : f(0)=0\}$. Since w is a solution of

$$\begin{aligned} i\partial _t w + w''=0 \text { and } w(t,0)=0, \end{aligned}$$

we obtain

$$\begin{aligned} i \partial _t \left\langle u_{\varepsilon } , w \right\rangle _{j} = u_{\varepsilon j}(t,0+) \overline{\partial _x w (t,0+)} - \left\langle (I+\varepsilon ^2 H)^{-1} F(u) , w \right\rangle _{j}. \end{aligned}$$

Integrating this on $[0,\tau )$ for $\tau \in I$ and taking $\varepsilon \rightarrow 0$, this completes the proof. $\square$