Dynamics of blow-up solutions for the Schrödinger–Choquard equation

Abstract

In this paper, we study the dynamics of blow-up solutions for the nonlinear Schrödinger–Choquard equation

$$i\psi_{t}+\Delta \psi =\lambda_{1}\vert \psi \vert ^{p_{1}}\psi +\lambda_{2}\bigl(I _{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr)\vert \psi \vert ^{p_{2}-2}\psi. $$

We first show existence of blow-up solutions and obtain a sharp threshold mass of global existence and blow-up for this equation with $\lambda_{1}>0$, $\lambda_{2}<0$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. Then we obtain some dynamical properties of blow-up solutions by the corresponding ground state of this equation with $\lambda_{1}=0$.

1 Introduction

In this paper, we will investigate the blow-up solutions of the nonlinear Schrödinger–Choquard equation

$$ \textstyle\begin{cases} i\psi_{t}+\Delta \psi =\lambda_{1}\vert \psi \vert ^{p_{1}}\psi +\lambda_{2}(I _{\alpha }\ast \vert \psi \vert ^{p_{2}}) \vert \psi \vert ^{p_{2}-2}\psi, \\ \psi (0,x) = \psi_{0} (x), \end{cases} $$

(1.1)

where $\psi (t,x):[0,T^{*})\times \mathbb{R}^{N} \rightarrow \mathbb{C}$ is a complex valued function and $0< T^{*}\leq \infty$, $N\geq 3$, $\psi_{0} \in H^{1}$, $0< p_{1}< \frac{4}{N-2}$, $1+\frac{ \alpha }{N}< p_{2}<1+\frac{2+\alpha }{N-2}$, $\lambda_{1},\lambda_{2} \in \mathbb{R}$, $I_{\alpha }:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is the Riesz potential defined by

$$I_{\alpha }(x)=\frac{\Gamma (\frac{N-\alpha }{2})}{\Gamma (\frac{ \alpha }{2})\pi^{N/2}2^{\alpha }\vert x\vert ^{N-\alpha }}, $$

with $\max \{0,N-4\}<\alpha <N$ and Γ is the Gamma function.

Our main motivation for studying Eq. (1.1) is the loss of scaling invariance for this equation. When $p_{2}>0$, there exists a scaling transform for the nonlinear Choquard equation,

$$ i\psi_{t}+\Delta \psi =\lambda_{2} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr)\vert \psi \vert ^{p_{2}-2}\psi, $$

(1.2)

which keeps it invariant. More precisely, the map

$$ \psi (t,x)\mapsto \lambda^{-\frac{\alpha +2}{2p_{2}-2}}\psi \biggl( \frac{t}{ \lambda^{2}},\frac{x}{\lambda } \biggr) $$

(1.3)

maps a solution to (1.2) to another solution to (1.2). When $p_{2}=1+\frac{2+\alpha }{N}$, the scaling transform (1.3) keeps the mass invariant. Thus, the nonlinearity $(I_{\alpha }\ast \vert \psi \vert ^{p_{2}})\vert \psi \vert ^{p_{2}-2}\psi$ is called $L^{2}$-critical.

When $\lambda_{1}=0$ and $p_{2}=2$, Eq. (1.1) simplifies to the Hartree equation. The Cauchy problem of (1.1) has been extensively investigated in [1–16]. The local well-posedness and global existence of (1.1) have been studied in [1]. Chen and Guo [3] studied the instability of standing waves. In the $L^{2}$-critical case, Miao et al. [10] studied the dynamical properties of the blow-up solutions. The soliton dynamics has been studied in [11].

When $\lambda_{1}=0$, $0<\alpha <N$ and $1+\frac{\alpha }{N}< p_{2}<\frac{N+ \alpha }{N-2}$, under the assumption that the local well-posedness holds for (1.1), Chen and Guo [3] derived the existence of blow-up solutions and the instability of standing waves. When $0<\alpha <N$ and $1+\frac{\alpha }{N}< p_{2}<1+\frac{2+\alpha }{N}$, Squassina et al. in [17] studied the soliton dynamics of (1.1) under the assumption that the solution ψ of (1.1) is in $C([0,\infty),H^{2})\cap C^{1}((0,\infty),L^{2})$. In [18], Feng and Yuan systematically studied the Cauchy problem (1.1) for general $\max \{0,N-4\}<\alpha <N$ and $2\leq p_{2}<\frac{N+\alpha }{N-2}$. More precisely, they studied the local well-posedness, global existence, the existence of blow-up solutions and the dynamics of blow-up solutions. The sharp threshold of global existence and blow-up, the instability of standing wave of (1.1) with $\lambda_{1}=0$ and a harmonic potential have been investigated in [19].

However, in the above papers, the scale invariance plays an important role in the study of the dynamics of blow-up solutions to (1.2); see [7, 10, 12, 14, 18, 20, 21]. Because there exists no scale invariance for (1.1), the study of blow-up solutions to (1.1) is a very interesting problem. On the other hand, as far as we know, the existence of blow-up solutions to (1.1) with $\lambda_{1}>0$, $\lambda_{2}<0$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$ has not been obtained yet. Hence, in this paper, we first show the existence of blow-up solutions and obtain the sharp threshold mass $\Vert u\Vert _{L^{2}}$ of global existence and blow-up for (1.1), where u is a ground state solution of the elliptic equation

$$ -\Delta u+ u-\bigl(I_{\alpha }\ast \vert u\vert ^{p}\bigr)\vert u\vert ^{p-2}u=0. $$

(1.4)

Then, for overcoming the difficulty of the loss of scale invariance, we apply the ground state solution u of (1.4) to describe the dynamical properties of blow-up solutions to (1.1), including $L^{2}$-concentration, limiting profile and blow-up rates.

This paper is organized as follows: in Sect. 2, we recall some preliminaries. In Sect. 3, we firstly show the existence of blow-up solutions to (1.1) with $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$, and then obtain the sharp threshold mass $\Vert u\Vert _{L^{2}}$ of global existence and blow-up. In Sect. 4, we will consider some dynamical properties of blow-up solutions to (1.1) with $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. Section 5 is a concluding section.

Notation

In this paper, we use the following notations. We always denote u the ground state solution of (1.4). $\Sigma:=\{\psi \in H^{1}, x\psi \in L^{2}\}$ is the energy space equipped with the norm $\Vert \psi \Vert _{\Sigma }:=\Vert \psi \Vert _{H^{1}}+\Vert x\psi \Vert _{L^{2}}$.

2 Preliminaries

In order to study the blow-up solutions to (1.1), we firstly make the following assumption about the local well-posedness of (1.1).

Assumption 1

Let $\psi_{0} \in H^{1}$, $N\geq 3$, $0< p_{1}<\frac{4}{N-2}$ and $1+\frac{\alpha }{N}< p_{2}<1+\frac{2+ \alpha }{N-2}$. Then there exist $T^{*}>0$ and a unique maximal solution $\psi \in C([0,T^{*}),H^{1})$. In addition, if $T^{\ast }< \infty$, then $\Vert \psi (t)\Vert _{H^{1}}\rightarrow \infty$ as $t\uparrow T^{ \ast }$. Moreover, the solution $\psi (t)$ satisfies

$$\begin{aligned}& \bigl\Vert \psi (t)\bigr\Vert _{L^{2}}=\Vert \psi_{0}\Vert _{L^{2}}, \end{aligned}$$

(2.1)

$$\begin{aligned}& E\bigl(\psi (t)\bigr)=E(\psi_{0} ), \end{aligned}$$

(2.2)

for all $0\leq t< T^{*}$, where $E(\psi (t))$ is defined by

$$\begin{aligned} E\bigl(\psi (t)\bigr):={} &\frac{1}{2} \int_{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t,x)\bigr\vert ^{2}\,dx+\frac{\lambda_{1}}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{p _{1}+2}\,dx \\ &{}+\frac{\lambda_{2}}{2p_{2}} \int_{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x)\bigl\vert \psi (t,x)\bigr\vert ^{p_{2}}\,dx. \end{aligned}$$

(2.3)

When $0< p_{1}<\frac{4}{N-2}$ and $2\leq p_{2}<1+\frac{2+\alpha }{N-2}$, this assumption can easily be proved by the Strichartz estimates and a fixed point argument; see [1, 18].

By the same argument as that in [1], one can easily derive the following lemma.

Lemma 2.1

([1])

Let $\psi_{0} \in \Sigma:=\{u\in H^{1}, xu\in L^{2} \}$. Assume that the solution $\psi (t)$ to (1.1) exists on the interval $[0,T^{*})$. Then $\psi (t) \in \Sigma$ for all $t\in [0,T ^{*})$. Moreover, let $J(t)=\int_{\mathbb{R}^{N}} \vert x\psi (t,x)\vert ^{2}\,dx$, then

$$ J'(t)=-4\operatorname{Im}\int_{\mathbb{R}^{N}} \psi (t,x)x\cdot \nabla \bar{\psi }(t,x)\,dx, $$

(2.4)

and

$$\begin{aligned} J''(t)= {}&8 \int_{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t,x)\bigr\vert ^{2}\,dx+\frac{4N \lambda_{1}p_{1}}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{p_{1}+2}\,dx \\ &{}+\lambda_{2}\frac{4p_{2}N-4N-4\alpha }{p_{2}} \int_{\mathbb{R}^{N}} \bigl(I _{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x)\bigl\vert \psi (t,x)\bigr\vert ^{p_{2}}\,dx. \end{aligned}$$

(2.5)

As a direct result of this lemma, we have the following lemma.

Lemma 2.2

If the solution $\psi (t)$ to (1.1) with $\psi_{0} \in \Sigma$ blows up at the finite time $T^{*}$, then there exists $C>0$ such that for all $t\in [0,T^{*})$

$$\int_{\mathbb{R}^{N}} \vert x\vert ^{2} \bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx\leq C. $$

Next, we summarize some results about the ground state of (1.4), which is very important in the study of blow-up solutions to (1.1).

Lemma 2.3

([17, 22])

Let $\alpha \in (0,N)$ and $1+\frac{\alpha }{N}< p<1+\frac{2+ \alpha }{N-2}$. Then (1.4) admits a ground state solution u in $H^{1}$. Moreover, let $u_{1}$ and $u_{2}$ be two any ground state solutions of (1.4), then $\Vert u_{1}\Vert _{L^{2}}=\Vert u_{2}\Vert _{L^{2}}$.

Finally, we recall a useful result which gives the best constant in a Gagliardo–Nirenberg type inequality; see [18].

Lemma 2.4

The best constant in the Gagliardo–Nirenberg type inequality

$$ \int_{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p}\bigr)\vert \psi \vert ^{p}\,dx \leq C_{\alpha,p} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla \psi \vert ^{2}\,dx \biggr) ^{\frac{Np-N-\alpha }{2}} \biggl( \int_{\mathbb{R}^{N}} \vert \psi \vert ^{2}\,dx \biggr) ^{\frac{N+\alpha -Np+2p}{2}} $$

(2.6)

is

$$C_{\alpha,p}=\frac{2p}{2p-Np+N+\alpha } \biggl( \frac{2p-Np+N+\alpha }{Np-N- \alpha } \biggr) ^{\frac{Np-N-\alpha }{2}} \Vert u\Vert _{L^{2}}^{2-2p}. $$

In particular, in the $L^{2}$-critical case, i.e., $p=1+\frac{2+ \alpha }{N}$, $C_{\alpha,p}=p\Vert u\Vert _{L^{2}}^{2-2p}$.

3 The sharp threshold mass of global existence and blow-up

From the local well-posedness of the nonlinear Schrödinger–Choquard equation, for small initial data $\psi_{0}$, the solution $\psi (t)$ to (1.1) exists globally, and the solution $\psi (t)$ may blow up for some large initial data. Therefore, whether there are some sharp thresholds of global existence and blow-up for (1.1) is a very interesting problem. In particular, the sharp thresholds of global existence and blow-up for nonlinear Schrödinger equations are pursued strongly (see [1, 2, 19, 23–25] and the references therein).

In the following, applying the inequality (2.6) and a scaling argument, we derive the existence of blow-up solutions to (1.1) and a sharp threshold of global existence and blow-up.

Theorem 3.1

Let $\psi_{0}\in H^{1}$, $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. Then we have:

• (i) If $\Vert \psi_{0}\Vert _{L^{2}}<\Vert u\Vert _{L^{2}}$, then the solution $\psi (t)$ to (1.1) exists globally.

• (ii) Let $\psi_{0}=c\rho^{\frac{N}{2}} u(\rho x)$ and $\vert x\vert \psi_{0} \in L^{2}$, where $\vert c\vert \geq 1$, and $\rho >0$ and satisfies

  $$ \frac{2\vert c\vert ^{p_{1}}\Vert u\Vert ^{p_{1}+2}_{L^{p_{1}+2}}}{(p_{1}+2)(\vert c\vert ^{2p _{2}-2}-1)\Vert \nabla u\Vert _{L^{2}}^{2}}< \rho^{2-\frac{N}{2}p_{1}}. $$

  (3.1)

Then the solution $\psi (t)$ to (1.1) blows up in finite time.

Remark

We see from Theorem 1.2 in [18] that the critical value about the initial data for global existence of (1.1) with $\lambda_{1}=0$ and (1.1) is the same.

Proof

(i) Firstly, by (2.3) and (2.6), we have

$$\begin{aligned} E(\psi_{0})={}&E\bigl(\psi (t)\bigr) \\ = {}&\frac{1}{2} \int_{\mathbb{R}^{N}} \bigl\vert \nabla\psi (t,x)\bigr\vert ^{2}\,dx-\frac{1}{2p_{2}} \int_{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) (t,x)\bigl\vert \psi (t,x)\bigr\vert ^{p_{2}}\,dx \\ &{}+\frac{1}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{p_{1}+2}\,dx \\ \geq {}& \biggl( \frac{1}{2}-\frac{\Vert \psi_{0}\Vert _{L^{2}}^{2p_{2}-2}}{2\Vert u\Vert _{L^{2}}^{2p_{2}-2}} \biggr) \bigl\Vert \nabla \psi (t)\bigr\Vert _{L^{2}}^{2}. \end{aligned}$$

It follows from $\Vert \psi_{0}\Vert _{L^{2}}<\Vert u\Vert _{L^{2}}$ and $E(\psi_{0})=E( \psi (t))$ that there exists a constant C such that $\Vert \nabla\psi (t)\Vert _{L^{2}}\leq C$ for all $t>0$. Therefore, the solution $\psi (t)$ to (1.1) exists globally.

(ii) Since $\vert x\vert \psi_{0}\in L^{2}$, $J(t)=\int_{\mathbb{R}^{N}} \vert x\psi (t,x)\vert ^{2}\,dx$ is well defined. We deduce from Lemma 2.1 that

$$ J''(t)=16E(\psi_{0})- \frac{16-4Np_{1}}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{p_{1}+2}\,dx. $$

(3.2)

Since $\psi_{0}(x)=c\rho^{\frac{N}{2}} u(\rho x)$ and the Pohoz̆aev identity of (1.4), i.e., $\frac{1}{2}\int_{\mathbb{R}^{N}} \vert \nabla u(x)\vert ^{2}\,dx=\frac{1}{2p_{2}}\int_{\mathbb{R}^{N}} (I_{\alpha } \ast \vert u\vert ^{p_{2}})(x)\vert u(x)\vert ^{p_{2}}\,dx$ (see [18]), it follows that

$$\begin{aligned} E(\psi_{0})= {}&\frac{\vert c\vert ^{2}\rho^{2}}{2} \int_{\mathbb{R}^{N}} \bigl\vert \nabla u(x)\bigr\vert ^{2}\,dx-\frac{\vert c\vert ^{2p_{2}}\rho^{2}}{2p_{2}} \int_{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert u\vert ^{p_{2}}\bigr) (x)\bigl\vert u(x)\bigr\vert ^{p_{2}}\,dx \\ &{}+\frac{\vert c\vert ^{p_{1}+2}\rho^{\frac{N}{2}p_{1}}}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert u(x)\bigr\vert ^{p_{1}+2}\,dx \\ = {}& {-}\frac{\vert c\vert ^{2}\rho^{2}}{2}\bigl(\vert c\vert ^{2p_{2}-2}-1\bigr)\Vert \nabla u\Vert _{L^{2}} ^{2}+\frac{\vert c\vert ^{p_{1}+2}\rho^{\frac{N}{2}p_{1}}}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert u(x)\bigr\vert ^{p_{1}+2}\,dx. \end{aligned}$$

Thus, it follows from (3.1) that $E(\psi_{0}) < 0$. We deduce from (3.2) that $J''(t)<16 E(\psi_{0}) < 0$. By a standard argument, the solution $\psi (t)$ to (1.1) with $\psi_{0}=c\rho^{ \frac{N}{2}} u(\rho x)$ blows up in finite time. □

4 Dynamics of blow-up solutions in the $L^{2}$-critical case

In this section, we study the dynamical properties of blow-up solutions for (1.1) with $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}< \frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. For this purpose, we firstly recall a refined compactness lemma which has been proved in [18] by the inequality (2.6) and the profile decomposition theory.

Lemma 4.1

Let $p_{2}=1+\frac{2+\alpha }{N}$. If $\{\psi_{n}\}_{n=1}^{\infty }$ is a bounded sequence in $H^{1}$ and satisfies

$$\limsup_{n\rightarrow \infty }\Vert \nabla \psi_{n}\Vert _{L^{2}}^{2}\leq M, \qquad \limsup_{n\rightarrow \infty } \int_{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert \psi_{n}\vert ^{p_{2}}\bigr)\vert u_{n}\vert ^{p_{2}}\,dx\geq m. $$

Then there exists $\{x_{n}\}_{n=1}^{\infty }\subset \mathbb{R}^{N}$, such that, up to a subsequence,

$$\psi_{n}(\cdot +x_{n})\rightharpoonup \Psi $$

with $\Vert \Psi \Vert _{L^{2}}\geq (\frac{m}{p_{2}M})^{\frac{1}{2p_{2}-2}}\Vert u\Vert _{L^{2}}$.

Theorem 4.2

($L^{2}$-concentration)

Assume that $\psi_{0}\in H^{1}$, $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. Let the solution $\psi (t)$ to (1.1) blow up at the finite time $T^{*}$. If $a(t):[0,T^{*}) \mapsto \mathbb{R}$ is a real-valued function and $a(t)\Vert \nabla\psi (t)\Vert _{L^{2}}\rightarrow \infty$ as $t\rightarrow T^{*}$. Then there exists $x(t)\in \mathbb{R}^{N}$ such that

$$ \liminf_{t\rightarrow T^{*}} \int_{\vert x-x(t)\vert \leq a(t)}\bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx \geq \int_{\mathbb{R}^{N}} \bigl\vert u(x)\bigr\vert ^{2}\,dx. $$

(4.1)

Proof

Set

$$\rho_{n}:=\Vert \nabla u\Vert _{L^{2}}/\bigl\Vert \nabla \psi (t_{n})\bigr\Vert _{L^{2}} \quad \mbox{and}\quad v _{n}(x):= \rho_{n}^{\frac{N}{2}}\psi (t_{n},\rho_{n} x), $$

where $\{t_{n}\}_{n=1}^{\infty }\subseteq [0,T^{*})$ and $t_{n}\rightarrow T^{*}$ as $n\rightarrow \infty$. Then the sequence $\{v_{n}\}$ satisfies

$$ \begin{aligned} &\Vert v_{n}\Vert _{L^{2}}=\bigl\Vert \psi (t_{n})\bigr\Vert _{L^{2}}=\Vert \psi_{0}\Vert _{L^{2}}, \\ &\Vert \nabla v_{n}\Vert _{L^{2}}= \rho_{n}\bigl\Vert \nabla \psi (t_{n})\bigr\Vert _{L^{2}}=\Vert \nabla u\Vert _{L^{2}}. \end{aligned}$$

(4.2)

It follows from (2.3) that

$$\begin{aligned} H(v_{n}) := &\frac{1}{2} \int_{\mathbb{R}^{N}} \bigl\vert \nabla v_{n}(x)\bigr\vert ^{2}\,dx-\frac{1}{2p _{2}} \int_{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert v_{n} \vert ^{p_{2}}\bigr) (x)\bigl\vert v_{n}(x)\bigr\vert ^{p _{2}}\,dx \\ = &\rho_{n}^{2} \biggl( \frac{1}{2} \int_{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t_{n},x)\bigr\vert ^{2}\,dx-\frac{1}{2p_{2}} \int_{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \bigl\vert \psi (t_{n})\bigr\vert ^{p_{2}}\bigr) (x)\bigl\vert \psi (t_{n},x)\bigr\vert ^{p_{2}}\,dx \biggr) \\ = &\rho_{n}^{2} \biggl( E(\psi_{0})- \frac{1}{p_{1}+2} \int_{\mathbb{R} ^{N}} \bigl\vert \psi (t_{n},x)\bigr\vert ^{p_{1}+2}\,dx \biggr). \end{aligned}$$

(4.3)

Hence, by the Gagliardo–Nirenberg inequality

$$\int_{\mathbb{R}^{N}} \bigl\vert \psi (x)\bigr\vert ^{p_{1}+2}\,dx \leq C\Vert \psi \Vert _{L^{2}} ^{p_{1}+2-\frac{Np_{1}}{2}}\Vert \nabla \psi \Vert _{L^{2}}^{\frac{Np_{1}}{2}}, $$

and $0< p_{1}<\frac{4}{N}$, it follows that

$$\begin{aligned} \bigl\vert H(v_{n})\bigr\vert &\leq \rho_{n}^{2} \biggl( \bigl\vert E(\psi_{0}) \bigr\vert +\frac{1}{p_{1}+2} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t_{n},x)\bigr\vert ^{p_{1}+2}\,dx \biggr) \\ &\leq \frac{\vert E(\psi_{0})\vert \Vert \nabla u\Vert _{L^{2}}^{2}}{\Vert \nabla \psi (t_{n})\Vert _{L^{2}}^{2}}+C\frac{\Vert \nabla \psi \Vert _{L^{2}}^{2} \Vert \nabla\psi (t_{n})\Vert _{L^{2}}^{\frac{Np_{1}}{2}}}{\Vert \nabla \psi (t_{n})\Vert _{L^{2}}^{2}} \rightarrow 0 \quad \mbox{as }n\rightarrow \infty. \end{aligned}$$

(4.4)

This yields $\int_{\mathbb{R}^{N}} (I_{\alpha }\ast \vert v_{n}\vert ^{p_{2}})\vert v_{n}\vert ^{p_{2}}\,dx\rightarrow p_{2}\Vert \nabla u\Vert _{L^{2}}^{2}$.

Set $m=p_{2}\Vert \nabla u\Vert _{L^{2}}^{2}$ and $M=\Vert \nabla u\Vert _{L^{2}}^{2}$. Then we deduce from Lemma 4.1 that there exist $V\in H^{1}$ and $\{x_{n}\}_{n=1}^{\infty }\subset \mathbb{R}^{N}$ such that, up to a subsequence,

$$ v_{n}(\cdot +x_{n})=\rho_{n}^{N/2} \psi \bigl(t_{n},\rho_{n}(\cdot + x_{n})\bigr) \rightharpoonup V \quad \mbox{weakly in }H^{1} $$

(4.5)

with

$$ \Vert V\Vert _{L^{2}}\geq \Vert u\Vert _{L^{2}}. $$

(4.6)

Therefore, we have

$$\begin{aligned} \liminf_{n\rightarrow \infty } \int_{\vert x\vert \leq r}\bigl\vert v_{n}(t_{n},x+ x_{n})\bigr\vert ^{2}\,dx &= \liminf_{n\rightarrow \infty } \int_{\vert x\vert \leq r}\rho_{n}^{N}\bigl\vert \psi \bigl(t_{n},\rho_{n}(x+ x_{n})\bigr)\bigr\vert ^{2}\,dx \\ &\geq \int_{\vert x\vert \leq r}\bigl\vert V(x)\bigr\vert ^{2}\,dx,\quad \mbox{for every }r>0. \end{aligned}$$

(4.7)

From the assumption on $a(t)$, we have

$$\frac{a(t_{n})}{\rho_{n}}=\frac{a(t_{n})\Vert \nabla \psi (t_{n})\Vert _{L ^{2}}}{\Vert \nabla u\Vert _{L^{2}}}\rightarrow \infty, \quad \mbox{as }n\rightarrow \infty. $$

Then $r\rho_{n}< a(t_{n})$ for sufficiently large n. Therefore, it follows from (4.5) that

$$\begin{aligned} &\liminf_{n\rightarrow \infty }\sup_{y\in \mathbb{R}^{N}} \int_{\vert x-y\vert \leq a(t_{n})}\bigl\vert \psi (t_{n},x)\bigr\vert ^{2}\,dx \\ &\quad \geq \liminf_{n\rightarrow \infty }\sup_{y\in \mathbb{R}^{N}} \int_{\vert x-y\vert \leq r\rho_{n}}\bigl\vert \psi (t_{n},x)\bigr\vert ^{2}\,dx \\ &\quad \geq \liminf_{n\rightarrow \infty } \int_{\vert x-x_{n}\vert \leq r\rho_{n}}\bigl\vert \psi (t_{n},x)\bigr\vert ^{2}\,dx \\ &\quad =\liminf_{n\rightarrow \infty } \int_{\vert x\vert \leq r}\rho_{n}^{N}\bigl\vert \psi \bigl(t_{n},\rho_{n}(x+ x_{n})\bigr)\bigr\vert ^{2}\,dx. \end{aligned}$$

This and (4.7) imply that

$$\liminf_{n\rightarrow \infty }\sup_{y\in \mathbb{R}^{N}} \int_{\vert x-y\vert \leq a(t_{n})}\bigl\vert \psi (t_{n},x)\bigr\vert ^{2}\,dx\geq \int_{ \mathbb{R} ^{N}}\bigl\vert V(x)\bigr\vert ^{2}\,dx\geq \int_{ \mathbb{R}^{N}}\bigl\vert u(x)\bigr\vert ^{2}\,dx. $$

Since the sequence $\{t_{n}\}_{n=1}^{\infty }$ is arbitrary, it follows that

$$ \liminf_{t\rightarrow T^{*}}\sup_{y\in \mathbb{R}^{N}} \int_{\vert x-y\vert \leq a(t)}\bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx \geq \int_{ \mathbb{R}^{N}}\bigl\vert u(x)\bigr\vert ^{2}\,dx. $$

(4.8)

Furthermore, for every $t\in [0,T^{*})$, the function $y\mapsto h(y)= \int_{\vert x-y\vert \leq a (t)}\vert \psi (t,x)\vert ^{2}\,dx$ is continuous and $h(y)\rightarrow 0$ as $\vert y\vert \rightarrow \infty$. Hence, there is $x(t)\in \mathbb{R}^{N}$ such that

$$\sup_{y\in \mathbb{R}^{N}} \int_{\vert x-y\vert \leq a (t)}\bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx= \int_{\vert x-x(t)\vert \leq a(t)}\bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx, $$

which, together with (4.8), implies (4.1). □

In the following, we will study some properties of blow-up solutions to (1.1) with $\Vert \psi_{0}\Vert _{L^{2}}=\Vert u\Vert _{L^{2}}$. When $p=2$ or $\alpha =2$, the uniqueness of the ground state of (1.4) plays an important role in the characterization of blow-up solutions to (1.2) in [7, 10]. However, the uniqueness of ground states of (1.4) with $0<\alpha <N$ and $1+\frac{\alpha }{N}< p _{2}<\frac{N+\alpha }{N-2}$ is not known, we cannot apply the method in [7, 10] to study the dynamics of the blow-up solutions.

Theorem 4.3

Assume that $\psi_{0} \in \Sigma$, $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. Let the solution $\psi (t)$ to (1.1) blow up at the finite time $T^{*}$ and $\Vert \psi_{0}\Vert _{L^{2}}=\Vert u\Vert _{L^{2}}$. Then there exists $x_{0}\in \mathbb{R}^{N}$ such that

$$ \bigl\vert \psi (t,x)\bigr\vert ^{2}\rightarrow \Vert u\Vert _{L^{2}}^{2}\delta_{x_{0}} \quad \textit{as }t \rightarrow T^{*} $$

(4.9)

in the sense of a distribution.

Proof

Firstly, it follows from Theorem 4.2 that for all $r>0$

$$ \liminf_{t\rightarrow T^{*}} \int_{\vert x-x(t)\vert < r} \bigl\vert \psi (t, x)\bigr\vert ^{2}\,dx \geq \Vert u\Vert _{L^{2}}^{2}. $$

(4.10)

This and (2.1) yield for all $r>0$

$$\Vert u\Vert _{L^{2}}^{2}=\Vert \psi_{0} \Vert _{L^{2}}^{2}=\bigl\Vert \psi (t)\bigr\Vert _{L^{2}}^{2} \geq \liminf_{t\rightarrow T^{*}} \int_{\vert x-x(t)\vert < r} \bigl\vert \psi (t, x)\bigr\vert ^{2}\,dx \geq \Vert u\Vert _{L^{2}}^{2}. $$

This implies

$$ \bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\rightarrow \Vert u\Vert _{L^{2}}^{2} \delta_{x=0} \quad \mbox{as }t\rightarrow T^{*}. $$

(4.11)

On the other hand, it follows from the inequality (2.6) and (4.3) that for any $\varepsilon >0$ and any real-valued function θ

$$\begin{aligned} H\bigl(e^{\pm i\epsilon \theta }\psi (t)\bigr) =&\frac{\epsilon^{2}}{2} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2} \bigl\vert \nabla \theta (x)\bigr\vert ^{2}\,dx \\ &{}\mp \epsilon \operatorname{Im}\int_{\mathbb{R}^{N}} \bar{\psi }(t,x)\nabla \psi (t,x) \cdot \nabla \theta (x)\,dx +H\bigl(\psi (t)\bigr) \\ \geq& \frac{1}{2} \int_{\mathbb{R}^{N}} \bigl\vert \nabla \bigl(e^{\pm i\epsilon\theta }\psi (t,x) \bigr)\bigr\vert ^{2}\,dx \biggl( 1-\frac{\Vert \psi_{0}\Vert _{L^{2}}^{2p_{2}-2}}{ \Vert u\Vert _{L^{2}}^{2p_{2}-2}} \biggr) = 0. \end{aligned}$$

This implies that

$$\begin{aligned}& \biggl\vert \mp \operatorname{Im}\int_{\mathbb{R}^{N}} \bar{\psi }(t,x)\nabla \psi (t,x) \cdot \nabla \theta (x)\,dx\biggr\vert \\& \quad \leq \biggl( 2H\bigl(\psi (t)\bigr) \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2} \bigl\vert \nabla \theta (x)\bigr\vert ^{2}\,dx \biggr) ^{1/2}. \end{aligned}$$

(4.12)

Therefore, this and $H(\psi (t))\leq E(\psi (t))=E(\psi_{0})$ yield

$$\begin{aligned} \biggl\vert \frac{d}{dt} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2}x_{j}\,dx\biggr\vert &\leq C\biggl\vert \int_{\mathbb{R}^{N}} \bar{\psi }(t,x)\partial_{j} \psi (t,x)\,dx \biggr\vert \\ &\leq C\biggl\vert \int_{\mathbb{R}^{N}} \bar{\psi }(t,x)\nabla \psi (t,x) \nabla x_{j}\,dx\biggr\vert \\ & \leq C \biggl( 2H\bigl(\psi (t)\bigr) \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2} \vert \nabla x_{j}\vert ^{2}\,dx \biggr) ^{1/2} \leq C, \end{aligned}$$

for every $j=1,2,\ldots,N$. This implies

$$\biggl\vert \int_{\mathbb{R}^{N}} \bigl\vert \psi (t_{m},x)\bigr\vert ^{2}x_{j}\,dx- \int_{\mathbb{R}^{N}} \bigl\vert \psi (t_{k},x)\bigr\vert ^{2}x_{j}\,dx\biggr\vert \leq C\vert t_{m}-t_{k} \vert \rightarrow 0 \quad \mbox{as }m,k\rightarrow \infty, $$

for every $j=1,2,\ldots,N$, where $\{t_{m}\}_{m=1}^{\infty },\{t_{k} \}_{k=1}^{\infty }\subseteq (0,T^{*})$ and $\lim_{m\rightarrow \infty }t_{m}=\lim_{k\rightarrow \infty }t_{k}=T^{*}$. Thus, we have

$$\lim_{t\rightarrow T^{*}} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2}x_{j}\,dx\quad \mbox{exists}, $$

for every $j=1,2,\ldots,N$. Set

$$ x_{0}=\lim_{t\rightarrow T^{*}} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2}x\,dx/ \Vert u\Vert _{L^{2}}^{2}, $$

(4.13)

it follows that

$$ \lim_{t\rightarrow T^{*}} \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2}x\,dx=\Vert u\Vert _{L^{2}}^{2}x_{0}. $$

(4.14)

In addition, we deduce from Lemma 2.2 and (4.11) that

$$\begin{aligned}& \int_{\mathbb{R}^{N}} \vert x\vert ^{2}\bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\,dx \\& \quad \leq C \int_{\mathbb{R}^{N}} \bigl\vert x+x(t)\bigr\vert ^{2}\bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\,dx+C\bigl\vert x(t) \bigr\vert ^{2} \int_{\mathbb{R}^{N}} \bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\,dx \\& \quad \leq C+C\bigl\vert x(t)\bigr\vert ^{2}\Vert \psi_{0}\Vert _{L^{2}}^{2} \\& \quad \leq C+C\limsup_{t\rightarrow T^{*}} \int_{\vert x\vert < 1}\bigl\vert x+x(t)\bigr\vert ^{2}\bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\,dx \\& \quad \leq C+C \int_{\mathbb{R}^{N}} \vert x\vert ^{2}\bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx\leq C. \end{aligned}$$

(4.15)

This implies

$$ \limsup_{t\rightarrow T^{*}}\bigl\vert x(t)\bigr\vert \leq \frac{\sqrt{C}}{ \Vert \psi_{0}\Vert _{L^{2}}} $$

(4.16)

and

$$\limsup_{t\rightarrow T^{*}} \int_{\mathbb{R}^{N}} \vert x\vert ^{2}\bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\,dx \leq C. $$

Thus, for any $\varepsilon >0$, there is $R_{0}$ such that

$$\limsup_{t\rightarrow T^{*}}\biggl\vert \int_{\vert x\vert \geq R_{0}} x\bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}\,dx\biggr\vert \leq \frac{C}{R_{0}}< \frac{\varepsilon }{2}. $$

We see from (4.11) that

$$\begin{aligned}& \limsup_{t\rightarrow T^{*}} \biggl\vert \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2}x\,dx-x(t) \Vert u\Vert _{L^{2}}^{2}\biggr\vert \\& \quad = \limsup_{t\rightarrow T^{*}} \biggl\vert \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2} \bigl(x-x(t)\bigr)\,dx\biggr\vert \\& \quad \leq \limsup_{t\rightarrow T^{*}} \biggl\vert \int_{\vert x\vert \leq R_{0}} \bigl\vert \psi \bigl(t,x+x(t)\bigr)\bigr\vert ^{2}x\,dx\biggr\vert +\frac{\varepsilon }{2} \leq \varepsilon. \end{aligned}$$

(4.17)

This and (4.14) imply that $\lim_{t\rightarrow T^{*}}x(t)=x_{0}$. Thus, it follows from (4.11) that

$$ \bigl\vert \psi (t,x)\bigr\vert ^{2}\rightarrow \Vert u\Vert _{L^{2}}^{2}\delta_{x=x_{0}} \quad \mbox{as }t \rightarrow T^{*} $$

in the sense of distribution. □

Finally, we study the blow-up rate of blow-up solutions to (1.1) with $\Vert \psi_{0}\Vert _{L^{2}}=\Vert u\Vert _{L^{2}}$.

Theorem 4.4

Assume that $\psi_{0} \in \Sigma$, $\lambda_{1}=1$, $\lambda_{2}=-1$, $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. Let the solution $\psi (t)$ to (1.1) blow up at the finite time $T^{*}$ and $\Vert \psi_{0}\Vert _{L^{2}}=\Vert u\Vert _{L^{2}}$. Then there exists a constant $C>0$ such that for all $t\in [0,T^{*})$

$$ \bigl\Vert \nabla \psi (t)\bigr\Vert _{L^{2}}\geq \frac{C}{T^{*}-t}. $$

(4.18)

Proof

Let $g \in C_{0}^{\infty }(\mathbb{R}^{N})$ be a nonnegative radial function satisfying

$$g(x)=g\bigl(\vert x\vert \bigr)=\vert x\vert ^{2}, \quad \mbox{if }\vert x\vert < 1 \quad \mbox{and}\quad \bigl\vert \nabla g(x)\bigr\vert ^{2}\leq Cg(x). $$

For $A>0$, we define $g_{A}(x)=A^{2}g(\frac{x}{A})$ and $h_{A}(t)= \int_{\mathbb{R}^{N}} g_{A}(x-x_{0})\vert \psi (t,x)\vert ^{2}\,dx$ with $x_{0}$ defined by (4.13).

It follows from (4.12) and $H(\psi (t))\leq E(\psi (t))=E(\psi _{0})$ that for every $t\in [0,T^{*})$

$$\begin{aligned} \biggl\vert \frac{d}{dt}h_{A}(t)\biggr\vert \leq& C \biggl\vert \int_{\mathbb{R}^{N}} \bar{\psi }(t,x)\nabla \psi (t,x)\nabla g_{A}(x-x_{0})\,dx\biggr\vert \\ \leq& 2\sqrt{H\bigl(\psi (t)\bigr)} \biggl( \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2} \bigl\vert \nabla g_{A}(x-x_{0})\bigr\vert ^{2}\,dx \biggr) ^{1/2} \\ \leq& 2\sqrt{E(\psi_{0})} \biggl( \int_{\mathbb{R}^{N}} \bigl\vert \psi (t,x)\bigr\vert ^{2} \bigl\vert g_{A}(x-x_{0})\bigr\vert \,dx \biggr) ^{1/2} \\ \leq &C\sqrt{h_{A}(t)}. \end{aligned}$$

(4.19)

This implies that there is a constant C such that $\vert \frac{d}{dt}\sqrt{h _{A}(t)}\vert \leq C$. Integrating on both sides with respect to time t on $[t_{1},t]$, we have

$$ \bigl\vert \sqrt{h_{A}(t)}-\sqrt{h_{A}(t_{1})} \bigr\vert \leq C\vert t-t_{1}\vert . $$

(4.20)

On the other hand, from (4.9), we have

$$h_{A}(t_{1})\rightarrow \Vert Q\Vert _{L^{2}}g_{A}(0)=0 \quad \mbox{as }t_{1}\rightarrow T^{*}. $$

Thus, let $t_{1}\rightarrow T^{*}$ in (4.20), we have $h_{A}(t) \leq C(T^{*}-t)^{2}$. Now fix $t \in [0,T^{*})$, it follows that

$$\lim_{A\rightarrow \infty }h_{A}(t)= \int_{\mathbb{R}^{N}} \vert x-x_{0}\vert ^{2} \bigl\vert \psi (t,x)\bigr\vert ^{2}\,dx\leq C\bigl(T^{*}-t \bigr)^{2}. $$

Thus, we deduce from the uncertainty principle that

$$\bigl\Vert \nabla \psi (t)\bigr\Vert _{L^{2}}\geq \frac{\int_{\mathbb{R}^{N}} \vert \psi (x)\vert ^{2}\,dx}{ ( \int_{\mathbb{R}^{N}} \vert x-x_{0}\vert ^{2}\vert \psi (x)\vert ^{2}\,dx) ^{1/2}} \geq \frac{C}{T^{*}-t},\quad \forall t\in [0,T^{*}). $$

This completes the proof. □

5 Conclusions

In this paper, we study the dynamics of blow-up solutions for the nonlinear Schrödinger–Choquard equation (1.1) with $0< p_{1}<\frac{4}{N}$ and $p_{2}=1+\frac{2+\alpha }{N}$. In the previous papers, the scale invariance played an important role in the study of the dynamics of blow-up solutions to nonlinear Schrödinger equations. Because there exists no scale invariance for Eq. (1.1), the study of blow-up solutions to (1.1) is an interesting problem. We must overcome the difficulty brought about by the loss of scale invariance. For (1.1), we find that the ground state solution u to (1.4) exactly describes the sharp threshold mass of global existence and blow-up, the dynamical properties of blow-up solutions, including $L^{2}$-concentration, limiting profile and blow-up rates.