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

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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 \(\lambda_{1}=0\).

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 [116]. 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 \(\Vert \psi \Vert _{\Sigma }:=\Vert \psi \Vert _{H^{1}}+\Vert x\psi \Vert _{L^{2}}\).

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., \(p=1+\frac{2+ \alpha }{N}\), \(C_{\alpha,p}=p\Vert u\Vert _{L^{2}}^{2-2p}\).

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, 2325] 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:

  1. (i)

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

  2. (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. □

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. □

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.

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Acknowledgements

This work is supported by the PhD scientific research start-up capital funded projects of Longdong University (XYBY05), the fundamental research funds for the Gansu universities (2015A-150).

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Correspondence to Cunqin Shi.

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MSC

  • 35Q55
  • 35A15

Keywords

  • Nonlinear Schrödinger–Choquard equation
  • Blow-up solutions
  • The dynamical properties