Abstract
In this paper, we study the dynamics of blow-up solutions for the nonlinear Schrödinger–Choquard equation
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 \(\lambda_{1}=0\).
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1 Introduction
In this paper, we will investigate the blow-up solutions of the nonlinear Schrödinger–Choquard equation
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
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,
which keeps it invariant. More precisely, the map
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
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
for all \(0\leq t< T^{*}\), where \(E(\psi (t))\) is defined by
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
and
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 \(t\in [0,T^{*})\)
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
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
is
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
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
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
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
Then there exists \(\{x_{n}\}_{n=1}^{\infty }\subset \mathbb{R}^{N}\), such that, up to a subsequence,
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
Proof
Set
where \(\{t_{n}\}_{n=1}^{\infty }\subseteq [0,T^{*})\) and \(t_{n}\rightarrow T^{*}\) as \(n\rightarrow \infty \). Then the sequence \(\{v_{n}\}\) satisfies
It follows from (2.3) that
Hence, by the Gagliardo–Nirenberg inequality
and \(0< p_{1}<\frac{4}{N}\), it follows that
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,
with
Therefore, we have
From the assumption on \(a(t)\), we have
Then \(r\rho_{n}< a(t_{n})\) for sufficiently large n. Therefore, it follows from (4.5) that
This and (4.7) imply that
Since the sequence \(\{t_{n}\}_{n=1}^{\infty }\) is arbitrary, it follows that
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
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
in the sense of a distribution.
Proof
Firstly, it follows from Theorem 4.2 that for all \(r>0\)
This and (2.1) yield for all \(r>0\)
This implies
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
This implies that
Therefore, this and \(H(\psi (t))\leq E(\psi (t))=E(\psi_{0})\) yield
for every \(j=1,2,\ldots,N\). This implies
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
for every \(j=1,2,\ldots,N\). Set
it follows that
In addition, we deduce from Lemma 2.2 and (4.11) that
This implies
and
Thus, for any \(\varepsilon >0\), there is \(R_{0}\) such that
We see from (4.11) that
This and (4.14) imply that \(\lim_{t\rightarrow T^{*}}x(t)=x_{0}\). Thus, it follows from (4.11) that
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 \(t\in [0,T^{*})\)
Proof
Let \(g \in C_{0}^{\infty }(\mathbb{R}^{N})\) be a nonnegative radial function satisfying
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 \(t\in [0,T^{*})\)
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
On the other hand, from (4.9), we have
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
Thus, we deduce from the uncertainty principle that
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.
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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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Shi, C., Liu, K. Dynamics of blow-up solutions for the Schrödinger–Choquard equation. Bound Value Probl 2018, 64 (2018). https://doi.org/10.1186/s13661-018-0985-z
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DOI: https://doi.org/10.1186/s13661-018-0985-z