Sharp threshold of global existence and mass concentration for the Schr¨odinger-Hartree equation with anisotropic harmonic conﬁnement

This article is concerned with the initial-value problem of a Schr¨odinger-Hartree equation in the presence of anisotropic partial/whole harmonic conﬁnement. Firstly, we get a sharp threshold for global existence and ﬁnite time blow-up at the ground state mass in the L 2 -critical case. Then, some new cross-invariant manifolds and variational problems are constructed to study blow-up versus global well-posedness criteria in the L 2 -critical and L 2 -supercritical cases. Finally, we research the mass concentration phenomenon of blow-up solutions and the dynamics of the L 2 -minimal blow-up solutions in the L 2 -critical case. The main ingredients of the proofs are the variational characterization of the ground state, a suitable reﬁned compactness lemma and scaling techniques. Our conclusions extend and compensate for some previous results.


Introduction
In this paper, we consider the initial-value problem of the following Schrödinger-Hartree equation in the presence of anisotropic partial/whole harmonic confinement where ϕ : [0, T ) × R N → C is a complex valued function, 0 < T ≤ ∞ and ϕ 0 is a given function in R N , 1 ≤ k ≤ N , ν i = 0 and ν i ∈ R (1 ≤ i ≤ k), λ > 0, 2 ≤ p < N +α N −2 , I α : R N → R is the Riesz potential defined by with 0 < α < N and Γ is the Gamma function.
Nonlinear Schrödinger equations of Hartree-type have a broad physical background.They often appear as models of quantum semiconductor devices [1].When k = N , Eq.(1.1), known as Schrödinger-Hartree equation with complete harmonic confinement, can be used to characterize Bose-Einstein condensation (BEC) in a gas with very weak two-body interactions, which was found in 23 N a or 87 Rb atomic systems [2].When 1 ≤ k < N , Eq.(1.1) is called nonlinear Hartree equation with partial confinement, arising also as a typical model to describe the BEC [3].When removing the harmonic confinement in Eq.(1.1), for N = 3, p = 2 and α = 2, Eq.(1.1) is used to describe electrons trapped in their own holes, which is similar to the Hartree-Fock theory of single component plasma to some extent [4].
When k = N , Eq.(1.1) with complete harmonic confinement has been well-studied.In the special case k = N , ν 1 = ν 2 = • • • = ν N and p = 2, Huang et al. [5] applied the Hamiltonian invariants and the Gagliardo-Nirenberg inequality of convolution type and scaling technique to investigate the sharp threshold of global existence and showed the stability of standing waves in the mass-critical case α = N − 2. Wang [6] proved the existence of blow-up solutions and studied the strong instability of standing waves by variational methods in the mass-supercritical case 2 < N − α < min{4, N }.It's worth mentioning that, Feng [7] derived the sharp threshold for global existence and finite time blow-up on mass for ν 1 = ν 2 = • • • = ν N and p = 1 + 2+α N ≥ 2 in Eq.(1.1), by using the variational characterization of the ground state solution to a nonlinear Schrödinger-Hartree equation without potential (see Eq.(2.6)).Moreover, in the general L 2 -supercritical case 1+ 2+α N ≤ p < N +α N −2 with 0 < α < N , Feng [7] obtained blow-up versus global well-posedness criteria by constructing some cross-invariant manifolds and variational problems and studied the stability and instability of standing waves.If the nonlinearity (I α * |ϕ| p )|ϕ| p−2 ϕ is replaced by |ϕ| p−1 ϕ, there exists extensive literatures on the Cauchy problem of nonlinear Schrödinger equation with complete harmonic potential, see e.g., [8][9][10].In particular, Shu and Zhang [9] and Zhang [10] derived the sharp criterion of global existence to Eq.(1.1) by constructing different cross-constrained variational problems and so-called invariant sets.
When 1 ≤ k < N , the main difference between nonlinear Schrödinger-type equation with partial harmonic confinement and complete confinement is that the embedding from natural energy space Σ (see Section 2) ) is lack of compactness, resulting the main difficulty on the study of corresponding Cauchy problem.Due to the fact, the existence of stable standing waves, global and blow-up dynamics, and sharp criterion of global existence to the nonlinear Schrödingertype equation with partial confinement have attracted considerable interest.There exists several studies in these directions to Eq.(1.1) with power type nonlinearity |x| −b |ϕ| p−1 ϕ (b ≥ 0), see [11][12][13][14][15][16] for example and the references therein.More precisely, for the case b = 0, Ardila and Carles [12] studied the criteria of blow-up and scattering below the ground state in the focusing L 2 -supercritical case.The papers [13,14] studied the sharp threshold for finite time blow-up and global existence in the mass-critical case by utilizing the variational characteristic of the ground state to a classical nonlinear elliptic equation without harmonic confinement and Hamilton conservation.It is worth noting that, by exploiting the refined compactness lemma proposed by Hmidi and Keraani [17] and the variational characterization of the ground state, Pan and Zhang [14] investigated the mass concentration properties and limiting profile of the blow-up solutions possessing small supercritical mass in the L 2 -critical case in dimension N = 2.More recently, when k = 1, that's, the harmonic potential is confined in one direction, Wang and Zhang [15] derived the sharp condition for global existence and blowup to the solutions by making using of the irregular variance identity and constructing cross-constrained variational problems and invariant manifolds of the evolution flow.Liu, He and Feng [16] studied the existence and stability of normalized standing waves for Eq.(1.1) with anisotropic partial confinement and inhomogeneous nonlinearity |x| −b |ϕ| p−1 ϕ (b > 0).As far as we know, there is no paper concerning the sharp threshold of global existence and mass concentration phenomenon to the blow-up solutions of nonlinear Schrödinger-type equation with Hartree nonlinearity (I α * |ϕ| p )|ϕ| p−2 ϕ and partial confinement, which are greatly pursued in physics.This is the main motivation for us to study these problems for Eq.(1.1).
In the absence of harmonic confinement in Eq.(1.1), the corresponding equation is also known as Choquard equation, whose Cauchy problem has also been extensively studied, see for instance [8,[18][19][20][21][22].In particular, by constructing invariant sets and using variational methods, Chen and Guo [18] obtained the existence of blow-up solutions for some suitable initial data and showed strong instability of standing waves in the case N = 3 and 2 < N − α < 3. Miao et al. [19] studied the mass concentration properties of blow-up solutions as well as the dynamics of blow-up solutions with minimal mass for Eq.(1.1) in the L 2 -critical case with α = 2 and N = 4.When p = 1 + 4 N (N = 3, 4), Genev and Venkov [20] gave a sharp sufficient condition of global existence to Eq.(1.1).Furthermore, they proved the existence of blow-up solutions and considered the blowup dynamics to the solutions in the L 2 -critical setting, i.e., p = 1 + 2+α N with α = 2. Notice that Feng and Yuan [21] not only considered the local and global well-posedness and finite time blow-up to the corresponding initial-value problem (1.1) in the general case 2 ≤ p < N +α N −2 with max{0, N − 4} < α < N , but also took into account the concentration phenomenon of blow-up solutions and the blow-up dynamics of blow-up solutions possessing minimal mass in the case p = 1 + 2+α N ≥ 2, by establishing a new refined compactness lemma with respect to the nonlocal nonlinearity (I α * |ϕ| p )|ϕ| p−2 ϕ.
To the best of our knowledge, there are few papers dealing with the global well-posedness and blow-up dynamics to the Cauchy problem (1.1) in the presence of anisotropic partial/whole harmonic confinement.Inspired by the literatures aforementioned, the purposes of this present article are devoted to investigate the sharp criterion of global existence and mass-concentration phenomenon of blow-up solutions as well as the dynamical properties to minimal mass blow-up solutions of Eq.(1.1) with anisotropic partial/whole confinement.To achieve these aims, the main difficulties that we will encounter come from the presence of anisotropic harmonic confinement k j=1 ω 2 j x 2 j and the nonlocal nonlinearity (I α * |ϕ| p )|ϕ| p−2 ϕ, resulting in the loss of scale invariance and pseudo-conformal transformation.Motivated by [7,13,14,23], we utilize the ground state to the nonlinear Schrödinger-Hartree equation (2.6), which is without any confined potential, to overcome the lack of the above two symmetries.Firstly, we get a sharp threshold for global existence and finite time blow-up at the ground state mass in the L 2 -critical case, which extend the global existence and blow-up results of Feng [7] to the case with anisotropic partial/complete confinement.Then, in the L 2 -critical and L 2 -supercritical cases, by constructing some new crossinvariant manifolds of the evolution flow and some variational problems associated to Eq.(1.1), we derive blow-up versus global well-posedness criteria for Eq.(1.1).In the present case, the constructed cross-invariant sets and variational problems are in light of Shu and Zhang [9], which differ from those of Feng [7], and some new criterions of global existence are derived.Finally, based on the ideas of [14,17,21], we research the mass concentration phenomenon of blow-up solutions and the dynamics of the L 2 -minimal blow-up solutions, including the precise mass-concentration and blow-up rate of the minimal mass blow-up solutions.The main ingredients of the proofs are the variational characterization of the ground state to Eq.(2.6), a refined compactness lemma established by Feng and Yuan [21] and scaling techniques.Our conclusions extend and compensate for some previous results of [7] and [21].
The rest of this paper is organized as follows.In section 2, some notations and preliminaries are given.Section 3 considers the sharp threshold for global existence and finite time blow-up of Eq.(1.1) in both the L 2 -critical and L 2 -supercritical cases.The last section focuses on the mass concentration phenomenon of blow-up solutions and the dynamics of the L 2 -minimal blow-up solutions.

Notations and Preliminaries
Throughout this paper, we use , and use C to stand for positive constants, which may vary from line to line.Without loss of generality, we assume λ = 1 in this and subsequent sections.
For Eq.(1.1), we equip the natural energy space with the inner product and the corresponding norm is given by The energy functional associated to Eq.(1.1) is defined as Let us now state the local well-posedness of Eq.(1.1) in energy space Σ according to [7,21].
Lemma 2.2.( [24]) Let 0 < λ < N and s, r > 1 be constants such that By inequality (2.4), we can obtain the following generalized Gagliardo-Nirenberg inequality . (2.5) Following Weinstein [25], Feng and Yuan [21] derived the best constant in the inequality (2.5) by discussing the existence of the minimizer of the functional where Q(x) is the ground state of the elliptic equation In particular, in the L 2 -critical case, C α,p = p Q(x) 2−2p 2 .It's known that ground state is of great importance in studying global existence and blow-up dynamics to the initial-value problem of nonlinear Schrödinger equation, in the following lemma we recall some existence results and properties of the ground state solution to Eq.(2.6).Lemma 2.4.([26]) Let α ∈ (0, N ) and N +α N < p < N +α N −2 .It follows that Eq.(2.6) admits a ground state solution Moreover, the following pohozaev identity holds. ) From (2.7) and (2.8), one has In order to study the blow-up phenomenon of Eq.(1.1), we also need the following lemma obtained in Weinstein [25].
Following the idea of Glassey [27] (see also Feng [7]), we will adopt the convexity method to study the existence of blow-up solutions.More precisely, we need to consider the variance and show that there exists time T > 0 such that V (T ) = 0.With some formal computations (which can be rigorously proved as in [8]), we have the following virial identities.Proposition 2.6.
for all t ∈ [0, T ).In particular, when p = 1 + 2+α N , we have (2.10) Using Lemma 2.5 and Proposition 2.6, we can easily get the following sufficient conditions on the existence of blow-up solutions.Corollary 2.7.Assume that max{0, N − 4} < α < N and max{1 ), and satisfy one of the following conditions: Then the corresponding solution ϕ(t, x) of Eq.(1.1) blows up in finite time.N ≥ 2 and Q(x) be the positive radially symmetric ground state solution of Eq.(2.6).If ϕ 0 ∈ Σ and ϕ 0 satisfies

Sharp threshold for global existence and blow-up
then the Cauchy problem (1.1) has a global solution ϕ(t, x) in C([0, ∞), Σ).Furthermore, we have for any 0 ≤ t < ∞, From (3.3) and (3.1), we have for all t ∈ [0, T ), where T is arbitrary and T < ∞, there exists C such that Then according to Proposition 2.1, ϕ(t, x) exists globally in time.Moreover, we have (1.1), Feng [7] proved that the solution ϕ(t, x) of Eq.(1.1) exists globally (see Theorem 3.2 in [7]).Theorem 3.1 can be viewed as the complement of the corresponding result of [7] for Eq.(1.1) with whole harmonic confinement.
(1.1), Feng [7] proved the existence of blow-up solutions (see Theorem 3.2 in [7]).When considering Eq.(1.1) in the presence of anisotropic partial/complete harmonic confinement, we derive the corresponding blow-up result by scaling approach, which differs from the method of Feng [7].
(ii) Theorems 3.1 and 3.3 declare that Q(x) 2 provides a sharp threshold for global existence and blow-up to Eq.(1.1) in terms of the initial data, which is called minimal mass for the blow-up solutions.
Similar to the proof above, we can also prove that K + , R − , R + are invariant manifolds.
In the below, we will use the cross-constrained variational approach to investigate the sharp condition of global existence for Eq.(1.1).
By Theorem 3.8, we can get another sufficient condition of global existence of Eq.(1.1).

Mass concentration and dynamics of the L 2 -minimal blow-up solutions
In this section, we are devoted to the dynamical properties of blow-up solutions to Eq.(1.1) with partial/whole harmonic confinement.We first study the mass concentration phenomenon and then the dynamics of the L 2 -minimal blow-up solutions, including the precise mass-concentration and blow-up rate to the blow-up solutions with minimal mass.
In order to study the dynamical properties for the blow-up solutions of Eq.(1.1), we recall the refined compactness lemma established by Feng and Yuan [21].Then, there exists {x n } ∞ n=1 ⊂ R N such that, up to a subsequence, Using the refined compactness lemma, we can establish the following concentration property to the blow-up solutions of Eq.(1.1).Theorem 4.2.(L 2 -concentration) Assume N − 2 ≤ α < N and p = 1 + 2+α N .Let ϕ(t, x) be a solution of Eq.(1.1) that blows up in finite time T , and s(t) be a real-valued nonnegative function on [0, T ) such that s(t) ∇ϕ 2 → ∞ as t → T .Then there exists a function x(t) ∈ R N for t < T such that lim inf where Q(x) is the ground state solution of Eq.(2.6).
Let {t n } ∞ n=1 be an arbitrary time sequence such that t n → T as n → ∞, and denote ρ n = ρ(t n ) and v n (x) = v(t n , x).By (2.2), (2.3) and (4.2), we obtain (4.3) For f (x) ∈ H 1 (R N ), we define the functional From (4.3), (2.1) and (4.2), one has that which yields, in particular, Take M = ∇Q(x) 2 2 and m = p ∇Q(x) 2 2 .Then by Lemma 4.1, there exists U (x) ∈ H 1 (R N ) and {x n } ∞ n=1 ⊂ R N such that, up to a subsequence, with U 2 ≥ Q(x) 2 .From (4.4), it follows that where Q(x) is the ground state solution of Eq.where Q(x) is the ground state solution of Eq.(2.6).

3. 1 .
The L 2 -critical case The aim of this subsection is mainly to consider the global existence and blowup of the solutions to Eq.(1.1) in the L 2 -critical case, i.e. p = 1+ 2+α N .The ground state mass Q(x) 2 gives a sufficient condition on the global existence of the solution to Eq.(1.1).Theorem 3.1.Let p = 1 + 2+α