Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential

The Cauchy problem of nonlinear Schrödinger equation with a harmonic potential for describing the attractive Bose-Einstein condensate under themagnetic trap is considered.We give some su cient conditions of global existence and nite time blow up of solutions by introducing a family of potential wells. Some di erent sharp conditions for global existence, and some invariant sets of solutions are also obtained here.


Introduction
In the present paper we study the Cauchy problem of nonlinear Schrödinger equation with a harmonic potential where p satis es (A) + n < p < n + n − for n ≥ ; + n < p < ∞ for n = , .
The equation in (1.1) may model the Bose-Einstein condensate with attractive inter-particle interactions under the magnetic trap [1,6,10,11]. Fujiwara [7] considered the above nonlinear Schrödinger equation with a general real-valued potential function V(x) which later studied in [5]. When |D α V(x)| is bounded for all α ≥ , the author of [7] gave the smoothness of Schrödinger kernel for potentials of quadratic growth. Further Yajima [20] showed that for super-quadratic potentials, the Schrödinger kernel is nowhere C . In addition, Oh [16] pointed out that the quadratic potentials are the highest order potentials for local well-posedness of the equation. Therefore V(x) = |x| is the critical potential for the local existence of the Cauchy problem.
On the global existence and nite time blow up of solutions for problem (1.1), there have been some results. Firstly, Oh [16] and Cazenave [4] established the local existence of solutions of problem (1.1) in the energy space. Then for < p < + n , Zhang [22] proved the global existence of solutions for any initial data in energy space. For the case p = + n , Zhang [21] gave a sharp condition of global existence of solution for problem (1.1). For the case p > + n , Cazenave [4], Carles [2,3] and Tsurumi and Wadati [18] showed that the solutions of problem (1.1) blows up in a nite time for some initial data, especially for a class of su ciently large initial data; but the solutions of problem (1.1) globally exist for other su ciently small initial data [2,3,18].
Chen and Zhang [5] studied problem (1.1) and gave a su cient condition of global existence of solutions in energy space. They proved that if p satis es (A), φ ∈ H(R n ) and satis es φ L ≤ h ∇φ L + φ L , then the solution φ of problem (1.1) exists globally and satis es Q is the ground state solution of equation In addition, in [17] Shu and Zhang gave a sharp condition for global existence of solution for problem (1.1). Moreover, in [23] Zhang studied Cauchy problem of following nonlinear Schrödinger equation with harmonic potential obtained a sharp condition for global existence and nite time blow up of solutions and discussed the instability of standing wave. These works motivated us to study on this and related problems [24][25][26][27][28][29][30][31][32].
In this paper we study problem (1.1), where p satis es (A). By introducing two families of sets W δ and V δ like [12][13][14], we not only give some di erent su cient conditions for global existence and nite time blow up of solutions which are completely di erent from that given in [2][3][4][5], [17], [18], [23] but also obtain a lot of di erent sharp conditions for global existence of solutions and some invariant sets of solutions for problem (1.1).
Throughout the present paper, the following notations are used for precise statements: L p (R n ) ( ≤ p ≤ ∞) denotes the usual space of all complex L p -functions on R n with norm φ L p (R n ) = φ p and φ L (R n ) = φ . H and H denote H (R n ) and H(R n ) respectively. H(R n ) and E(φ) are de ned by (1.2) and (1.4) respectively. Propostion 1.1. [4,8,9,16] Assume that < p < n+ n− for n ≥ and < p < ∞ for n = , , and φ ∈ H (R n ). Then the problem (1.1) admits a unique solution φ(t) ∈ C([ , T); H(R n )) for some T ∈ [ , ∞) (Maximal existence time), and φ(t) satis es the following two conservation laws: for all t ∈ [ , T). Furthermore, we have the following alternative: T = ∞ or T < ∞ and Then φ(t) ∈ H for ≤ t < T and Propostion 1.3. [19] Let < p < n+ n− for n ≥ and < p < ∞ for n = , . Then the best constant C * > of the Gagliardo-Nirenberg's inequality, is given by (1.8)

Preliminaries
In this section we shall give some necessaries lemmas and by using them we introduce two families W δ and V δ . For problem (1.1) with φ ≠ we de ne we can obtain the following lemma. Next we discuss the relations between ∇φ and the sign of I δ (φ), which are crucial for obtaining the main results in this paper.

Lemma 2.2. Let p satisfy (A). Assume that φ ∈ H and
Proof.
As it is well known that in space H (R n ), Poincaré inequality does not hold, so that one can not use the important fact that ∇u is equivalent to u H . In order to overcome this di culty, we introduce the space H(R n ), so that by (1.5) and (1.7) the norms ∇φ and φ H are equivalent in some sense again.
In the following Lemma 2.4 we estimate the value of d(δ) and give its expression by d( ), which palys an important role in the proof of the main results of this paper.
(2.1) From (2.1) and From the arbitrariness of ε we obtain From φ λ ∈ N and the de nition of d( ) we have From (2.4) and the arbitrariness of ε we get From(2.2) and (2.5) we obtain (ii) of this lemma.
Proof. Conclusions (i) and (ii) follow from (ii) in Lemma 2.4 immediately. Conclusion (iii) follows from (ii) in Lemma 2.4 and De nition 2.6. For problem (1.1) with φ ≠ we de ne Remark 2.7. Lemma 2.4 shows that the depth d of the potential well W de ned by De nition 2.6 depends on φ and d → +∞ as φ → . This property of d is completely di erent from the depth of other known potential wells de ned for other nonlinear evolution equations. In addition, W or any W δ do not include φ = .

Invariant sets and vacuum isolating of solutions
In this section we discuss the invariant sets and vacuum isolating of solutions for problem (1.1). First we consider the case < E(φ ) < d, and the case E(φ ) = d will be considered later.   T). Arguing by contradiction, we suppose that there exists a t ∈ ( , T) such that φ(t ) ∈ ∂W δ for some δ ∈ [δ , δ ], i.e. I δ (φ(t )) = or J(φ(t )) = d(δ). From (1.6) we get Now we consider the invariant sets of solutions of problem (1.1) with E(φ ) = d.
we get φ ∈ W. Next we prove that φ(t) ∈ W for t ∈ ( , T). Arguing by contradiction, we assume that there exists a t ∈ ( , T) such that φ(t ) ∈ ∂W, i.e. I(φ(t )) = or J(φ(t )) = d. From φ(t ) = φ ≠ we get |x|φ(t ) ≠ . Hence from The proof is similar to that of part (i) of this lemma.

global existence and nite time blow up of solutions
In this section we shall prove the global existence and nite time blow up of solutions and give some sharp conditions for global existence and nite time blow up of solutions for problem (1.1) which are completely di erent from those given in [2][3][4][5], [17,18,23].
Since I(φ ) > gives E(φ ) > , the following corollary is the improvement of Theorem 4.1.

So by Corollary 4.2, problem (1.1) admits a unique global solution φ(t) ∈ C [ , ∞); H and φ(t) ∈ W δ for
In the following theorem we give two results on global existence of solutions for problem (1.1) regarding ∇φ .
Theorem 4.6 is proved.
Remark 4.7. The proof of Theorem 4.6 strongly depends on the fact that φ(t) ∈ V δ for some δ > a, where φ(t) is the solution of problem (1.1) with E(φ ) < d, I(φ ) < . Therefore the introducing of V δ is crucial for the proof of Theorem 4.6.
Proof. Let φ(t) be any solution of problem (1.1) with E(φ ) = d, I(φ ) < and F ( ) ≤ . Let us prove T < ∞. Arguing by contradiction, again suppose T = +∞. From Theorem 3.2 we have φ(t) ∈ V for ≤ t < ∞. Hence we get From this and F ( ) ≤ it follows that for any t > we have F (t ) < and Hence there exists a T > such that F(t) > for ≤ t < T and lim t→T F(t) = .
The remainder of this proof is same as the proof of Theorem 4.6.
From  Note that from Lemma 2.4 we have From this we get the following sharp condition that only depends on ∇φ .  Proof. This corollary follows from Corollary 4.14 and the fact that (4.5), (4.6) and (4.7) are equivalent to E(φ ) < d , ∇φ < r(a) and ∇φ ≥ r(a) respectively.
Finally we give another series of sharp conditions for globally existence and nite time blow up of solution for problem (1.1) as follows: