Concentration of blow-up solutions for the Gross-Pitaveskii equation

: We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 supercritical case. Finally, in the L 2 critical case, a delicate limit of blow-up solutions is analyzed.

The attractive Bose-Einstein condensate are known to be metastable in spatially localized systems, provided that the number of condensed particles, say N , is below a critical value N c , while they are unstable for ≥ N N c , which corresponds to wave's collapse.From the mathematical studies, the wave's collapse is called blow-up, i.e., there exists a finite time < +∞ T such that = +∞ → ψ t x lim , with the inner product defined by [9,57].To understand how dynamical properties of blow-up phenomenon for nonlinear Schrödinger equations is interesting and fascinating [2,14,16,18,22,35,40,44,56,63].
In the present article, we impose the initial data  to equation (1.1) and study the dynamical properties of blow-up solutions to Cauchy problems (1.1) and (1.2).
The mathematical aspect research for equation (1.1) dates its history back to 1989 in the study by Oh [43], and Although the existence of blow-up solutions and sharp criteria of blow-up and global existence are widely studied [5,13,57,58], exploring dynamical properties of blow-up solutions is an intriguing and challenging problem due to the harmonic potential x 2 | | .
Particularly, in the L 2 critical case: = + p 2 D 4 , Carles [11] constructed a crucial transformation of solutions between equation (1.1) and the canonical critical semilinear Schrödinger equation Thus, Li and Zhang [32] obtained the sharp upper and lower blow-up rates for equation (1.1) in terms of Merle and Raphaël's blow-up rates for equation (1.3) in studies by Merle and Raphaël [40,41].Zhu et al. [60] investigated the limiting profile of blow-up solutions in Σ.But in the L 2 supercritical case: > + p 2 D 4 , there is no transformation, the dynamical properties of blow-up solutions, including blow-up rate and concentration are interesting and extensively open.
On the other hand, the presence of the harmonic potential breaks scaling and translation symmetries, introducing new challenges into the analysis for equation (1.1).The well known results, including stability of standing waves, concentration for the nonlinear Schrödinger equations with a potential, have dramatically changed, comparing with the canonical nonlinear Schrödinger equation without any potential; see [5,11,20,43,57].Particularly, our work also fits in the context of the recent work on dispersive equations in the presence of broken symmetries, which have attracted a great deal of interest in recent years; see [4,[23][24][25][26][27]29,30,46,51,54].
Before summarizing our results in this article, we now introduce the following profile decomposition of bounded sequences in , which is the main tool in this article.
2 *, and . Then there exist a subsequence of { } of sequences in D and a sequence such that the following properties: where the remaining term ≔ v v x n l n l ( ) is small in the following sense: The profile decomposition of bounded sequences in , is first established in the author's Ph.D thesis in the study by Zhu [62], to prove the existence of ground state solutions of (1.8) and find the best constant in (1.2) by the variational argument.We should point out that the profile decomposition of bounded sequences in H 1 was first proposed by Hmidi and Keraani [21].In fact, the profile decomposition tool is less technical and simpler, and there are two main advantages of the profile decomposition: one is that the decomposition form of bounded sequences is given, and it can the aim functionals.The other is that the decomposition is almost orthogonal, and the norms of bound sequences have similar decomposition.The profile decomposition argument has been successfully applied to find the best constant of generalized Gagliardo-Nirenberg inequalities [21,61,63].
Then, in the L 2 supercritical case: > + p 2 D 4 , we introduce a new stationary equation: We obtain the following new sharp Gagliardo-Nirenberg inequality, which is sharp in the sense: let = v W in (1.9), then "≤" is "=" by the Pohozaev's identities for equation (1.8).
where W is a ground state solution of (1.8).
It is well known that the blow-up theory for equation (1.3) in H 1 has been developed during the last two decades by the energy arguments and variational arguments; see previous studies [12,28,40,57] and the references therein.Essentially, the blow-up theory is connected to the standing wave through the following Gagliardo-Nirenberg inequality proposed by Weinstein in [55].It reads that for and R is the unique positive radially symmetric solution of , whose existence and uniqueness are proved in the studies by Kwong [31] and Strauss [49], respectively  ( ) , then there exists where W is a ground state solution of (1.8).Here, we denote − 1 2 be any real number − ε with any sufficiently small > ε 0.
In particular, when = + p 2 D 4 , we see that = s 0 c .Then, the rate of H ˙sc -concentration phenomenon in (1.12) will recover Merle and Tsutsumi's concentration results [42,52] for the canonical semilinear Schrödinger equation (1.3): there exists where Q x ( ) is the unique positive radially symmetric solution of To our surprise, the concentration rate of blow-up solutions in (1.12) is the same to (1.13), and it does not changed with respect to p.This article is organized as follows.In Section 2, we give a review of Cauchy problems (1.1) and (1.2), and state some preliminaries.In Section 3, by establishing the profile decomposition of bounded sequences in , we give the proof of Theorem 1.2.In Section 4, we give the proof of Theorem 1.3.In Section 5, we further investigate the limiting profile of blow-up solutions in the L 2 critical case.

Review of the Cauchy problem
In this article, we denote L q D ( ), ‖ ‖ , H 1 and ∫ ⋅ x d , respectively.′ q is the conjugate of real number q which satisfies the condition + = . z R and z I are the real part and imaginary part of the complex number z, respectively.z denotes the complex conjugate of the complex number z.
First, we review the well-posedness of Cauchy problems (1.1) and (1.2 where is the propagator of H . Fujiwara [17] found that the corresponding Schrödinger kernel k t x y , , ( )to H has the following explicit format: With this result, one can easily deduce that ↦ t S t ( ) is a strongly continuous from ′ L q to L q for ≥ q 2, and when Thus, one can obtain the following Strichartz estimate for the Schrödinger operator H .A pair is called . Define the Strichartz norms: . We can see the following Strichartz estimates.
admissible, then we have the following estimates: Oh [43] established the local well-posedness for a class of nonlinear Schrödinger equations with a general potential including the harmonic potential.Here, we review the following the local well-posedness of Cauchy problems (1.1) and (1.2) in Σ; see the study by Cazenave [12] for a review.
is the solution of Cauchy problems (1.1) and (1.2).Then, , and there exists a positive constant C 0 such that for all times t, ) where Q x ( ) is the unique positive radially symmetric solution of equation (1.14).
Proof.From the local well-posedness, is well-defined and continuous for all times t.After some computations, we have After integrating, we deduce that (2.5) holds for all times t.
Concentration of blow-up solutions for the G-P equation  5 4 , we obtain that J t ( ) has the following explicit form: is arbitrary, we deduce that for all times ) .Theorem 1.2 is a consequence of the profile decomposition of the bounded sequences in Proposition 1.1.The author has proved Proposition 1.1 in his PhD thesis [62].To keep the self-contained, here, we give the proof of Proposition 1.1 in detail.
Proof of Proposition 1.1.According to the fact that any bounded sequence has a weakly convergent subsequence, we shall use the weakly limit points as the profiles to decompose the sequence { }, and μ v ( ) be the set of functions obtained as weak limits of subsequences of the translated . Then, there exist a subsequence And up to extracting a subsequence, v n can be written as follows: and also (1.6) are true.More precisely, the construction of { } are given in the following.Indeed, if = η v 0 ( ) , we can take = V 0 j for all j, otherwise, we choose . By the definition of μ v ( ), there exists a subsequence x n 1 of D such that up to extracting a subsequence, Take the transformation . By Brézis and Lieb's lemma, we obtain 1 and repeat the same process.We can find . By Brézis and Lieb's lemma, we obtain . This is a contradiction, and (3.3) is true.Thus, an argument of iteration and orthogonal extraction allows us to construct the families { } satisfying the afore- mentioned claims.From the convergence of ( ) Therefore, we prove that (1.6) and (3.1) are true.Next, we shall prove that (1.5) for all ∈ r p, where ˆdenotes the Fourier transformation.We can decompose v n l : where * is the convolution and δ is the Dirac function.It follows from the Sobolev embedding Yielding by the assumption that 2* and the assumption that Moreover, from the Hölder interpolation inequality, we deduce that for all l , according to the definition of χ K , we see that for all In view of the definition of μ v ( ), we see that By using the Parseval identity and the Hölder inequality, we deduce that with > ε 0 sufficiently small.We see that as → +∞ l .Injecting this into (3.6),we deduce that ) vanish.More precisely, we shall prove that for all ≠ j k, . Indeed, by using the Hölder inequality and Sobolev embedding theorem, we obtain Concentration of blow-up solutions for the G-P equation  7 By the orthogonality condition: for every ≠ k j, Collecting (3.9) and (3.10), we prove that (1.7) is true.This completes the proof.□ At the end of this section, we shall apply the profile decomposition in Proposition 1.1 and the variational argument to finish the proof of Theorem 1.2.
Proof of Theorem 1.2.First, we define the variational problem and study the properties of the corresponding minimizers.Define By the assumption: Then, by the Hölder interpolation estimate and Sobolev embedding, we deduce that Hence, the functional J v ( ) has a positive lower bound, and the variational problem (3.11) is well-defined.Now, we investigate the Euler-Lagrange equation to variational problem (3.11) Furthermore, from (3.13), we see that any minimizer of J v ( ) is a corresponding solution of equation (1.8).Since any smaller H ˙sc -norm solution would correspond to a lower value of J v ( ), the Pohozhaev identities show that it is in fact a minimal H ˙sc -norm solution of equation (1.8).Therefore, to prove the existence of a ground state, it suffices to prove the existence of a minimizer for J v ( ).Next, we use Lemma 1.1 to prove the existence of a minimizer for J v ( ), which implies the existence of a ground state solution to equation (1.8).Without loss of generality, we can inquire the minimizing sequence to variational problem (3.11) satisfying 1 and 1 as .
does not satisfy (3.14), then one can take the following transformation: . The new sequence By applying Lemma 1.1, we see that v x n ( ) can be decomposed by (1.6), we deduce that . Thus, we obtain the following estimate.
On the other hand, for all ≥ j 1, we deduce that ‖ ‖ for every ≥ j 1.Hence, there exists only one term ≠ V 0 . Therefore, we prove that V j 0 is the minimizer of J v ( ).It follows from (3.13) that V j 0 is the solution of Because ⋆ J is a fixed real number, after rescaling, we can check that (3.17) and (1.8) are equivalent.Hence, the existence of ground state solutions of (3.17) implies that of (1.8).
Finally, to give the exact expression of ⋆ J by W , we take , where

| |
, where W is a ground state solution of (1.8).Since ‖ ‖ .This completes the proof of Theorem 1.2.□ Remark 3.1.In Theorem 1.2, we prove the existence of nontrivial solutions of (1.8), and by the rescaling trick, we can see that all solutions of (1.8) has the same H ˙sc -norm, which is a fixed number.And we call the nontrivial solution of equation (1.8) with the same H ˙sc -norm is a ground state solution.In particular, when = s 0 c , the new sharp Gagliardo-Nirenberg inequality (1.9) degenerates to inequality (1.10).

Blow-up rate and concentration
By the Strichartz estimates in Lemma 2.1 generated by the Schrödinger operator with a harmonic potential: | | , we obtain the following lower bound of blow-up rate for Cauchy problems (1.1) and (1.2), in terms of Cazenave's arguments in [12].
Proof of (i) in Theorem 1.3.First, define the Schrödinger operator with a harmonic potential: e itH , where

( ) ( ) ( )
. By applying the Strichartz estimates (2.2)-(2.4) in Lemma 2.1, we deduce that , ; Then, by the conservation of energy, there exists C 0 defined in (2.5), and > C 0 such that By applying the Leibniz's law and Young inequality, we deduce that for all Now, by injecting (4.3) and (4.4) into the last term in (4.2), we obtain that for all ∈ t T 0, [ ), Concentration of blow-up solutions for the G-P equation  9 Finally, we denote . Substitute (4.5) into (4.2).There exists a positive constant ).Now, take = τ τ 0 in (4.6), yields is true for all < < < +∞ t T 0 and so (1.11) is true.□ To prove (ii) in Theorem 1.3, we prove the following refined compactness results, which is the applications of the profile decomposition of bounded sequences in ∩ H H ˙ṡ .
then, there exists a sequence such that up to a subsequence and W is the ground state solution of equation (1.8).
Proof.By extracting a subsequence, we may replace limsup in the assumption in Proposition 4.1 by lim.According to the profile decomposition in Proposition 1.1, the sequence = +∞ v n n 1 { } can be written up to a subse- quence as follows: And (1.5), (1.6) and (1.7) are true.Moreover, by (1.5), we deduce that .By using the new Gagliardo-Nirenberg inequality (1.9), we deduce that From the hypothesis (4.7) and (1.6), we have {‖ ‖ } is attained.Thus, by injecting these estimates into (4.11),we . This implies that there exists a satisfying the lower bound.Next, we will prove that which implies the sequence { } and the function V j 0 now fulfill the condition of Proposition 4.1.Indeed, by a change of variables in (4.9), we have By applying the pairwise orthogonality of family = +∞ x n j n 1 { } to (4.13), we see that as for ≠ j j 0 .Denote v ˜l to be the weak limit of v ˜nl and take weak limit in (4.13) as → +∞ n .We have Finally, by using (1.5), we deduce that , and by the uniqueness of weak limits, we obtain: = v ˜0 l for ≥ l j 0 .Then, (4.12) is true, so is (4.8).This completes the proof.□ At the end of this section, we shall finish the proof of Theorem 1.3 by the refined compactness result in Proposition 4.1.
, where T is the blow- up time.Since ψ t ( ) is the blow-up solutions to Cauchy problems (1.1) and (1.2) at the finite time , as → +∞ n , where W is the ground state solution of equation (1.8).We take the scaling transformation ‖ ‖ , there exists a subsequence t n n 1 { } is an arbitrary sequence approaching T , by letting → +∞ K , we see that Furthermore, because the function is continuous and goes to 0 at infinity for every Then, we can obtain (1.12) by injecting the aforementioned identity into (4.17).□

Dirac function concentration
In this section, we will continuously investigate the dynamical properties of blow-up solutions to Cauchy problem (1. Here, we use the definition of Dirac function = δ x 0 to obtain that (5.5) is true in distribution.From (2.5) , and there exists an

x x ψ t x x y t x ψ t x y t x ψ t y y t y y ψ t y y t y t y ψ t y y t y y ψ t y y t y y ψ t y y t y t y
B (5.7) Due to > M M 0 , there exists > η 0 (small enough), ).From (5.5) and (5.7), we can complete the proof of Claim (5.6).By (2.7), we deduce that for all ∈ . By combining this with (5.6), we obtain where T is the blow-up time.By comparing (2.6) with (5.8), we see that = − → y t y lim t T

( )
. Next, we shall prove Claim (5.10) by the following two cases.
Case a: = y 0 1 . For any > ε 0, we decompose that for all ∈ t T 0, [ ) where A 1 and A 2 are two positive constants, which are determined below.For the first term I , by using (5.17)Substitute (5.16) and (5.17) into (5.15).We have

| | | |
By combining the aforementioned estimations with (5.14), we can deduce that (5.10) is true in this case.Finally, we return the proof of (ii) in Theorem 5.2.After some computation, we deduce that  Remark 5.3.Theorem 5.2 parallels the existing results in the study by Merle [39] for the minimal mass blow-up solutions to the canonical nonlinear Schrödinger equation (1.3).But, as we will see, our argument is by exploring the variational characteristic of (1.14) and the order of infinitesimal term ( ) | | | ( )| as the time t goes to the blow-up time T , and it is a very simple and direct way to study the delicate profile of blow-up solutions for the nonlinear Schrödinger equation with a potential, which has potential applications for the nonlinear Schrödinger equations without a translation from themselves to the canonical nonlinear Schrödinger equation (1.3).Of course, these results rely in an essential way on those results.
6) is true by the continuity of ψ t x , ( ). □3 A new variational structureIn the sequel, we set = − − ) Therefore, we can obtain (1.5) by collecting (3.4),(3.5),and(3.8).Finally, we shall prove (1.7).Without loss of generality, we assume every V j is continuous and compactly supported.By the inequality: Then,(5.4) holds for all < < t T 0 .This completes the proof.□Concentration of blow-up solutions for the G-P equation  15 Oh established the local well-posedness in the natural energy space Σ: Let ≥ Concentration of blow-up solutions for the G-P equation  11 [60]d (1.2) in the L 2 -critical case: = + p 2 D 4 .Zhu et al.[60]proved the following limiting result.goes to the blow-up time T , we obtain a delicate profile of blow-up solutions to Cauchy problems (1.1) and (1.2), as follows.
Collecting(5.11),(5.12), and (5.13), we derive that for any > ε 0, there exists For I and III , we can estimate them as in Case a.For sufficiently small A 1 and sufficiently large A 2 , we deduce that for all ∈