Blow-up results for systems of nonlinear Schr\"odinger equations with quadratic interaction

We establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our analysis extends to general NLS systems with quadratic interactions, and it also provides improvements of known results in the radial case.


Introduction
In this paper, we investigate the existence of blowing-up solutions for the Cauchy problem for the following system of nonlinear Schrödinger equations with quadratic interaction where the wave functions u, v : R × R d → C are complex scalar functions, the parameters m, M are two real positive quantities, and λ, µ ∈ C are two complex coupling constants.
Multi-components systems of nonlinear Schrödinger equations with quadratic-type interactions appear in the processes of waves propagation in quadratic media. They model, for example, the Raman amplification phenomena in a plasma, or they are used to describe other phenomena in nonlinear optics. We refer the readers to [6,7,21,22] for more insights on these kind of physical models.
In the case of the so-called mass-resonance condition, namely provided that the condition is satisfied, the system (1.1) can be viewed, see [14], as a non-relativistic limit of the following system of nonlinear Klein-Gordon equations 1 2c 2 m ∂ 2 t u − 1 2m ∆u + mc 2 2 u = −λvu, 2 , as the speed of light c tends to infinity.
To the best of our knowledge, the first mathematical study of the system (1.1) is due to Hayashi, Ozawa, and Tanaka [14], where, among other things, they established the local well-posedness of the system (1.1), and they proved that, in order to ensure the conservation law of the total charge, namely the sum (up to some constant) of the L 2 norm of u and v, it is natural to consider the condition ∃ c ∈ R\{0} such that λ = cµ. (1.3) Moreover, if we assume that λ, µ satisfy (1.3) for some c > 0 and λ, µ = 0, by the change of variablẽ the system (1.1) can be written (by dropping the tildes) as i∂ t u + ∆u = −2vu, i∂ t v + κ∆v = −u 2 , (1.4) where κ = m M is the mass ratio. Note that κ = 1 2 in the mass-resonance case (1.2). The system (1.4) satisfies the conservation of mass and energy defined respectively by For the purpose of our paper, we define the kinetic energy 5) and the potential energy by hence we rewrite the total energy as E(u(t), v(t)) = 1 2 T (u(t), v(t)) − P (u(t), v(t)).
We also introduce the following functional defined in terms of T and P : Even if the we will use G evaluated at time-dependent solutions, it is worth mentioning that G is the Pohozaev functional which is strictly related to the time-independent elliptic equations (1.9) and (1.10) below. Another crucial property of (1.4) is that (1.4) is invariant under the scaling u λ (t, x) := λ 2 u(λ 2 t, λx), v λ (t, x) := λ 2 v(λ 2 t, λx), λ > 0. (1.8) A direct computation gives This shows that (1.8) leaves theḢ γc -norm of initial data invariant, where According to the conservation laws of mass and energy, (1.4) is called mass-critical, mass and energy intercritical (or intercritical for short), and energy-critical if d = 4, d = 5, and d = 6, respectively.
In the present paper, we restrict our attention to the dimensions d = 4, 5, 6, and we are interested in showing the formation of singularities in finite or infinite time for solutions to the initial value problem associated to (1.4), with initial data (u, v)(0, ·) =: (u 0 , v 0 ) ∈ H 1 (R d ) × H 1 (R d ).
As well-known, the existence of blowing-up solutions to the Schrödinger-type equations is closely related to the notion of standing wave or static (in the energy-critical case) solutions. Therefore, before stating our main results, we recall some basic facts about the existence of ground states for (1.4).
First of all, we recall that by standing waves solutions we mean solutions to (1.4) of the form where φ, ψ are real-valued functions satisfying (1.9) In [14], Hayashi, Ozawa, and Tanaka showed the existence of ground states related to (1.9), i.e. non-trivial solutions to (1.9) that minimizes the action functional over all non-trivial solutions to (1.9). It is worth mentioning that this existence result holds whenever d ≤ 5, and not only for d = 4, 5. When d = 6, i.e. the energy-critical case, (1.4) admits a static solution of the form (1.10) The existence of ground states related to (1.10) was shown in [14] (see also [25,Section 3]). Here by a ground state related to (1.10), we mean a non-trivial solution to (1.10) that minimizes the energy functional over all non-trivial solutions of (1.10).

Main results
We are now ready to state our first result about the blow-up of solutions in the mass and energy intercritical case in anisotropic spaces. To this aim, we introduce some notation. Denote Here Σ d stands for the space of cylindrical symmetric functions with finite variance in the last direction. We also introduce the following blow-up conditions: As for the usual Schrödinger equation, the conditions expressed in (BC 5d ) are the counterpart of the conditions in the dichotomy leading to global well-posedness & scattering ((SC 5d )) or blow-up ((BC 5d )). In the energy critical case, the previous conditions in (BC 5d ) will be replaced by analogous inequalities, see (BC 6d ) below. Since in this paper we are concerned only with the blow-up dynamics of solutions to (1.4), we will not use the modified conditions for the scattering theory.
2.1. Intercritical case. Our first result concerns a finite time blow-up for (1.4) in the intercritical case d = 5.
Let us give some comments on the previously known blow-up results for the system (1.4). The formation of singularities in finite time for negative energy and radial data was shown by Yoshida in [29], while for non-negative energy radial data a proof was recently given by Inui, Kishimoto, and Nishimura. Specifically, they proved in [19] the blow-up for radial initial data satisfying By a variational characterization, we show in Lemma 3.1 that (BC 5d ) and (2.2) are indeed equivalent. Thus a version of Theorem 2.1 for radial solutions would be an interchangeable restatement of the result obtained in [19].
Despite our approach relies on the classical virial identities, we need to precisely construct suitable cylindrical cut-off functions enabling us to get enough decay (by means of some Sobolev embedding for partially radial functions) to close our estimates. With respect to the classical NLS equation, we will use an ODE argument instead of a concavity argument to prove our results, by only using the first derivative in time of suitable localized quantity, see Section 3. We refer the reader to the early work of Martel [23] in the context of the NLS equation in anisotropic spaces, and the more recent papers [3,16,17]. See also our recent paper [1] in the context of NLS system with cubic interaction.
For sake of completeness, we report now known blow-up and long time dynamics results for (1.4) in the intercritical case.
If κ = 1 2 , Hamano, Inui, and Nishimura [13] established the scattering for radial data below the mass-energy threshold. The proof is based on the concentration/compactness and rigidity scheme in the spirit of Kenig and Merle [20]. Wang and Yang [28] extended the result of [13] to the non-radial case provided that κ belongs to a small neighbourhood of 1 2 . Their proof made use of a recent method of Dodson and Murphy [11] using the interaction Morawetz inequality. Noguera and Pastor [26] proved that if (u 0 , v 0 ) ∈ H 1 × H 1 satisfies (SC 5d ), then the corresponding solution to (1.4) exists globally in time.
Remark 2.1. From a pure mathematical perspective, distinguishing the cases κ = 1 2 and 0 < κ = 1 2 plays a role in the virial identities related to (1.4). Under the mass-resonance condition, namely κ = 1 2 , some terms in the virial identities disappear, and the study of the dynamics of solutions is easier due to these cancellations. This is no more the case in the non-mass-resonance setting, i.e. when κ = 1 2 . We refer the reader to [25,Introduction] for an exhaustive list of references in which the effects of the mass and non-mass resonance conditions on the dynamics of solutions to systems similar to (1.4) were studied.

2.2.
Energy-critical case. Our next Theorem deals with a blow-up result in the energy-critical case d = 6.
Then the corresponding solution to (1.4) blows-up in finite time.
It is worth mentioning that finite time blow-up with negative energy radial data was established in [29], while for non-negative energy radial data, the blow-up result was shown in [19] for data satisfying Since we will prove in Lemma 3.3 that (BC 6d ) is equivalent to (2.3), our result restricted to a radial framework, would be equivalent to the one in [19].
Remark 2.2. If κ = 1 2 , the blow-up result with negative energy and finite variance data was shown in Hayashi, Ozawa, and Tanaka, see [14].
for all t ≥ t 0 , where C > 0 and t 0 ≫ 1 depend only on κ, M (u 0 , v 0 ), and E(u 0 , v 0 ). A similar statement holds for negative times.
Under the assumption of Theorem 2.3, the blow-up or grow-up result along one time sequence was proved in [19,Theorem 1.2]. More precisely, if T * = ∞, then there exists a time sequence By performing a more careful analysis, our argument yields to a stronger result with respect to the one in [19]. Indeed, we are able to show a growth rate for the kinetic energy of the form (2.4) which in turn implies the grow-up result along an arbitrary diverging sequence of times. We would like to mention that this grow-up result along any diverging time sequence, is also an interesting open problem related to the usual mass-supercritical focusing cubic 3D NLS, see the weak conjecture of Holmer and Roudenko in [15]. Remark 2.3. In the case κ = 1 2 and for radial data with negative energy, the finite time blow-up was shown by the first author in [8]. For the long time dynamics in the mass-critical case we refer to [18].
We give now the following blow-up or grow-up result for anisotropic solutions to (1.4).
Then the corresponding solution to (1.4) either blows-up forward in finite time, i.e. T * < ∞, or T * = ∞ and there exists a time sequence t n → ∞ such that (u(t n ), v(t n )) H 1 ×H 1 → ∞ as n → ∞. If we assume κ = 1 2 , then either T * < ∞ or T * = ∞ and there exists a time sequence t n → ∞ such that ∂ 4 u(t n ) L 2 → ∞ as n → ∞. A similar statement holds for negative times.

2.4.
Extensions to a general system of NLS with quadratic interactions. We conclude this section by listing some extensions of the previous Theorems for general NLS systems with quadratic interactions.
In dimension d = 5 and d = 6, namely in the mass-supercritical and the energy-critical case, respectively, the results above can be extended -provided that some structural hypothesis are satisfied -to the following initial value problem for general system of NLS with quadratic interactions: where u j : R × R d → C, the parameters a j , b j , c j are real coefficients satisfying a j > 0, b j > 0 and c j ≥ 0, and the functions f j grow quadratically for all j = 1, . . . , N . More precisely, under the assumptions (H1)-(H8) in [25], Theorems 2.1 and 2.2 can be stated for (2.5) as well, with the necessary modifications. In particular, the set of conditions (H1)-(H8) in [25] (see also [24]) ensure that (2.5) is local well-posed, there exist ground states (along with stability and instability properties), and the mass and the energy are conserved. Here the mass is defined by where the real parameters s j > 0 satisfy and u is the compact notation for (u 1 , . . . , u N ). The energy is instead defined by where F : C N → C is such that f j = ∂z j F + ∂ zj F for any j ∈ {1, . . . , N }.
In d = 5, we denote by φ = (φ 1 , . . . , φ N ) the ground state related to the system of elliptic equations i.e. φ is a non-trivial real-valued solution of (2.6) that minimizes the action functional over all non-trivial real-valued solutions to (2.6), where Under the assumptions (H1)-(H8) in [25] , ground states related to (2 If we denote by G(ω, c) the set of ground states related to (2.6), where c = (c 1 , . . . , c N ), then G(ω, c) = ∅ provided that (2.7) is satisfied. In particular, G(1, 0) = ∅. Moreover, the following Gagliardo-Nirenberg inequality is achieved by a ground state φ ∈ G(ω, c). We refer the reader to [24, Section 4] for more details on ground states related to (2.6). By adapting the arguments presented in this paper, we can prove the result in Theorem 2.1 provided that we replace (BC 5d ) with Similarly, in d = 6, we can prove the result in Theorem 2.2 provided that we replace (BC 6d ) with Here ϕ is a ground state related to (2.9) if it is a non-trivial real valued solution to (2.9) that minimizes the functional E 0 over all non-trivial real-valued solutions of (2.9). Note that blow-up results similar to Theorems 2.1 and 2.2 for radial solutions to (2.5) were established in [25]. Thus our extensions are for anisotropic solutions.
As pointed-out in [25], the non-mass-resonance condition 0 < κ = 1 2 for (1.4) in Theorems 2.1 and 2.2, corresponds to the following analogous condition for (2.5): In the mass-critical case d = 4, we have the following blow-up results for (2.5).
The proof of this result follows from a similar argument as that for Theorem 2.3 using a refined localized virial estimates (see Lemma A.1). Our result is new even under the mass-resonance condition (2.11). Note that the finite time blow-up for (2.5) in the mass-critical case d = 4 was proved in [24,Theorem 5.11] only for finite variance solutions.
Then the corresponding solution to (2.5) either blows-up forward in finite time, i.e. T * < ∞, or T * = ∞ and there exists a time sequence t n → ∞ such that Similarly to Theorem 2.4, the proof of Theorem 2.6 is based on refined localized virial estimates for anisotropic solutions to (2.5) (see Lemma A.2). Therefore, we will omit the details of the proof.
The paper is organized as follows. In Section 3, we recall some useful properties of ground states related to (1.9) and (1.10). We also prove some variational estimates associated to blow-up conditions given in Theorems 2.1 and 2.2. Section 4 is devoted to various localized virial estimates for radial and anisotropic solutions to (1.4). The proofs of our main results are given in Section 5. Finally, we prove in Appendix A some localized virial estimates for the general system (2.5) of NLS with quadratic interactions.

Variational analysis
In this section, we report some useful properties of ground states related to (1.9) and (1.10). Then we use them to get some a-priori uniform-in-time estimates for the Pohozaev functional evaluated at the solutions to the corresponding time-dependent equations.
This result was first shown by Hayashi It follows that .
When d = 4, we have Although the uniqueness (up to symmetries) of ground states related to (1.9) is not known yet, (3.3) shows that the mass of ground states does not depend on the choice of a ground state (φ, ψ).
In the case d = 5, we have and In particular, the quantities do not depend on the choice of a ground state (φ, ψ).
When d = 6, we have the following Sobolev type inequality: It was shown in [25,Theorem 3.3] that the sharp constant in (3.7) is achieved by a ground state (φ, ψ) related to (1.10), i.e. .
Using the following identity T (φ, ψ) = 3P (φ, ψ), we see that and This shows in particular that E(φ, ψ) and T (φ, ψ) do not depend on the choice of a ground state (φ, ψ).

Variational estimates.
In this section, we characterize the blow-up region defined in (BC 5d ) (see (BC 6d ) for the energy critical case) in terms of the sign of the Pohozaev functional G defined in (1.7). For similar analysis in the context of the classical NLS equation, we refer to our previous works [2,10].
Proof. The proof is similar to that of Lemma 3.2 using (3.8) and (3.9). We thus omit the details.

Localized virial estimates
In this section we prove the preliminary and fundamental estimates we need for the proof of our main Theorems. We start with the following virial identity (see e.g. [28, (4.34)]). (4.1) Then we have for all t ∈ [0, T * ), The above identity can be checked by formal computations. The rigorous proof can be done by performing a standard approximation trick (see e.g. [4, Section 6.5]).
Remark 4.1. From now on we denote r = |x|.
where G is as in (1.7). (2) If ϕ is radially symmetric, then using the fact that where r = |x|. In particular, we have (3) If ϕ is radial and (u, v) is also radial, then (4) Let d ≥ 3 and denote x = (y, x d ) with y = (x 1 , . . . , x d−1 ) ∈ R d−1 and x d ∈ R. Let ψ : R d−1 → R be a sufficiently smooth and decaying function. Set ϕ(x) = ψ(y) Let χ : [0, ∞) → [0, ∞) be a sufficiently smooth function satisfying Given R > 1, we define, by rescaling, the radial function ϕ R : In the mass-critical case, we have the following refined (with respect to the one in [19]) radial localized virial estimate.
for some constant C > 0 depending only on κ and M (u 0 , v 0 ), where Proof. By Item (3) of Remark 4.1, we have for all t ∈ [0, T * ), (4.6) By the fact that ∆ 2 ϕ R L ∞ R −2 together with the conservation of mass, we get the decay Furthermore, by using that ϕ ′′ R (r) ≤ 2, and by noting that G(u(t), v(t)) = 2E(u(t), v(t)) if d = 4, (4.6) can be controlled by We estimate where we have used the conservation of mass in the last estimate. Note that θ 2,R (x) = 0 for |x| ≤ R. As ∇θ 2,R L ∞ 1, the conservation of mass implies that It follows that The proof is complete.
Next we derive localized virial estimates for cylindrically symmetric solutions. To this end, we introduce and set ϕ R (x) := ψ R (y) + x 2 d .
for some constant C > 0 depending only on d, κ, and M (u 0 , v 0 ).
We also have the following refined localized virial estimate which will be used in the mass-critical problem.  Proof. Estimating as in the proof of Lemma 4.3, we have By the conservation of mass, we see that By the Hölder's inequality, we havê The first term is treated in (4.12). For the second term, as ϑ 2,R (y) = 0 for |y| ≤ R, we use the radial Sobolev embedding (4.13) to havê where we have used the conservation of mass and ∇ϑ 2,R L ∞ x 1 to get the last estimate. Collecting the above estimates, we prove (4.15).

Proof of the main results
We are now able to prove the main results stated in Section 2.
5.1. The intercritical case. The proof of Theorem 2.1 is done by performing an ODE argument, by using the a-priori estimates we proved in the previous Section.
Proof of Theorem 2.1. Let (u 0 , v 0 ) ∈ Σ 5 × Σ 5 satisfy (BC 5d ). We will show that T * < ∞. Assume by contradiction that T * = ∞. By Lemma 3.2, there exist positive constants ε and c such that for all t ∈ [0, ∞). On the other hand, by Lemma 4.3, we have for all t ∈ [0, ∞), where ϕ R is as in (4.3) and M ϕR (t) is as in (4.1). It follows from (5.1) and (5.2) that for all t ∈ [0, ∞), By choosing R > 1 sufficiently large, we get for all t ∈ [0, ∞). Integrating the above inequality, we see that M ϕR (t) < 0 for all t ≥ t 0 with some t 0 > 0 sufficiently large. We infer from (5.3) that for all t ≥ t 0 . On the other hand, by the Hölder's inequality and the conservation of mass, we have , v(t)).
We see that z(t) is non-decreasing and non-negative. Moreover, For t 1 > t 0 , we integrate (5.7) over [t 1 , t] to obtain , ∀t ≥ t 1 .
This shows that z(t) → +∞ as t ր t * , where In particular, we have M ϕR (t) ≤ −Az(t) → −∞ as t ր t * . Thus the solution cannot exist for all time t ≥ 0. Therefore it must blow-up in finite time. ✷

5.2.
The energy-critical case. The proof is done by an ODE argument as well, similarly to the intercritical case.
Proof of Theorem 2.2. The proof is similar to that of Theorem 2.1 using (3.20) and (4.9). Thus we omit the details. If T * < ∞, then we are done. Suppose that T * = ∞. We will show that there exists a constant C > 0 depending only on κ, M (u 0 , v 0 ), and E(u 0 , v 0 ) such that for all t ≥ t 0 , where t 0 ≫ 1, namely that (2.4) holds true. Let ϕ R be as in (4.3) and M ϕR (t) as in (4.1). By Lemma 4.2 and the conservation of energy, we have for all t ∈ [0, ∞), where θ 1,R and θ 2,R are as in (4.5).
It remains to find a suitable cut-off function χ (as defined in (4.2)) so that (5.8) holds. For the choice of such a function, we are inspired by [27]. Let It is easy to see that χ satisfies (4.2). Define ϕ R as in (4.3). We will show that (5.8) is satisfied for this choice of ϕ R . Indeed, we have θ 1,R (r) = 2 − ϕ ′′ R (r) and where the latter follows from the fact that We infer, from the definition of ϕ R , that By choosing R > 1 sufficiently large, we see that (5.8) is fulfilled. • When r > (1 + 1/ √ 3)R, we see that ζ ′ (r/R) ≤ 0, so θ 1,R (r) = 2 − ϕ ′′ R (r) ≥ 2. We also have θ 2,R (r) ≤ C for some constant C > 0. Thus by choosing R > 1 sufficiently large, we have (5.8).
It follows that, for all t ∈ [0, ∞), Integrating the above inequality, there exists t 0 > 0 sufficiently large so that V ϕR (t 0 ) < 0 which is impossible.
By the conservation of mass, we have Thus we get where θ 1,R and θ 2,R are as in (4.5). By the assumption (H6) in [25], the radial Sobolev embedding and the conservation of mass, we estimate Reˆθ 2,R F ( u(t))dx ≤ˆθ 2,R |F ( u(t))|dx Thanks to the conservation of mass and the fact that ∇θ 2,R L ∞ 1, we have ∇(θ 2,R u j (t)) L 2 θ 2,R ∇ j u(t) L 2 + 1 which implies that for some constant C > 0 depending only on κ, a = (a 1 , . . . , a N ), and M( u 0 ), where ϑ 1,R and ϑ 2,R are as in (4.16).