Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with $\chi^3$ nonlinear response

We study the asymptotic dynamics for solutions to a system of nonlinear Schr\"odinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.


Introduction
In this paper, we consider the Cauchy problem for the following system of nonlinear Schrödinger equations with cubic interaction with initial datum (u, v)| t=0 = (u 0 , v 0 ). Here u, v : R × R 3 → C, u 0 , v 0 : R 3 → C, and the parameters γ, µ are strictly positive real numbers. The system (1.1) is the dimensionless form of a system of nonlinear Schrödinger equations as derived in [29] (see also [30]), where the interaction between an optical beam at some fundamental frequency and its third harmonic is investigated. More precisely, from a physical point of view, (1.1) models the interplay of an optical monochromatic beam with its third harmonic in a Kerrtype medium (we refer to [28] for the latter terminology, as well as for a sketch of the derivation of (1.1)).
Models such as in (1.1) arise in nonlinear optics in the context of the so-called cascading nonlinear processes. These processes can indeed generate effective higher-order nonlinearities, and they stimulated the study of spatial solitary waves in optical materials with χ 2 or χ 3 susceptibilities (or nonlinear response, equivalently).
Let us mention, following [10], the difference between χ 2 (quadratic) and χ 3 (cubic) media. The contrast basically reflects the order of expansion (in terms of the electric field) of the polarization vector, when decomposing the electrical induction field appearing in the Maxwell equations as the sum of the electric field E and the polarization vector P. Indeed, for "small" intensities of the electric field, the polarization response is linear, while for "large" intensities of E, the vector P has a non-negligible nonlinear component, denoted by P nl . Thus, when considering the Taylor expansion for P nl , one gets the presence of (at least) quadratic and cubic terms whose coefficients χ j , which depend on the frequency of the electric field E, are called j-th optical susceptibility. For j = 2, 3, they are usually denoted by χ 2 and χ 3 . Therefore quadratic media arise from approximation of the type P nl ∼ χ 2 E 2 , and similarly one can define cubic media. The so-called non-centrosymmetric crystals are typical examples of χ 2 materials. Moreover, it can be shown, see [15], that isotropic materials have χ 2n = 0 susceptibility, namely even orders of nonlinear responses are zero. In the latter case, the leading-order in the expansion of P nl is cubic, and these kind of isotropic materials are called Kerr-materials. See the monographs [5,15,31] for more discussions. In addition, we refer to [1,6,7,10,19,24,29,30,36], and references therein, for more insights on physical motivations and physical results (both theoretical and numerical) about (1.1) and other NLS systems with cubic and quadratic interactions. Models as in (1.1) are therefore physically relevant, and they deserve a rigorous mathematical investigation. In particular, we are interested in qualitative properties of solutions to (1.1).
Our main goal is to understand the asymptotic dynamics of solutions to (1.1), by establishing conditions ensuring global existence and their long time behavior, or leading to formation of singularities in finite time. Let us mention since now on, that once the Strichartz machinery has been established, and this is nowadays classical, local well-posedness of (1.1) at the energy regularity level (i.e. H 1 (R 3 ), mathematically speaking) is relatively straightforward to get (see below for a precise definition of the functional space to employ a fixed point argument).
The dynamics of solution of NLS-type equation is intimately related to the existence of ground states (see below for a more precise definition). The analysis of solitons is a very important physical problem, and the main difference between χ 2 media and χ 3 media, is that, in the latter case, the cubic nonlinearity is L 2 supercritical, while in the former quadratic nonlinearities are L 2 subcritical. The last two regimes dramatically reflect the possibility for the problem to be globally well-posed, and the stability/instability properties of the solitons are different. See [10] for further discussions, and a rigorous analysis for solitons in quadratic media.
Regarding system (1.1), existence of ground states and their instability properties were established in a recent paper by Oliveira and Pastor, see [28]. Our aim is to push forward their achievements to obtain a qualitative description of solutions to (1.1), by giving sharp thresholds, defined by means of quantities linked to the ground state, are sufficient to guarantee a linear asymptotic dynamics for large time (i.e. scattering) or finite time blow-up of the solutions.
Let us start our rigorous mathematical discussion about (1.1). The existence of solutions is quite simple to obtain. As said above, it is well-known that (1.1) is locally well-posed in H 1 (R 3 ) × H 1 (R 3 ), (see e.g., [8]). More precisely, for (u ) for any Strichartz L 2 -admissible pair (q, r), i.e., 2 q + 3 r = 3 2 , for 2 ≤ r ≤ 6. See Section 2. In addition, the maximal times of existence obey the blow-up alternative, i.e., either T + = ∞, or T + < ∞ and lim tրT + (u(t), v(t)) H 1 (R 3 )×H 1 (R 3 ) = ∞, and similarly for T − . When T ± = ∞, we call the solution global. Solutions to (1.1) satisfy conservation laws of mass and energy, namely where It is worth introducing since now the Pohozaev functional and, for later purposes, we rewrite the functionals P (see (1.4)) by means of its density: namely The previous conservation laws can be formally proved by usual integration by part, then a rigorous justification of them can be done by a classical regularization argument, see [8].
In order to introduce other invariance of the equations, let us give the following definition.
Definition 1.1. We say that the initial-value problem (1.1) satisfies the mass-resonance condition provided that γ = 3.
For γ = 3, (1.1) has the Galilean invariance: namely, if (u, v) is a solution to (1.1), then is also a solution to (1.1) with initial data (e ix·ξ u 0 , e 3ix·ξ v 0 ). As, in this paper, we are interested in long time behavior of solutions to (1.1), let us recall the notion of scattering. Definition 1.2. We say that a global solution (u(t), v(t)) to (1.1) are linear Schrödinger propagators.
Note that the set of initial data such that solutions to (1.1) satisfy (1.8) is non-empty, as solutions corresponding to small H 1 (R 3 ) × H 1 (R 3 )-data do scatter (see Section 2). As already mention above, it is well-known that the dynamics of nonlinear Schrödinger-type equations is strongly related to the notion of ground states. Hence, we recall some basic facts about ground state standing waves related to (1.1). By standing waves, we mean solutions to where ω ∈ R is a frequency and (f, g) is a real-valued solution to the system of elliptic equations (1.10) It was proved by Oliveira and Pastor, see [28], that solutions to (1.10) exist, provided that Moreover, a non-trivial solution (φ, ψ) to (1.10) is called ground state related to (1.10) if it minimizes the action functional over all non-trivial solutions to (1.10). Under the assumption (1.11), the set of ground states related to (1.10) denoted by is not empty, where A ω,µ,γ is the set of all non-trivial solutions to (1.10). In particular, G(0, 3γ, γ) = ∅.
It was shown (see [28,Theorem 3.10] where (φ, ψ) ∈ G(0, 3γ, γ), then the corresponding solution to (1.1) exists globally in time. The proof of this result is based on a continuity argument and the following sharp Gagliardo-Nirenberg inequality This type of Gagliardo-Nirenberg inequality was established in [28,Lemma 3.5]. Note that in [28], this inequality was proved for real-valued H 1 -functions. However, we can state it for complexvalued H 1 -functions as well since P (f, g) ≤ P (|f |, |g|) and ∇(|f |) We are now in position to state our first main result. The following theorem provides sufficient conditions to have scattering of solutions. More precisely, for data belonging to the set given by conditions (1.13) and (1.14), solutions to (1.1) satisfy (1.8), for some scattering state (u ± , v ± ).
. Assume that the initial data satisfies (1.13) and (1.14). Provided that Our proof of the scattering results is based on the recent works by Dodson and Murphy [13] (for non-radial solutions) and [12] (for radial solutions), using suitable scattering criteria and Morawetz-type estimates. In the non-radial case, we make use of an interaction Morawetz estimate to derive a space-time estimate. In the radial case, we make use of localized Morawetz estimates and radial Sobolev embeddings to show a suitable space-time bound of the solution.
Let us highlight the main novelties of this paper, regarding the linear asymptotic dynamics. For the classical focusing cubic equation in H 1 (R 3 ), scattering (and blow-up) below the mass-energy threshold, was proved by Holmer and Roudenko in [17] for radial solutions, by exploiting the concentration/compactness and rigidity scheme in the spirit of Kenig and Merle, see [18]. The latter scattering result has been then extended to non-radial solution in Duyckaerts, Holmer, and Roudenko [14]. To remove the radiality assumption, a crucial role is played by the invariance of the cubic NLS under the Galilean boost, which enables to have a zero momentum for the soliton-like solution. As observed in Remark 1.1, equation (1.1) lacks the Galilean invariance unless γ = 3. Hence we cannot rely on a Kenig and Merle road map to achieve our scattering results, and we instead build our analysis on the recent method developed by Dodson and Murphy, see [12,13]. In the latter two cited works, Dodson and Murphy give alternative proofs of the scattering results contained in [14,17], which avoid the use of the concentration/compactness and rigidity method. They give a shorter proofs, though quite technical, based on Morawetz-type estimates. In our work, by borrowing from [12,13], we prove interaction Morawetz and Morawetz estimates for (1.1), and we prove Theorem 1.1 for non-radial solutions which do not fit the massresonance condition, as well as for radially symmetric solutions. In this latter case, instead, we only need (localized) Morawetz estimates, which are less involved with respect to the interaction Morawetz ones, as we can take advantage of the spatial decay of radial Sobolev functions.
Our second main result is about formation of singularities in finite time for solutions to (1.1). We state it for two classes of initial data. Indeed, besides the fact that these initial data must satisfy the a-priori bounds given by (1.13) and (1.16) -the latter (see below) replacing the condition (1.14) yielding to global well-posedness -they can belong either to the space of radial function, or to the anisotropic space of cylindrical function having finite variance in the last variable. The Theorem reads as follows.
we assume moreover that (1.13) holds and (1.16) If the initial data satisfy then the corresponding solution to (1.1) blows-up in finite time.
Let us now comment previous known results about blow-up for (1.1) and the one stated above, and highlight the main novelties of this paper regarding the blow-up achievements with respect to the previous literature.
In the mass-resonance case, i.e., γ = 3, and provided µ = 3γ = 9, the existence of finite time blow-up solutions to (1.1) with finite variance initial data was proved in [28, Theorems 4.6 and 4.8]. More precisely, they proved that if satisfying either E 9 (u 0 , v 0 ) < 0 or if E 9 (u 0 , v 0 ) ≥ 0, they moreover assumed that where (φ, ψ) ∈ G(0, 9, 3), then the corresponding solution to (1.1) blows-up in finite time. The proof of the blow-up result in [28] is based on the following virial identity (see Remark 3.3) where Using (1.17), the finite time blow-up result follows from a convexity argument. For the powertype NLS equation, this kind of convexity strategy goes back to the early work of Glassey, see [16], for finite variance solutions with negative initial energy. See the works by Ogawa and Tsutsumi [27] for the removal of the finiteness hypothesis of the variance, but with the addition of the radial assumption. See the already mentioned paper [17] for an extension to the cubic NLS up to the mass-energy threshold, of the results by Glassey, and Ogawa and Tsutsumi.
If we do not assume the mass-resonance condition, or we do not assume that µ = 3γ, the identity (1.17) ceases to be valid. Thus the convexity argument is no-more applicable in our general setting. The proof of Theorem 1.2 above relies instead on an ODE argument, in the same spirit of our previous work [11], using localized virial estimates and the negativity property of the Pohozaev functional (see Lemma 5.1). We point-out that our result not only extends the one in [28] to radial and cylindrical solutions, but also extends it to the whole range of µ, γ > 0. It worth mentioning that blow-up in a full generality, i.e. for infinite-variance solutions with no symmetric assumptions, is still an open problem even for the classical cubic NLS.
We conclude this introduction by reporting some notation used along the paper, and by disclosing how the paper is organized.
1.1. Notations. We use the notation X Y to denote X ≤ CY for some constant C > 0. When X Y and Y X (possibly for two different universal constants), we write X ∼ Y, or equivalently, we use the 'big O' notation O, e.g., with the usual modifications when either r or q are infinity. When q = r, we simply write and if q = r, we simply write . The L p (R 3 ) spaces, with 1 ≤, p ≤ ∞, are the usual Lebesgue spaces, as well as spaces W k,p (R 3 ) spaces, and their homogeneous versions, are the classical Sobolev spaces. To lighten the notation along the paper, we will avoid to write R 3 (unless necessary), as we are dealing with a threedimensional problem.
1.2. Structure of the paper. This paper is organized as follows. In Section 2, we state preliminary results that will be needed throughout the paper, and we will prove some coercivity conditions which play a vital role to get the scattering results. In Section 3, we introduce localized quantities, and we derive localized virial estimates, Morawetz and interaction Morawetz estimates which will be the fundamental tools to establish the main results. The latter a-priori estimates will be shown in both radial and non-radial settings. In Section 4, we give scattering criteria for radial and non-radial solutions. We eventually prove, in Section 5, the scattering results and the blow-up results, by employing the tools developed in the previous Sections. We conclude with the Appendixes A and B, devoted to the proofs of some results used along the paper.

Preliminary tools
In this section, we introduce some basic tools towards the proof of our main achievements. Specifically, we give a small data scattering result, as well as useful properties related to the ground states. We postpone the proof of some of the following results to the Appendix A.
2.1. Small data theory. We have the following small data scattering result, which will be useful in the sequel.
then the solution scatters forward in time.
Proof. See Appendix A.

Variational analysis.
We first recall some basic properties of ground states in G(0, 3γ, γ) and then show a coercivity condition (see (2.8)), which play a vital role to get scattering results. It was shown in [28, Lemma 3.5] that any ground state (φ, ψ) ∈ G(0, 3γ, γ) optimizes the Gagliardo-Nirenberg inequality (1.15), that is Using the Pohozaev identities (see [28,Lemma 3.4]) we have To employ some Morawetz estimates in the proof of the scattering theorem, we will also use the following refined Gagliardo-Nirenberg inequality.
We conclude this preliminary section by giving the following two coercivity results.
(1.14). Then the corresponding solution to (1.1) exists globally in time and satisfies for all t ∈ R.
Proof. See Appendix A.

Virial and Morawetz estimates
This section is devoted to the proof of virial-type, Morawetz-type, and interaction Morawetztype estimates, which will be crucial for the proof of the main Theorems 1.1 and 1.2.
3.1. Virial estimates. We start with the following identities. In what follows we use the Einstein convention, so repeated indices are summed.
Lemma 3.1. Let µ, β, γ > 0, and (u, v) be a H 1 -solution to (1.1). Then the following identities hold: where N is as in (1.6). In particular, we have Proof. See Appendix B.
A direct consequence of Lemma 3.1 is the following localized virial identity related to (1.1).
Lemma 3.2. Let µ, γ > 0, and ϕ : R 3 → R be a sufficiently smooth and decaying function. Let Then we have for all t ∈ (−T − , T + ), The following Corollary is easy to get.
Proof. See Appendix B.
We now aim to construct precise localization functions that we will use to get the desired main results of the paper. Let ζ : [0, ∞) → [0, 2] be a smooth function satisfying We define the function ϑ : For R > 0, we define the radial function ϕ R : We readily check that, ∀x ∈ R 3 and ∀r ≥ 0, We are ready to state the first virial estimate for radially symmetric solutions.
for some constant C > 0 depending only on µ, γ, and We rewrite, using G − K + 3P = 0, As ∆ 2 ϕ R L ∞ R −2 , the conservation of mass implies that The latter, together with ϕ ′′ R (r) ≤ 2 for all r ≥ 0, ∆ϕ R L ∞ 1, ϕ R (x) = |x| 2 on |x| ≤ R, and Hölder's inequality, yield where we have used the fact that (see (1.6)) To estimate the last term, we recall the following radial Sobolev embedding (see e.g., [9]): for a radial function f ∈ H 1 (R 3 ), we have (3.11) Thanks to (3.11) and the conservation of mass, we estimatê It follows that The proof is complete.
Proof. By (3.7), we have for all t ∈ (−T − , T + ), As ψ ′′ R (ρ) ≤ 2 and ∆ y ψ R L ∞ 1, the Hölder's inequality implies that which, by the conservation of mass, implies that sup z∈R u(t, z) 2 By the radial Sobolev embedding (3.11) with respect to the y-variable, we havê L 2 x . The latter and (3.15) give (3.14). The proof is complete.
Let R > 1 be a large parameter. We define the following radial functions where χ R (x) := χ x R and ω 3 is the volume of unit ball in R 3 . We also define the functions We collect below some properties of the above functions.
Here repeated indices are summed. Moreover, by using integration by parts, we readily see that We are able to prove the following interaction Morawetz estimates, which will play a fundamental role for the proof of the scattering theorem in the non-radial framework.

(3.35)
Hence ψ R − φ R is radial and non-negative, by the Cauchy-Schwarz inequality, we infer that On the other hand, as χ R is radial and non-negative, we have where Notice that B(u, v) is invariant under the gauge transformation (u(t, x), v(t, x)) → (u ξ (t, x), v ξ (t, x)) := (e ix·ξ u(t, x), e iγx·ξ v(t, x)) for any ξ ∈ R 3 . Indeed, we see that provided that the denominator is non-zero; otherwise we can define ξ(t, z, R) ≡ 0. With this choice of ξ, we haveˆχ Combining this with (3.36), we infer that Therefore, by the above identity, (3.26), (3.27), (3.29), and (3.34), we get Now, we consider (3.37). Sincê (3.42) Thus, substituting (3.42) in (3.37) and using Lemma 2.4 with χ R instead of Γ R , we see that there exists ν > 0 such that By the conservation of mass and the fact that ∆(χ R ) L ∞ R −2 , the absolute value of the second term in the right hand side can be bounded by This implies that By (3.19), the conservation of mass, (2.5), and Sobolev embedding, we have where we have used the fact that , Using (3.19), we see that where we have used the fact that We thus get Finally, as |γ − 3| < η and |∇Θ R (x)| R, we infer from the conservation of mass, (2.5), and Sobolev embedding that which shows (3.25) by choosing σ = ǫ, J = ǫ −3 , R 0 = ǫ −1 , T 0 = e ǫ −3 , and η = e −ǫ −3 . The proof is complete.
3.3. Morawetz estimates. Radial setting. We now turn our attention to the proof of the radial version of the Morawetz estimate which will be essential in the proof of the scattering theorem in the radially symmetric setting. In this context, we take advantage of the radial Sobolev embedding to get some spatial decay.

Scattering criteria
In this section, we give scattering criteria for solution to (1.1) in the spirit of Dodson and Murphy [12,13] (see also [33]). Let us start with the scattering criterion for non-radial solutions.
for some constant E > 0. Then there exist ǫ = ǫ(E) > 0 small enough and T 0 = T 0 (ǫ, E) > 0 sufficiently large such that if for any a ∈ R, there exists t 0 ∈ (a, a + T 0 ) such that (u(t), v(t)) then the solution scatters forward in the time.
Proof. By Lemma 2.1, it suffices to show that there exists T > 0 such that To prove (4.3), we first write By Sobolev embedding, Strichartz estimates, and the monotone convergence theorem, there exists T 1 > 0 sufficiently large such that if T > T 1 , then We take a = T 1 and T = t 0 , where a and t 0 are as in (4.2), we write where To estimate H 2 , we observe that
By choosing ǫ small enough, we get (4.5). By Sobolev embedding and Strichartz estimates, we see that which together with (4.2) and (4.5) imply On the other hand, we claim that In fact, we notice that which, by Strichartz estimates, implies Moreover, as we have from the dispersive estimate (A.2) and Young's inequality that By interpolation, we get which proves (4.7). Collecting (4.4), (4.6), and (4.7), we obtain (4.3), and the proof is complete.
Let us give now an analogous of the previous Criterion in the radial setting.
then the solution scatters forward in time.
Proof. Let ǫ > 0 be a small constant. By Lemma 2.1, it suffices to show the existence of T = T (ǫ) > 0 such that To show this, we follow the argument of [12,Lemma 2.2]. By the Strichartz estimates and the monotone convergence theorem, there exists T = T (ǫ) > 0 sufficiently large such that As in the proof of Proposition 4.1, we write where By (4.9) and enlarging T if necessary, we havê where ̺ R (x) = ̺(x/R) with ̺ : R 3 → [0, 1] a smooth cut-off function satisfying Using the fact (see Lemma 3.1) that ∂ t (|u| 2 + 3γ|v| 2 ) = −2∇ · Im(u∇u) − 6∇ · Im(v∇v), (4.8), and ∇̺ R L ∞ (R 3 ) R −1 , an integration by parts and the Hölder inequality yield Taking R sufficient large such that R −1 ǫ − 1 4 ≪ ǫ 2 , we infer from (4.12) that This inequality implies that ǫ. (4.14) Thanks to the radial Sobolev embedding (3.11), we have from (4.8) and (4.14) that A similar estimate holds for v. In particular, we get Moreover, we have from the local theory that By Sobolev embedding and Strichartz estimates, we see that that On the other hand, the same argument developed in the proof of (4.7) shows that Collecting (4.11), (4.16), and (4.17), we prove (4.10), and the proof is complete.

Proofs of the main Theorems
By exploiting the tools obtained in the previous parts of the paper, we are now able to prove the scattering for non-radial and radial solutions to (1.1) given in Theorem 1.1. See [26,33,34] for analogous results for NLS systems of quadratic type.
Proof of Theorem 1.1 for radial solutions. We fix ǫ > 0 and R as in Proposition 4.2. From (3.48) and the mean value theorem, we infer that there exist sequences of times t n → ∞ and radii R n → ∞ such that  Choosing n sufficiently large so that R n ≥ R, the Hölder inequality yieldŝ |x|≤R |u(t, x)| 2 + 3γ|v(t, x)| 2 dx R

5.2.
Proof of the blow-up results. It remains to prove the blow-up results as stated in Theorem 1.2. Let us start with the following observation.
We are now able to provide a proof of Theorem 1.2. To the best of our knowledge, the strategy of using an ODE argument -when classical virial estimates based on the second derivative in time of (localized) variance break down -goes back to the work [4], where fractional radial NLS is investigated. See instead [11,22] for some blow-up results for quadratic NLS systems.
Proof of Theorem 1.2. We only consider the case of radial data, the one for Σ 3 -data is treated in a similar manner using (3.14). Let (u 0 , v 0 ) ∈ H 1 × H 1 be radially symmetric and satisfy either E µ (u 0 , v 0 ) < 0 or if E µ (u 0 , v 0 ) ≥ 0, we assume that (1.13) and (1.16) hold. Let (u, v) be the corresponding solution to (1.1) defined on the maximal time interval (−T − , T + ). We only show that T + < ∞ since the one for T − < ∞ is similar. Assume by contradiction that T + = ∞. By Lemma 5.1, we have for ε > 0 sufficiently small, there exists c = c(ε) > 0 such that For t 1 > t 0 , we integrate over [t 1 , t] to obtain , ∀t ≥ t 1 .
In particular, we have M ϕ R (t) ≤ −Az(t) → −∞ as t ր t * , hence K(u(t), v(t)) → +∞ as t ր t * . Thus the solution cannot exist for all time t ≥ 0. The proof is complete.
Appendix A. Proofs of Lemmas 2.1, 2.2, 2.3, and 2.4 Let I ⊂ R be an interval containing zero. We recall that a pair of functions (u, v) ∈ C(I, H 1 (R 3 ))×C(I, H 1 (R 3 )) is called a solution to the problem (1.1) if (u, v) satisfies the Duhamel formula (u(t), v(t)) = (S 1 (t)u 0 , S 2 (t)v 0 ) + iˆt  The linear operators S 1 and S 2 introduced in (1.9) satisfy the following dispersive estimates: for j = 1, 2, and 2 ≤ r ≤ ∞, for all t = 0, which in turn yield the following Strichartz estimates: for any interval I ⊂ R and any Strichartz L 2 -admissible pairs (q, r) and (m, n), i.e., pairs of real numbers satisfying we have, for j = 1, 2, where (m, m ′ ) and (n, n ′ ) are Hölder conjugate pairs. We refer the readers to the boos [8,23,32] for a general treatment of the Strichartz estimates for NLS equations.
We are ready to prove Lemma 2.1.
In the following, we provide the proofs for Lemmas 2.2, 2.3, and 2.4.