On blow up for the energy super critical defocusing non linear Schr\"odinger equations

We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal C^\infty$ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression in the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $\mathcal C^\infty$ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper \cite{MRRSprofile}.

(1.1) in dimension d ≥ 3 for an integer nonlinearity p ∈ 2N * + 1 and address the problem of its global dynamics. We begin by giving a quick introduction to the problem and its development.
1.1. Cauchy theory and scaling. It is a very classical statement that smooth well localized initial data u 0 yield local in time, unique, smooth, strong solutions. For the global dynamics, two quantities conserved along the flow (1.1) are of the utmost importance: mass: The scaling symmetry group u λ (t, x) = λ 2 p−1 u(λ 2 t, λx), λ > 0 acts on the space of solutions by leaving the critical norm invariant Accordingly, the problem (1.1) can be classified as energy subcritical, critical or supercritical depending on whether the critical Sobolev exponent s c lies below, equal or above the energy exponent s = 1. This classification also reflects the (in)/ability for the kinetic term in (1.2) to control the potential one via the Sobolev embedding H 1 ֒→ L q .
1.2. Classification of the dynamics. We review the main known dynamical results which rely on the scaling classification.
Energy subcritical case. In the energy subcritical case s c < 1, the pioneering work of Ginibre-Velo [22] showed that for all u 0 ∈ H 1 , there exists a unique strong solution u ∈ C 0 ([0, T ), H 1 ) to (1.1) and identified the blow up criterion Conservation of energy, which is positive definite and thus controls the energy norm H 1 , then immediately implies that the solution is global, T = +∞. In fact, it can be shown in addition that these solutions scatter as t → ±∞, [23].
Energy critical problem. In the energy critical case s c = 1, the criterion (1.3) fails and the energy density could concentrate. For the data with a small critical norm, Strichartz estimates allow one to rule out such a scenario, [10]. The large data critical problem has been an arena of an intensive and remarkable work in the last 20 years. For large spherically symmetric data in dimensions d = 3, 4, the energy concentration mechanism was ruled out by Bourgain [7] and Grillakis [25] via a localized Morawetz estimate. In Bourgain's work, a new induction on energy argument led to the statements of both the global existence and scattering. These results were extended to higher dimensions by Tao, [59].
A new approach was introduced in Kenig-Merle [29] in which, if there exists one global non-scattering solution, then using the concentration compactness profile decomposition [2,45], one extracts a minimal blow up solution and proves that up to renormalization, such a minimal element must behave like a soliton. The existence of such objects is ruled out using the defocusing nature of the nonlinearity, which is directly related to the non existence of solitons for defocusing models.
In all of these large data arguments, the a priori bound on the critical norm provided by the conservation of energy played a fundamental role. Let us note that in the energy critical focusing setting, the concentration of the critical norm is known to be possible via type II (non self similar) blow up with soliton profile, see e.g [33,40,52,53,51,50].
Energy supercritical problem. In the energy supercritical range s c > 1, local in time unique strong solutions can be constructed in the critical Sobolev space H sc , [10,31]. Kenig-Merle's approach, [30,31], gives a blow up criterion T < +∞ =⇒ lim sup t↑T u(t, ·) H sc = +∞, but the question of whether this actually happens for any solution remained completely open. On the other hand, the main difficulty in proving that T = ∞ for all solutions is that there are no a priori bounds at the scaling level of regularity H sc .
1.3. Qualitative behavior for supercritical models. The question of global existence or blow up for energy supercritical models is a fundamental open problem in many nonlinear settings, both focusing and defocusing. For focusing problems, the existence of finite energy type I (self similar) blow up solutions is known in various instances, see e.g [19,37,35,15], and solitons have been proved to be admissible blow up profiles in certain type II (non self-similar) blow up regimes in all three settings of heat, wave and Schrödinger equations, see e.g. [28,41,14,48,38]. There are also several examples of supercritical problems with positive definite energy (wave maps, Yang-Mills) which admit smooth self-similar profiles and thus provide explicit blow up solutions, [57,5,18].
On the other hand, for defocusing problems, soliton-like solutions are known not to exist and admissible self similar solutions are expected not to exist. For a simple defocusing model like the scalar nonlinear defocusing heat equation, a direct application of the maximum principle ensures that bounded data yield uniformly bounded solutions which are global in time and in fact dissipate. We recall again that for the energy critical problems, blow up occurs in the focusing case, where solitons exist, and it does not in the defocusing case where solitons are known not to exist.
This collection of facts led to the belief, as explicitly conjectured by Bourgain in [6], that global existence and scattering should hold for the energy supercritical defocusing Schrödinger and wave equations. Indications of various qualitative behaviors supporting different conclusions have been provided (we give a highly incomplete list) in numerical simulations e.g. [12,49], in model problems showing blow up e.g. [60,61], in examples of global solutions e.g. [32,4], in logarithmically supercritical problems e.g. [62,58,13], and in ill-posedness and norm inflation type results e.g. [24,34,1,63].
The behavior of solutions in other supercritical models such as the ones arising in fluid and gas dynamics is extremely interesting and not yet well understood. We will not discuss it here. 1.4. Statement of the result. We assert that in dimensions 5 ≤ d ≤ 9 the defocusing (NLS) model (1.1) admits finite time type II (non self similar) blow up solutions arising from C ∞ well localized initial data. The singularity formation is based neither on soliton concentration nor self similar profiles, but on a new front scenario producing a highly oscillatory blow up profile. Theorem 1.1 (Existence of energy supercritical type II defocusing blow up). Let (d, p) ∈ { (5,9), (6,5), (8,3), (9, 3)}, (1.4) and let the critical blow up speed be Then there exists a discrete sequence of blow up speeds (r k ) k≥1 with 2 < r k < r * (d, ℓ), lim k→+∞ r k = r * (d, ℓ) such that any all k ≥ 1, there exists a finite co-dimensional manifold of smooth initial data u 0 ∈ ∩ m≥0 H m (R d , C) with spherical symmetry such that the corresponding solution to (1.1) blows up in finite time 0 < T < +∞ at the center of symmetry with u(t, ·) L ∞ = c p,r,d (1 + o t→T (1)) (T − t) Comments on the result.
1. Hydrodynamical formulation. The heart of the proof of Theorem 1.1 is a study of (1.1) in its hydrodynamical formulation, i.e. with respect to its phase and modulus variables. The key to our analysis is the identification of an underlying compressible Euler dynamics. The latter arises as a leading order approximation of a "front" like renormalization of the original equation. In this process, the Laplace term applied to the modulus 1 of the solution is treated perturbatively in the blow up regime. This is one of the key insights of the paper. The approximate Euler dynamics furnishes us with a self-similar solution, which requires very special properties and is constructed in the companion paper [42] and which, in turn, acts as a blow up profile for the original equation. The existence of these blow up profiles is directly related to the restriction on the parameters (1.4) which we discuss in comment 3 below. Let us recall that there is a long history of trying to use the hydrodynamical variables in (NLS) problems and exploit a connection with fluid mechanics, going back to Madelung's original formulation of quantum mechanics in hydrodynamical variables, [36]. Geometric optics and the hydrodynamical formulation were used to address ill-posedness and norm inflation in the defocussing Schrödinger equations, [24,1]. There is also a recent study of vortex filaments in [3] and its dynamical use of the Hasimoto transform. The scheme of proof of Theorem 1.1 will directly apply to produce the first complete description of singularity formation for the three dimensional compressible Navier-Stokes equation in the companion paper [43]. Here, the blow up profile (ρ P , Ψ P ) is, after a suitable transformation, picked among the family of spherically symmetric, smooth and decaying as Z → +∞ self-similar solutions to the compressible Euler equations. The interest in self-similar solutions for the equations of gas dynamics goes back to the pioneering works of Guderley [26] and Sedov [56] (and references therein,) who in particular considered converging motion of a compressible gas towards the center of symmetry. However, the rich amount of literature produced since then is concerned with non-smooth self-similar solutions. This is partly due to the physical motivations, e.g. interests in solutions modeling implosion or detonation waves, where self-similar rarefaction or compression is followed by a shock wave (these are self-similar solutions which contain shock discontinuities already present in the data), and, partly due to the fact that, as it turns out, global solutions with the desired behavior at infinity and at the center of symmetry are generically not C ∞ . This appears to be a fundamental feature of the self-similar Euler dynamics and, in the language of underlying acoustic geometry, means that generically such solutions are not smooth across the backward light (acoustic) cone with the vertex at the singularity. The key of our analysis is to find those non-generic C ∞ solutions and to discover that this regularity is an essential element in controlling suitable repulsivity properties of the associated linearized operator. This is at the heart of the control of the full blow up. A novel contribution of the companion paper [42] is the construction of C ∞ spherically symmetric self-similar solutions to the compressible Euler equations with suitable behavior at infinity and at the center of symmetry for discrete values of the blow up speed parameter r in the vicinity of the limiting blow up speed r * (d, ℓ) given by (1.5).
3. Restriction on the parameters. There is nothing specific with the choice of parameters (1.4), and clearly the proof provides a full range of parameters. Two main constraints govern these restrictions. First of all, a fundamental restriction in order to make the Eulerian regime dominant is the constraint which provides a non empty set of nonlinearities iff As a result, the case of dimensions d = 3, 4 is not amenable to our analysis at this point, and the existence of blow up solutions for d = 3, 4 remains open. The second restriction concerns the existence of C ∞ smooth blow up profiles with suitable repulsivity properties of the associated linearized operator, as addressed in [42], see section 2.2 and remark 2.3 for detailed statements. In particular, a non degeneracy condition S ∞ (d, ℓ) = 0 for an explicit convergent series required. An elementary numerical computation is performed in [42] to check the condition in the range (1.4).

4.
Behavior of Sobolev norms. The conservation of mass and energy imply a uniform H 1 bound on the solution. This can also be checked directly on the leading order representation formulas (1.7), (1.8). For higher Sobolev norms, a computation, see Appendix D, shows that the blow up solutions of Theorem 1.1 break scaling, i.e., we can find and the critical Sobolev norm u(t, ·) H sc blows up polynomially.

Stability of blow up.
The blow up profiles of Theorem 1.1 have a finite number of instability directions, possibly none. Local asymptotic stability in the interior of the backward light cone (of the acoustical metric associated to the Euler profile) from the singularity relies on an abstract spectral argument for compact perturbations of maximal accretive operators. Related arguments have been used in the literature for the study of self-similar solutions both in focusing and defocusing regimes, for example [8,21,46,44,16] for parabolic and [19] for hyperbolic problems. The key to the control of the nonlinear flow in the exterior of the light cone is the propagation of certain weighted scale invariant norms. This generalizes a Lyapunov functional based approach developed in [41]. Counting the precise number of instability directions is an independent problem, disconnected to the nonlinear analysis of the blow up, and remains to be addressed.
6. Oscillatory behavior. The constructed solutions are smooth at the blow up time away from x = 0: (1.10) As in the cases for blow up problems in the focusing setting, see e.g. [39], the profile outside the blow up point has a universal behavior when approaching the singularity What is unusual, and together with potential non-genericity perhaps responsible for difficulties in numerical detection of the blow up phenomena, is the highly oscillatory behavior. This appears to be a deep consequence of the structure of the self-similar solution to the compressible Euler equation and the coupling of phase and modulus variable in the blow up regime, generating an anomalous Euler scaling. The heart of our analysis is to show that after passing to the suitable renormalized variables provided by the front, the highly oscillatory behavior (1.11) becomes regular near the singularity and can be controlled with the monotonicity estimates of energy type, without appealing to Fourier analysis.
7. Type I blow up. The existence of self similar solutions to the defocusing energy supercritical (NLS) decaying at infinity is an open problem. Such solutions are easily ruled out for the heat equation using the maximum principle, and we refer to [32] for further discussion in the case of the wave equation.
The paper is organized as follows. In section 2, we present the "front" renormalization of the flow which makes the Euler dynamics dominant, and recall all necessary facts about the corresponding self similar profile built in [42]. Theorem 1.1 reduces to building a global in time non vanishing solution to the renormalized flow (2.25) written in hydrodynamical variables. In section 2.4 we detail the strategy of the proof. In section 3, we introduce the functional setting related to maximal accretivity (modulo a compact perturbation) of the corresponding linear operator which leads to a statement of exponential decay in a neighborhood of the light cone for the space of solutions (modulo an a priori control of a finite dimensional manifold corresponding to the unstable directions.) In section 4, we describe our set of initial data and the set of bootstrap assumptions which govern the analysis. In sections 5, 6, 7, we close the control of weighted Sobolev norms and the associated pointwise bounds. In section 8, we close the exponential decay of low Sobolev norms by relying on spectral estimates and finite speed of propagation arguments.
Acknowledgements. The authors would like to thank N. Burq (Orsay), V. Georgescu (Cergy-Pontoise) and L. Vega (Bilbao) for stimulating discussions at the early stages of this work. P.R. is supported by the ERC-2014-CoG 646650 SingWave. P.R would like to thank the Université de la Côte d'Azur where part of this work was done for its kind hospitality. I.R. is partially supported by the NSF grant DMS #1709270 and a Simons Investigator Award. J.S is supported by the ERC grant ERC-2016 CoG 725589 EPGR.
Notations. The bracket r = 1 + r 2 . The weighted scalar product for a given measure g: (1.12) The infinitesimal generator of dilations Λ = y · ∇.

Front renormalization, blow up profile and strategy of the proof
In this section we introduce the hydrodynamical variables to study (1.1) and the associated renormalization procedure which makes the compressible Euler structure dominant. We collect from [42] the main facts about the existence of smooth spherically symmetric self-similar solutions to the compressible Euler equations which will serve as blow up profiles.
2.1. Hydrodynamical formulation and front renormalization. We renormalize the flow and, for non vanishing solutions, write the equivalent hydrodynamical formulation in phase and modulus variables. We begin with the standard self-similar renormalization where we freeze the scaling parameter at the self-similar scale In the defocusing case, (2.1) has no obvious type I self similar stationary solution, or type II soliton like solutions, [41], but, it turns out, that it admits approximate front like solutions. Their existence relies on a specific phase and modulus coupling and anomalous scaling. We introduce the parameters and claim: Lemma 2.1 (Front renormalization of the self similar flow). Define geometric parameters and introduce the renormalization with the phase and modulus In these variables (1.1) becomes, on [τ 0 , +∞): Separating the real and imaginary parts yields the self-similar equations (2.1): We now renormalize according to with a fixed choice of parameters in the modulation equations We now compute from (2.2): and (2.4) is proved.

Blow up profile and Emden transform.
A stationary solution (ρ P , Ψ P ) to (2.4) in the limiting Eulerian regime b = 0 satisfies the profile equation We supplement it with the boundary conditions: We now show that the system (2.7), (2.8) is equivalent to the corresponding system of equations describing self-similar solutions of the Euler equations. We define the Emden variables: or equivalently (2.11) The system (2.10) is exactly the one describing spherically symmetric self-similar solutions to the compressible Euler equation, [56] (and the references therein). For an explicit derivation see Appendix A. It is analyzed in [42], following pioneering work of Guderley, Sedov and others. Let 2.12) and the determinants (2.14) Solution curves w = w(σ) of the above system can be examined through its phase portrait in the (σ, w) plane. The shape of the phase portrait depends crucially on the polynomials ∆, ∆ 1 , ∆ 2 and the parameters (r, d, ℓ). It is not hard to see that there is a unique solution with the normalization ρ P (0) = 1, Ψ P (0) = 0, (2.15) at x = 0, which is also C ∞ in the vicinity of x = 0, but the heart of the matter is the global behavior of this unique solution.
In particular, any such solution with the required asymptotics as x → +∞ needs to pass through the point P 2 which lies on the so called sonic line 2 : ∆ = 0 but where also ∆ 1 (P 2 ) = ∆ 2 (P 2 ) (2.16) (there are potentially two such points). It turns out that at P 2 the solution experiences an unavoidable discontinuity of high derivatives, except for discrete values of the speed r. The following structural proposition on the blow up profile is proved in the companion paper [42].
2. Passing through P 2 : the solution passes through P 2 with C ∞ regularity.
3. Large Z asymptotic: the solution admits the asymptotics as Z → +∞: or equivalently with non zero constants c σ , c P . Similar asymptotics hold for all higher order derivatives. 4. Non vanishing: there holds ∀Z ≥ 0, ρ P > 0. 20) then there exists c = c(d, ℓ, r) > 0 such that

Repulsivity inside the light cone: let
6. Repulsivity outside the light cone: example where this holds, but a larger range of parameters can be directly extracted from [42], and the conclusion of Theorem 1.1 would follow. In particular and since we are working with non vanishing solutions, the fact that the non linearity is an odd integer can be relaxed as in [43].
From now on and for the rest of this paper, we assume (1.4). We observe from direct check that there holds: Recalling (2.2), we may therefore assume from (2.17) that the blow speed r = r k satisfies 2.3. Linearization of the renormalized flow. We look for u solution to (1.1) and proceed to the decomposition of Lemma 2.1. We are left with finding a global, in self similar time τ ∈ [τ 0 , +∞), solution to (2.4): with non vanishing density ρ T > 0. We define We linearize ρ T = ρ P + ρ, Ψ T = Ψ P + Ψ and compute, using the profile equation (2.7), for the first equation: and for the second one: We arrive at the exact (nonlinear) linearized flow (2.25) Theorem 1.1 is therefore equivalent to exhibiting a finite co-dimensional manifold of smooth well localized initial data leading to global, in renormalized τ -time, solutions to (2.25).

2.4.
Strategy of the proof. We now explain the strategy of the proof of Theorem 1.1. step 1 Wave equation and propagator estimate. After the change of variables Φ = ρ P Ψ, we may schematically rewrite the linearized flow (2.25) in the form where Q, H 1 , H 2 , H 3 are explicit potentials generated by the profile ρ P , Ψ p . During the first step the b 2 ∆ term is treated perturbatively. We commute the equation with the powers of the laplacian ∆ k and obtain for X k = ∆ k X We then show that, provided k is large enough, M k is a finite rank perturbation of a maximally dissipative operator with a spectral gap δ > 0. The topology in which maximal accretivity is established depends on the properties of the wave equation 3 encoded in (2.28) and is based on weighted Sobolev norms with weights vanishing on the light cone corresponding to the point P 2 of the profile. Indeed, the principal part of the wave equation is roughly of the form vanishes on the light cone Z = Z 2 corresponding to the P 2 point. The corresponding propagation estimates for the wave equation produce an priori control of the solution in the interior of the light cone Z < Z 2 , modulo an a priori control of a finite number of directions corresponding to non positive eigenvalues of M k . An essential structural fact of this step is the C ∞ regularity of the profile. Indeed, we claim that for a generic non C ∞ solution at P 2 , the number of derivatives required to show accretivity of the linearized operator is always strictly greater than the regularity of the profile at P 2 . As a result such profiles may be completely unstable and are not amenable to our analysis. The C ∞ regularity obtained in [42] is therefore absolutely fundamental. The analytic properties leading to the maximality of the linearized operator will be consequences of (2.21), (2.22). We note that the coercivity constant in (2.21) degenerates as r → r * , and the number of derivatives needed for accretivity is inversely proportional to this constant. This is a manifestation of a completely new nonlinear effect: the problem sees a scaling which depends on the chosen self similar profile. step 2 Extension slightly beyond the light cone. Exponential decay estimates provided in the first step yield control in the interior of the light cone Z < Z 2 only. It turns out that the analysis of the first step can be made more robust and extended 4 slightly beyond the light cone, all the way to a spacelike hypersurface Z = Z 2 + a, 0 < a ≪ 1, even though it is complicated by the dependence of the underlying wave equation on variable coefficients or, equivalently, on non constancy of the Q(Z) term in (2.27). We can revisit the first step by producing a new maximal accretivity structure for a norm which does not generate in the zone Z < Z 2 + a, 0 < a ≪ 1. propagation estimates recovers exponential decay in the extended zone Z < Z 2 + a. Once decay has been obtained strictly beyond the light cone, a simple finite speed of propagation argument allows us to propagate decay to any compact set Z < Z 0 , Z 0 ≫ 1. step 3 Loss of derivatives. The decay obtained in step 2 relies on energy estimates compatible with the wave propagation and the Eulerian structure of approximation. The full evolution however is that of the Schrödinger equation and contains the b 2 ∆ term on the right hand side of (2.26). Such a term leads to an unavoidable loss of one derivative. However, this loss comes with a b 2 smallness in front. We then argue as follows. We pick a large enough regularity level k m = k m (r, d) ≫ 2k 0 , where k 0 is the power of the laplacian used for commutation in step 2, and derive a global Schrödinger like energy identity on the full flow (2.25). The choice of phase and modulus as basic variables turns the equation quasilinear and makes this identity rather complicated and unfamiliar. An essential difficulty, which is deeply related to step 2, is that at the highest level of derivatives, the non trivial space dependence of the profile measured by Q(Z) = ρ p−1 P (Z) in (2.27) produces a coupling term and a non trivial quadratic form. The condition (2.22) implies that the corresponding quadratic form is definite positive for k m large enough. step 4 Closing estimates. As explained above, we work with a linearized nonlinear equation, i.e., obtained after subtracting off the profile, written in terms of the phase and modulus unknowns (Ψ, ρ), in renormalized self-similar variables (τ, Z), where the singularity corresponds to (τ = ∞, Z = 0), a special light cone is (τ, Z = Z 2 ) and where in the original variables (t, r) the region r ≥ 1 corresponds to Z ≥ e µτ .
First, outside the singularity r ≥ 1, we modify the profile by strengthening its decay to make it rapidly decaying and of finite energy. Relative to the self-similar variables this modification happens at Z ∼ e µτ , far from the singularity, and as a result is harmless. Then, we run two sets of estimates. First, we employ wave propagation like estimates which go initially just slightly beyond the special light cone and then extend to any compact set in Z. These estimates are carried out at a sufficiently high level of regularity with ∼ 2k 0 derivatives. The number k 0 emerges from the linear theory and is determined by the (conditional) positivity of a certain quadratic form responsible for maximal accretivity.
Then, we couple these estimates to global Schrödinger like estimates which take into account previously ignored b 2 ∆ and take care of global control. These estimates are carried out at all levels of regularity up to k m derivatives with k m ≫ k 0 . They are carefully designed weighted L 2 type estimates. The weights depend on the number of derivatives k: at first, their strength grows with k but by the time we reach the highest level of regularity k m the weight function is identically = 1. The latter has to do with a well-known fact that even for a linear Schrödinger equation estimates, use of weights leads to a derivative loss (∆ is not self-adjoint on a weighted L 2 space.) Therefore, our highest derivative norm should correspond to an unweighted L 2 estimate. Of course, this last estimate also sees a positivity condition (2.22) responsible for the coercivity of an appearing quadratic form.
These global weighted L 2 bounds then allow us to prove pointwise bounds for the solution and its derivatives which, in turn, allow us to control nonlinear terms. The obtained sets of weighted L ∞ bounds on derivatives recover in particular the non vanishing assumption required of the solution. We should note that while all the local (in Z) norms decay exponentially in τ , the global norms are merely bounded.
In the original (t, r) variables this means that the perturbation decays inside and slightly beyond the backward light cone from the singular point but does not decay away from the singularity. This is, of course, entirely consistent with the global conservation of energy for NLS.
The whole proof proceeds via a bootstrap argument which also involves a Brouwer type argument to deal with unstable modes, if any, arising in linear theory of step 1. This is what produces a finite co-dimension manifold of admissible data.

Linear theory slightly beyond the light cone
Our aim in this section is to study the linearized problem (2.25) for the exact Euler problem b = 0. We in particular aim at setting up the suitable functional framework in order to apply classical propagator estimates which will yield exponential decay on compact sets Z 1 modulo the control of a finite number of unstable directions.
3.1. Linearized equations. Recall the exact linearized flow (2.25) which we rewrite: ) and obtain equivalently using (2.24): and the nonlinear terms: We transform (3.2) into a wave equation for Φ and compute: In this section we focus on deriving decay estimates for (3.2).
Remark 3.1 (Null coordinates and red shift). We note that the principal symbol of the above wave equation is given by the second order operator In the variables of Emden transform this can be written equivalently as The two principal null direction associated with the above equation are so that Q = LL We observe that at P 2 , we have L = ∂ τ and the surface Z = Z 2 is a null cone. Moreover, the associated acoustical metric is for which ∂ τ is a Killing field (generator of translation symmetry). Therefore, Z = Z 2 is a Killing horizon (generated by a null Killing field.) We can make it even more precise by transforming the metric g Q into a slightly different form by defining the coordinate s: and then the coordinate x * : and s + x * and s − x * are the null coordinates of g Q . The Killing horizon Z = Z 2 corresponds to x * = −∞ and ∆ ∼ e Cx * for some positive constant C. In this form, near Z 2 the metric g Q resembles the 1 + 1-quotient Schwarzschild metric near the black hole horizon. The associated surface gravity κ which can be computed according to This is precisely the repulsive condition (2.21) (at P 2 ). The positivity of surface gravity implies the presence of the red shift effect along Z = Z 2 both as an optical phenomenon for the acoustical metric g Q and also as an indicator of local monotonicity estimates for solutions of the wave equation Q ϕ = 0, [17]. The complication in the analysis below is the presence of lower order terms in the wave equation as well as the need for global in space estimates.
which yields the (T, Φ) equation We rewrite these equations in vectorial form then for 0 < a < a * small enough, there exists a C 1 map a → Z a with (3.10) Proof of Lemma 3.2. We recall the notations of the Emden transform: Let the variables w = w 2 + W, σ = σ 2 + Σ, then near P 2 : Then, in our range of parameters, and we have which imply see Lemma 2.8 and Lemma 2.9 in [42]. We compute This yields and hence (3.17) and hence by the implicit function theorem applied to the function F (a, Z) = D a (Z) at (a, Z) = (0, Z 2 ) where D 0 (Z 2 ) = 0, we infer for all a small enough the existence of a locally unique solution Z a to D a (Z a ) = 0 (3.18) Furthermore, Z a is C 1 in a neighborhood of a = 0 and its derivative is given by We now observe Thus, (1 − a)(1 − w) − σ is increasing on (0, Z a ] and vanishes at Z = Z a so that D a (Z) < 0 on (0, Z a ).
Moreover, we have in view of the behavior of σ and w as Z → 0 + , see Lemma 3.1 in [42], This concludes the proof of (3.10).

3.4.
Commuting with derivatives. We define Lemma 3.3 (Commuting with derivatives). Let k ∈ N. There exists a smooth measure g defined for Z ∈ [0, Z a ] such that the following holds. Let the elliptic operator where M k satisfies the following pointwise bound (3.21) Moreover, g > 0 in [0, Z a ) and admits the asymptotics:

22)
with c g > 0 (3.23) for all k ≥ k 1 large enough and 0 < a < a * small enough.
Proof. This is a direct computation.
which together with the commutator formulas We then use to compute similarly: Recalling the definition of the operator (3.8), we obtain (3.20), (3.21) with step 2 Equation for the measure. We compute using (3.11), (3.9): and hence: We compute the measure and hence the relation: Equivalently: Asymptotics of the measure. We now solve (3.25). Near the origin, the normalization (2.15) and (3.11) yield We computeÃ and hence which, recalling (3.10), yields: and we may therefore choose explicitly: To compute the behavior near Z a , recall from (3.18) (3.19) that we have The fundamental computation is then at P 2 using (3.16): for 0 < a < a * small enough and k ≥ k 1 large enough 5 . Inserting this into (3.27) yields (3.22).

3.5.
Hardy inequality and compactness. We let k ≥ k 1 large enough so that (3.23) holds and extend the measure g by zero for Z ≥ Z a . We let χ be a smooth cut off function supported strictly inside the light cone |Z| < Z 2 with be a Hermitian scalar product, where we recall the notation (1.12). We let H Φ be the completion of D Φ for the norm associated to (3.28). We claim the following compactness subcoercivity estimate: Furthermore, there exists c > 0 and a sequence µ n > 0 with lim n→+∞ µ n = +∞ Proof. This is a classical Hardy and Sobolev based argument.
step 1 Interior estimate. Let Z 0 < Z a which will be chosen close enough to Z a in step 2. Then, we have Since −Z 2 D a and g are smooth and satisfy −Z 2 D a > 0 and g > 0 for some c Z 0 > 0. Thus, to prove (3.29), it remains to consider the region (Z 0 , Z a ). This will be done in step 2 and step 3.
step 2 Hardy inequality with loss. Let 0 < ν ≪ 1, we claim the lossy Hardy bound for all Φ ∈ D Φ : where we used the fact that 0 < ν < 1 and Z a − Z 0 = δ. Using again 0 < ν < 1 and Z a − Z 0 = δ, as well as Fubini and the fact that ∂ Z g < 0 on (Z 0 , Z a ) for Z 0 close enough to Z a so that g is decreasing on Letting δ = δ(ν) small enough and estimating from (3.19) yields (3.31).
step 3 Sharp Hardy. We now claim the sharp Hardy inequality for f ∈ D Φ : then integrating by parts: where we used (3.32). The bound (3.33) now follows using Hölder. Together with step 1 and step 2, this concludes the proof of (3.29).
step 4 Compactness. We now turn to the proof of (3.30) which follows from a standard compactness argument. Let us consider T ∈ L 2 g . Then from (3.29), the antilinear form h → (T, h) g , is continuous on H Φ , and hence by Riesz, there exists a unique L( 34) and the linear map L is bounded from L 2 g to H Φ . For any 0 < δ < Z a , we have in view of (3.29) Relying on the smallness of δ 1−ν 2 for the first term, and Rellich for the second one, we easily infer that and hence interchanging the roles of T 1 , T 2 : and L is selfadjoint on L 2 g . Since L > 0 from (3.34), we conclude that L is a diagonalizable with a non increasing sequences of eigenvalues λ n > 0, lim n→+∞ λ n = 0, and let (Π n,i ) 1≤i≤I(n) be an L 2 g orthonormal basis for the eigenvalue λ n . The and the minimization problem then the infimum is attained in view of (3.35) at Φ ∈ A n and, by a standard Lagrange multiplier argument: Letting h = Π i,j implies α i,j = 0 and hence from (3.34): which together with our orthogonality conditions implies and hence On the other hand, from Rellich and an elementary compactness argument, for all Summing the two inequalities yields for all δ > 0 small and ǫ smaller still: Together with (3.36), this implies for any Φ satisfying the orthogonality conditions 3.6. Accretivity. We now turn to the proof of the accretivity of the operator M.
Hilbert space. Recall (3.28). We define the space of test functions , C), and let H 2k be the completion of D 0 for the scalar product: which is a coercive Hermitian form from (3.29).
Unbounded operator. Following (3.8), we define the operator ) equipped with the domain norm. We then pick suitable directions (X i ) 1≤i≤N ∈ H 2k and consider the finite rank projection operator The aim of this section is to prove the following accretivity property: There exist k * ≫ 1 and 0 < c * , a * ≪ 1 such that for all k ≥ k * , ∀0 < a < a * small enough, there exist ∀X ∈ D(M), ℜ −MX, X ≥ c * ak X, X (3.39) and maximal: Remark 3.6. We recall that maximal dissipative operators are closed.
Proof of Proposition 3.5. given R > R * (k) large enough, we define the space of test functions In steps 1 to 3 below, we prove (3.39) for X ∈ D R so that all integrations by parts in steps 1 to 3 are justified, and all boundary terms at Z = Z a vanish due to the vanishing of g at Z = Z a . In steps 4 and 5, for any smooth F on [0, Z a ], we show existence and uniqueness of a solution X ∈ H 2k to (−M + R)X = F for R > R * (k) large enough. In step 6, we prove that D R is dense in D(M). In step 7, we conclude the proof of (3.39) and (3.40). step 1 Main integration by parts. Let X ∈ D R for R > R * (k) large enough. We aim at proving (3.39) and split the computation in two: In step 1, we consider the principal part. We compute from (3.20): For the second term: We have therefore obtained the formula: where we have defined We now claim the following lower bounds on A 5 , A 6 : there exist universal constants k * ≫ 1, 0 < c * , a * ≪ 1 such that for all k ≥ k * and 0 < a < a * , Proof of (3.44). Recall (3.25), (3.26): and hence from (3.42): from the fundamental coercivity bound (2.21), and hence for Z ≤ Z a and a < a * small enough: for some c * independent of k, a. Similarly: arguing as above. This concludes the proof of (3.44).
We lower bound from (3.44): The smoothness and boundedness of the profile together with (3.25), (3.26) ensure that The collection of above bounds yields: We conclude using (3.30) with N = N (a, k) large enough and its analogue for T : Therefore, The linear from and similarly for T i , and (3.39) follows for X ∈ D R .
step 4 ODE formulation of maximality. Our goal, in steps 4 to step 6, is to prove that forall R > 0 large enough, Solving for T : we look for Φ -solution to the second order elliptic equation: Recalling We therefore define , for Z > Z 2 .
In view of the above, we have obtained the elliptic equation: 49) with T recovered by (3.47). As Z → Z 2 , we have from (3.17): and hence 7 The choice of the lower limits Z 2 2 and 2Z2 in the definition of F± is arbitrary but dictate the choice of the constants C± in such a way as to ensure that lim The additional degree of freedom in the choice of C± is used to fix an overall normalization of ̟.
Since the profile passes through P 2 in a C ∞ way, we obtain the development of the measure at P 2 : for any M ≥ 1, for R > R * large enough. Note that the above choice of C ± is made to fix the normalization constant in front of |Z 2 − Z| c̟ to be equal to 1.
step 5 Solving (3.49). We analyze the singularity of (3.49) at P 2 using a change of variables.
with similar estimates for derivatives. Hence the potential term in (3.52) can be expanded in Y and estimated as Y → +∞ for R large enough: for some universal constantsd j , where C > 0 is independent of R. Therefore, by an elementary fixed point argument, (3.52) withH = 0 admits a basis of solutions Ψ − 1 and Ψ − 2 with the following behavior as Y → +∞ with similar estimates for derivatives. The sequences (c j,1 ) j=1,2 are uniquely determined inductively from (3.52) withH = 0 using the expansion of the potential (3.55).
To the right of P 2 , we let We construct a similar basis of homogenous solutions Ψ + 1 and Ψ + 2 as Y → +∞ with asymptotics given by: with the sequences c j,1 , c j,2 the same as in (3.56).
and is a solution to the homogeneous equation (3.52). Let now Φ rad (Z) be the radial solution to the homogeneous problem associated to (3.49) with Φ rad (0) = 1. Then the wronskian is given by In particular, if T rad is given by (3.47) with , we may apply the analysis in steps 1 to 4 for R > R * (k) large enough and (3.39) holds for X rad , i.e.
and hence X rad = 0 a contradiction. This concludes the proof of W 0 = 0. Inner solution of the inhomogeneous problem. (Φ rad , Φ 1 ) is then a basis for the homogeneous problem corresponding to (3.49). As a consequence, the only solution 8 We add constantC+ to match the asymptotic expansion of Y in terms of (Z2 −Z). In principle, it is unnecessary as it influences the terms of order R and higher while we only need the universality of the expansion up to the order √ R.
For a smooth H, Φ is smooth on [0, Z 2 ) and we study its regularity at Z 2 . In Y variables we obtain for some Y 0 large enough: and hence with similar estimates for derivatives. In particular, a smooth function H(Z) yields expansion forH(Z): Conversely, an expansion of the form We therefore have proved that for Outer solution of the inhomogeneous problem. We argue similarly, considering the basis Furthermore, Φ admits at Z = Z 2 the following asymptotic expansion analogous to (3.58) The asymptotic expansion is uniquely determined from the equation (3.49) and the first coefficientc 0,Φ . We now recall that the function Φ 1 belongs to C √ R [0, Z a ] and Φ 1 (Z 2 ) = 1. By adding Φ 1 to the above expansion, we obtain another solution in which we can force the conditionc (3.58). As a result, the asymptotic expansions of the inner and outer solutions are matched to order √ R, so that the constructed solution is C √ R at Z 2 . Finally, we have shown that given any smooth function H on [0, Z a ], there exists a unique solution Φ to (3.49) In particular, with T recovered by (3.47) and smooth for Z = Z 2 and C 10 , we have now proved that, in fact, there exists a unique solution X = (Φ, T ) to (−M + R)X = F on [0, Z a ] in H 2k , which concludes the proof of (3.46).
From step 5, for each integer n, there exist a unique Z n ∈ D R solution to and hence Thus, to conclude, it remains to check that Z n converges to X in H 2k . To this end, since Z n ∈ D R , (3.39) holds for Z n − Z q and thus: In view of the convergence of (Y n ) in H 2k , we deduce that Z n is a Cauchy sequence in H 2k and hence converges, i.e. Indeed, since F ∈ H 2k , by density, there exists .
Using (3.39) and arguing as in step 6, we have for R sufficiently large In view of the convergence of (F n ) in H 2k , we deduce that X n is a Cauchy sequence in H 2k and hence converges, i.e.
On the other hand, since (−M + R)X n = F n convergence to F in H 2k , we infer which concludes the proof of (3.59). Finally, (3.39) and a classical and elementary induction argument ensures that the maximality property (3.40) is implied by: Indeed, let R > 0 large enough and F ∈ H 2k . Since A is a bounded operator, for R large enough, from (3.59) and (3.39), Therefore, for any F ∈ H 2k , solution X to (3.59) is unique. Therefore, (−M + R) −1 is well defined on H 2k with the bound We recall the following classical lemma.
We now recall from Hille-Yoshida's theorem that a maximally dissipative operator A 0 generates a strongly continuous semigroup T 0 on H, and so does A 0 + K for any bounded perturbation K. Let us now recall the following classical properties of strongly continuous semigroup T (t). Let σ(A) denote the spectrum of A, i.e., the complement of the resolvent set.
Let w ess denote the essential growth bound of the semigroup: (3.61) Moreover, each eigenvalue λ ∈ Λ w (A) has finite algebraic multiplicity m a λ : ∃k λ ∈ Z such that AX, Y = X, A * Y = 0. We claim the following corollary. Lemma 3.9 (Perturbative exponential decay). Let T 0 be the strongly continuous semigroup generated by a maximal dissipative operator A 0 , and T be the strongly continuous semi group generated by A = A 0 + K where K is a compact operator on H. Then for any δ > 0, the following holds: has finite algebraic multiplicity k λ . In particular, the subspace V δ (A) is finite dimensional; (ii) We have Λ δ (A) = Λ δ (A * ) and dimV δ (A * ) = dimV δ (A). The direct sum decomposition is preserved by T (t) and there holds: iii) The restriction of A to V δ (A) is given by a direct sum of (m λ × m λ ) λ∈Λ δ (A) matrices each of which is the Jordan block associated to the eigenvalue λ and the number of Jordan blocks corresponding to λ is equal to the geometric multiplicity of λm g λ = dimker(A − λI). In particular, m a λ ≤ m g λ k λ . Each block corresponds to an invariant subspace J λ and the semigroup T restricted to J λ is given by the nilpotent matrix Proof. This is a simple consequence of Proposition 3.8. Let now λ ∈ σ(A) with ℜ(λ) > 0, then the formula and invertibility of (A 0 − λ) imply that λ belongs to the spectrum of the Fredholm operator Id + (A 0 − λ) −1 K. Therefore, λ is an eigenvalue of A. On the other hand, ℜ(λ) > δ implies ℜ(λ) > δ > 0 ≥ w ess (T ), and hence, by (3.61), there are finitely many eigenvalues with ℜ(λ) > δ. In fact, Proposition 3.8 also directly shows that each some λ is an eigenvalue and implies the rest of (i).
step 2 The first statement of (ii) is standard. We already explained that the subspaces V δ (A) and D(A) ∩ V ⊥ δ (A * ) are invariant for A. To prove the direct decomposition we recall that the subspace V δ (A) is the image of H under the spectral projection P δ (A) associated to the set Λ δ (A): where Γ is an arbitrary contour containing the set Λ δ (A). There is a direct decomposition On the other hand, the adjoint is the spectral projection of A * associated to the set Λ δ (A). The result is now immediate. LetT be the semigroup on U generated byÃ = A. Then for all X ∈ D(A) ∩ U , T (t)X ∈ C 1 ([0, +∞), D(Ã)) is the unique strong solution to the ode This implies thatT (t)X = T (t)X for all X ∈ D(A) ∩ U and thus for all X ∈ U by continuity of the semigroup. By Proposition 3.8 the growth bound ofT satisfies w 0 (T ) ≤ max{w ess (T ), s(Ã)}.
We first argue that w ess (T ) ≤ 0.
To prove that we note that we already established that w ess (T ) ≤ 0. We then fix ε > 0 and, for any t ≥ 0 choose a compact operator K(t) ∈ K(H) on H such that, log T (t) − K(t) H→H < εt + logM for some constant M which may depend on ε. The restrictionK(t) = P K(t) of K(t) to U is a compact operator on U . Then, for any t ≥ 0 where C P denotes the norm of the projector P . The desired conclusion follows.
Finally, part (iii) is completely standard.
We will use Lemma 3.9 in the following form.
Lemma 3.10 (Exponential decay modulo finitely many instabilities). Let δ > 0 and let T 0 be the strongly continuous semigroup generated by a maximal dissipative operator A 0 , and T be the strongly continuous semigroup generated by A = A 0 −δ+K where K is a compact operator on H. Let the (possibly empty) finite set where U and V are invariant subspaces for A and V is the image of the spectral projection of A associated to the set Λ. Then there exist C, δ g > 0 such that Proof. We apply Lemma 3.9 toÃ = A + δ = A 0 + K with generates the semi group T . Hence the set be the invariant decomposition ofÃ (and of A) associated to the set Λ δ

4
. Clearly, where O δ is the image of the spectral projection of A associated with the set Λ δ 4 \ Λ. By Lemma 3.9, Let now X ∈ U . Since U δ is invariant by T and (3.65) yields exponential decay on U δ , we assume X ∈ O δ . O δ is an invariant subspace of A generated by the eigenvalues λ with the property that − 3 4 δ ≤ ℜ(λ) < 0. Let δ g > 0 be defined as From part (iii) of Lemma 3.9, This concludes the proof of Lemma 3.10.
Our final result in this section is to set up a Brouwer type argument for the evolution of unstable modes.
Then, for any x in the ball x ≤ e − 3δg 5 t 0 , we have X(t) ≤ e − δg 2 t , t 0 ≤ t ≤ t 0 + Γ (3.66) for some large constant Γ (which only depends on A and t 0 .) Moreover, there exists x * ∈ V in the same ball as a above such that ∀t ≥ t 0 , Proof. According to Lemma 3.9 the subspace V can be further decomposed into invariant subspaces on which A is represented by Jordan blocks. We may therefore assume that V is irreducible and corresponds to a Jordan block of A of length m λ associated with an eigenvalue λ with ℜ(λ) ≥ 0 and restrict A to V . We decompose A as A = λI + N, where N has the property that N m λ −1 = 0, and The claim (3.66) follows from the growth on the Jordan block: and hence the size of constant Γ is determined from the inequality a sufficient condition being Γ ≤ t 0 2 δ g 10ℜ(λ) + 5δ g which can be made arbitrarily large by a choice of t 0 . We now define a new variable Since N and A commute, Since t 0 was chosen to be sufficiently large, we can assume that ∀t ≥ t 0 60 . We now run a standard Brouwer type argument for Y . For any y such that y ≤ 1 we define the exit time t * to be the first time such that Y (t * ) = 1. If for some y, t * = ∞, we are done. Otherwise, assume that for all y ≤ 1, t * < ∞ and define the map Φ : B → S as Φ(y) = Y (t * ) mapping the unit ball to the unit sphere. Note that Φ is the identity map on the boundary of B. To prove continuity of Φ we compute This is the outgoing condition which implies continuity. The Brouwer argument applies and shows that such Φ does not exist. We now reinterpret the result in terms of X. We have shown existence of x such that the corresponding solution X(t) has the property that ∀t ≥ t 0 , Now e −tN is an invertible operator with the inverse given by e tN and its norm bounded by Ct m λ −1 . The result follows immediately. We note that the resulting solution X(t) has initial data X(t 0 ) in the ball X(t 0 ) ≤ e − 3δg 5 t 0 .

Set up and the bootstrap
In this section we describe a set of smooth well localized initial data which lead to the conclusions of Theorem 1.1. The heart of the proof is a bootstrap argument coupled to the classical Brouwer topological argument of Lemma 3.11 to avoid finitely many unstable directions of the corresponding linear flow. Since our analysis relies essentially on the phase-modulus decomposition of solutions of the Schrödinger equation, our chosen data needs to give rise to nowhere vanishing solutions to (1.1) (at least for a sufficiently small time.) 4.1. Renormalized variables. Let u(t, x) ∈ C([0, T ), ∩ k≥0 H k ) be a solution to (1.1) such that u(t, x) does not vanish at any (t, x) ∈ [0, T ) × R d . This will be a consequence of our choice of initial data and suitable bootstrap assumptions. We introduce for such a solution the decomposition of Lemma 2.1 with the renormalized space and times Here, 0 < e < 1 is the fixed front speed such that Up to a constant the phase can more explicitly be written in the form Our claim is that given τ 0 = −logT large enough, we can construct a finite co-dimensional manifold of smooth well localized initial data u 0 such that the corresponding solution to the renormalized flow (2.23) is global in renormalized time τ ∈ [τ 0 , +∞), bounded in a suitable topology and nowhere vanishing. Upon unfolding (4.1), this produces a solution to (1.1) blowing up at T in the regime described by Theorem 1.1.

Stabilization and regularization of the profile outside the singularity.
The spherically symmetric profile solution (ρ P , Ψ P ) has an intrinsic slow decay as Z → +∞ which need to be regularized in order to produce finite energy non vanishing initial data.

Stabilization of the profile. Recall the asymptotics (2.19) and the choice of parameters (4.3), (4.2) which yield
For Z = √ b λ x ≫ 1, i.e., outside the singularity: We see that far away from the singularity the profile u P is stationary. It is precisely this property that will allow us to dampen the tail of the profile below and construct solutions arising from rapidly decaying (in particular, finite energy) initial data.
2. Dampening of the tail. We dampen the tail outside the singularity x ≥ 1, i.e., Z ≥ Z * as follows. Let then the asympotics (4.4) imply the existence of a limiting profile for x ≥ 1: We then pick once and for all a large integer n P ≫ 1 and define a smooth non decreasing connection K(x) for some large enough universal constant n P = n P (d) ≫ 1.
We then define the dampened tail profile in original variables for |x| ≥ 10 , (4.7) and hence in renormalized variables: (4.8) Let we have the equivalent representation: Note that by construction for j ∈ N * : and The obtained dampened profile for Z ≥ Z * will be denoted (ρ D , Ψ P ), Q D = ρ p−1 D .

Initial data.
We now describe explicitly an open set of initial data which will be considered as perturbations of the profile (ρ D , Ψ P ) in a suitable topology. The conclusions of Theorem 1.1 will hold for a finite co-dimension set of such data.
We pick universal constants 0 < a ≪ 1, Z 0 ≫ 1 which will be adjusted along the proof and depend only on (d, ℓ). We define two levels of regularity where k m denotes the maximum level of regularity required for the solution and k 0 is the level of regularity required for the linear spectral theory on the compact set [0, Z a ].
0. Variables and notations for derivatives. We define the variables and specify the data in the (ρ, Ψ) variables. We will use the following notations for derivatives. Given k ∈ N, we note Initializing the Brouwer argument. We define the variables adapted to the spectral analysis according to (3.1), (3.5): and recall the scalar product (3.37). For 0 < c g , a ≪ 1 small enough, we choose k 0 ≫ 1 such that Proposition 3.5 applies in the Hilbert space H 2k 0 with the spectral gap ∀X ∈ D(M), ℜ (−M + A)X, X ≥ c g X, X . (4.13) Hence M = (M − A + c g ) − c g + A and we may apply Lemma 3.10: is a finite set corresponding to unstable eigenvalues, V is an associated (unstable) finite dimensional invariant set, U is the complementary (stable) invariant set and P is the associated projection on V . We denote by N the nilpotent part of the matrix representing M on V: Then there exist C, δ g > 0 such that (3.64) holds: We now choose the data at τ 0 such that 2. Bounds on local low Sobolev norms. Let 0 ≤ m ≤ 2k 0 and let the weight function Then: 4. Pointwise assumptions. We assume the following interior pointwise bounds (4.20) for some small enough universal constant c 0 , and the exterior bounds: for some large enough universal C 0 (d, r, p). Note in particular that (4.20), (4.21) ensure for 0 < b 0 < b * 0 ≪ 1 small enough: and hence the data does not vanish.

Global rough bound for large Sobolev norms.
We pick a large enough constant k m (d, r, ℓ) and consider the global Sobolev norm , (4.23) then we require: The bound above is actually implied by the pointwise assumptions. for some large enough k c (d, p). To ensure non vanishing, we first note that since inf |x|≤10 |u 0 (x)| > 0, the continuity of u in time ensures inf |x|≤10 |u(t, x)| > 0 for t ∈ [0, T ], T small enough. For |x| ≥ 10, we estimate from the flow |r n P |u(t, x)| − r n P |u 0 || ≤ t 0 r n P (∆u − u|u| p−1 )dt and hence from our choice of initial data, the non vanishing of u(t, x) follows on a time interval where for some sufficiently small universal constant 0 < δ ≪ 1. Using spherical symmetry we can replace the above by for an arbitrarily small ǫ > 0. Our initial data u 0 belongs to the space Existence of the desired time interval [0, T ) now follows from a local well-posedness for NLS in weighted Sobolev spaces which is (essentially) in [27]. We may therefore introduce the hydrodynamical variables (4.1) on such a small enough time interval and will bootstrap the smallness bound which ensures non vanishing: for some sufficiently small 0 < δ = δ(k m ) ≪ 1.
1. Global weighted Sobolev norms. Pick a small enough universal constant 0 <ν < ν * (k m ) ≪ 1, we define and let the continuous function: with the continuity requirement at m 0 : In particular, α < 1. We note that for all We also define the functioñ where β is computed through the continuity requirement at 2km 3 : We will choose n P ≪ k m , e.g. n P = km 30 , so that in particular, We also note thatσ (m − 1) ≤σ(m) + β. We then define the weighted Sobolev norm: Let us briefly sketch the proof. For d = 2, we compute so that for f regular at the origin, As a result we obtain The desired claim for the term χ 2,m,σ ρ p−2 D ρ T |∇ 2ρ | 2 ρ, Ψ 2 2,σ then follows from the above inequality and the condition (2.8) ρ P (0) = ρ D (0) = 1 together with the regularity of the profile ρ P and the non vanishing bound (4.27). The statement for other terms and higher derivatives follows by iteration.
Remark 4.5. We note that the assumption (4.41) implies that We will prove the bootstrap proposition 4.4 under the weaker assumption (4.44). Specifically, we will define [τ 0 , τ * ] to be the maximal time interval on which (4.44) holds and will show that both the bounds

Control of high Sobolev norms
We first turn to the global in space control of high Sobolev norms. This is an essential step to control the b dependence of the flow and the dissipative structure which can neither be treated by spectral analysis nor perturbatively.
We claim an improvement of the bound (4.34), controlling all but the highest weighted Sobolev norm. The rest of this section is devoted to the proof of Proposition 5.1.

5.1.
Algebraic energy identity. We derive the energy identity for high Sobolev norms which in the hydrodynamical formulation has a quasilinear structure.

By construction
withẼ supported in Z ≥ 3Z * . The linearized flow

Note that the potentials
remain the same in these equations: they are not affected by the profile localization introduced by passing from ρ P to ρ D . We recall the Emden transform formulas (2.24): which, using (2.18), (2.19), yield the bounds: Our main task is now to produce an energy identity for (5.3) which respects the quasilinear nature of (5.3) and does not loose derivatives. step 2 Equation for derivatives. We recall the notation for the vector ∂ k : We use For the second equation: step 3 Algebraic energy identity. Let χ be a smooth function. We compute: We compute: Similarly: This yields the algebraic energy identity: Given σ ∈ R, we recall the notation then there exists c km > 0 such that for all 0 <ν <ν(k m ) ≪ 1 and b 0 < b 0 (k m ) ≪ 1, for all 1 ≤ k ≤ k m −1, I k := I k,σ(k) given by (5.11) satisfies the differential inequality We claim that Lemma 5.2 implies Proposition 5.1.
Proof of (5.16). Indeed, and hence (5.16) follows from σ(k)+k ≤ σ(m)+m and ξ k (x) ≥ ξ m (x) for 0 ≤ k ≤ m.  Indeed, the claim follows by interpolating the local decay bootstrap bound (4.38) and the bound (4.37) for the highest Sobolev norm for Z ≤ Z * c := (Z * ) c and using the global weighted Sobolev bound for (4.34) We will also use the bound for the damped profile from (4.7), (4.8) and (4.9): We will also use the bound which follows from and α + β ≤ 1.
step 2 Energy identity. We run (5.10) with with ξ k (x) = 1 for x ≤ 1 and ξ k (x) = x 2σ(k) for x > 1, and estimate all terms. In our notations From (4.28), (4.29) and recalling m 0 = 4km 9 + 1: which we will use below. The following additional inequality will be of particular significance (b = (Z * ) 2−r ): step 3 Leading order terms. In what follows, we will systematically use the standard Pohozhaev identity: which becomes in the case of spherically symmetric functions Cross terms. We consider We compute: The last 2 terms require an integration by parts: where in penultimate inequality we used the pointwise bound (4.39). We now estimate the source term from(5.25): (5.27) and hence, using (5.18), We estimate similarly, The remaining cross terms are estimated as follows.
where we used that p ≥ 1 and a trivial bound |ρ D | 1. Similarly, The other remaining cross term is estimated using an integration by parts: ρ k terms. We compute using (5.5): where we used the interpolation bound (5.18). Similarly, using that χ k,k,σ = Z χ k,k−1,σ and |ρ k | ≤ |∇ρ k−1 | as well as (5.5), (5.18) gives Next using we estimate after an integration by parts: We use the pointwise bootstrap bound (4.39) For the nonlinear term, we use the Pohozhaev identity (5.26) and the pointwise bound (5.28) to estimate by the interpolation bound (5.18) Note that the last term in the case k = 0 should be treated with the help of the boundρ ρ D and the estimate (5.27). For k = 0, we simply use |ρ k | ≤ |∇ρ k−1 |. We recall that by definition of the norm: Hence, by the interpolation bound, For the nonlinear term, we integrate by parts and use (5.28): From Pohozhaev (5.26) and (5.5): Integrating by parts and using (5.5), (5.30): We now claim the fundamental behavior and Assume (5.30), (5.31), we obtain Proof of (5.30). From (4.9): which yields: and (5.31) follows from (5.28).
Ψ (k) terms. Integrating by parts: where we used (5.27). Next: Similarly, using ∂ Z H 2 = O 1 Z r : where we also used that r > 2, k = 0 and Then using (5.28): and from (5.26), (5.28): We now carefully compute from (5.26) again: Hence the final formula recalling (5.31): Loss of derivatives terms. We integrate by parts the non linear term which must loose derivatives: We now use (5.20) for 0 ≤ k ≤ k m − 1 which implies
Conclusion for linear terms. The collection of above bounds yields: step 4 F 1 terms. We recall (5.7) and claim the bound: Source term induced by localization. Recall (5.2) From the proof of (5.30) Therefore, using the profile equation for ρ P , we obtaiñ Similarly, from (5.25): [∂ k , H 1 ] term. We use (5.5) to estimate: Hence: where we used the bootstrap bound (4.39), the decay of b 2 and (5.25).
We decompose ρ T = ρ D +ρ and control the ρ D term using the bound The corresponding contribution to (5.44): where we used (5.27) in the last step.
-case m q ≤ 4km 9 , then from (4.39): and hence, from (5.19) and (5.24), the contribution of this term We now assume m q ≥ 4km 9 + 1 and recall m q ≤ j 1 + 1 ≤ k + 1 ≤ k m . -case m q−1 ≤ 4km 9 , then from (4.39): If m q ≤ k then On the other hand, if m q = k + 1, then, using (5.20) where we used the interpolation bound p−1 and the condition which follows from r > 2.

Pointwise bounds
We are now in position to close the control of the pointwise bounds (4.39). We start with inner bounds |x| 1: Lemma 6.1 (Interior pointwise bounds). For all 0 ≤ k ≤ 2km 3 : where d 0 is a smallness constant depending on data.
Proof. We integrate (5.13) in time and obtain, by choosing 0 <ν ≪ c + c km , ∀0 ≤ m ≤ k m − 1: for some small constant d 0 , which can be chosen to be arbitrarily small by increasing τ 0 . Below, we will adjust d 0 to remain small while absorbing any other universal constant.
Similar to the above, we also have the following exterior bounds for |x| ≥ 1: Lemma 6.2 (Exterior pointwise bounds). There holds: where d 0 is a smallness constant depending on data.
Proof. We recall (5.22) and (5.23) and, in the case 0 ≤ k ≤ 2km 3 , bound We observe from (4.9), (4.28), (5.12) and b = Z * 2−r that for Z ≥ Z * Similarly, Now, for a spherically symmetric function u, Z ≥ Z * and an arbitrary λ > 0 We apply this to u = Z kρ(k) where we used the already proved interior bounds (6.1). This, together with (6.2), immediately implies the exterior bound for ∂ k Z ρ and 1 ≤ k ≤ 4km 9 + 1. The corresponding bound for ∂ k Z Ψ is obtained similarly using u = Z k Ψ (k) and λ =ν. To prove the result for ρ in the case of k = 0 we note that we could run the above argument for u = Z k+ν 2ρ (k) (Z * ) 2ν ρ D and λ =ν 2 , which would imply It is a stronger estimate and, more crucially in the case of k = 1, an estimate which can be integrated in Z to produce the desired bound for k = 0. Similar argument applies to Ψ. Finally, the regime 4km 9 + 1 ≤ k ≤ 2km 3 can be treated by a combination of the argument above and the corresponding interior one. We note that the weight function ξ k , which determines the Z Z * dependence and thus relevant in the exterior, remains exactly the same in the whole range 0 ≤ k ≤ 2km 3 + 1. We omit the details.

Highest Sobolev norm
In this section we improve the bootstrap bound (4.37) on the highest unweighted Sobolev norm of (ρ, Ψ). Specifically, for (see (4.23)) we will establish the following Proof of Proposition 7.1. This follows from the global unweighted quasilinear energy identity. We let k m = 2K m , K m ∈ N and denote in this section k = k m ,ρ (k) = ∆ Kmρ , Ψ (k) = ∆ Km Ψ.
We recall the notation (5.11) step 1 Control of lower order terms. We recall the notation: Observe from (4.29) that for m 0 ≤ m ≤ k m − 1: and the same holds for 0 ≤ m ≤ m 0 for k m large enough. Hence, and a similar estimate for Ψ, imply, using (6.2): By Remark 4.2 we can replace (up to the lower order terms controlled as above) I km with We claim: there exist k * m (d, r, p), c d,r,p > 0 such that for all k m > k * m (d, r, p), there holds: Integrating the above in time, using (4.24), (7.4), yields (7.2).
We use [∆ Km , Λ] = k m ∆ Km and recall (C.1): which gives: where ∇ j = ∂ α 1 1 . . . ∂ α d d , j = α 1 + · · · + α d denotes a generic derivative of order j. Using (C.1) again: For the second equation, we have similarly: and We then run the global quasilinear energy identity similar to (5.10) with χ = 1 and obtain: We now estimate all terms in (7.13). The proof is similar to that one of Proposition (5.2) with two main differences: the absence of a cut-off function χ, and a priori control of lower order derivatives from (7.4). The challenge here is to avoid any loss of derivatives and to compute exactly the quadratic form at the highest level of derivatives. The latter will be shown to be positive on a compact set in Z provided k m > k * m (d, r, p) ≫ 1 has been chosen large enough.
In what follows, below, we will use δ > 0 as a small universal constant and will assume that the pointwise bounds (6.1) obtained on the lower order derivatives of ρ and Ψ are dominated by δ. On the set Z ≤ Z * , this will often be a source of smallness, while for Z ≥ Z * , we may use the bootstrap bounds (4.39) and the δsmallness will be generated by extra powers of Z. We also note that from (7.6) the quadratic form is expected to be proportionate to k m I km . Choosing k m large will allow us to dominate other quadratic terms without smallness but with the uniform dependence on k m . The notation will allow dependence on k m , while O will indicate a bound independent of k m . As before, d 0 (as well as d) will denote small constants, dependent on the data (or, more precisely, on τ 0 ), that can be made arbitrarily small. In particular, we will use ρ, Ψ km−1,σ(km−1) ≤ d 0 . (7.14) The constants δ ≫ d 0 will be assumed to be smaller than any power of k m , so that our calculations will be unaffected by combinatorics generated by taking k m derivatives of the equations.
step 3 Leading order terms.
Cross term. Recall (5.26): Letting g = g 1 + g 2 yields a bilinear off-diagonal Pohozhaev identity: We may therefore integrate by parts the one term in (5.10) which has too many derivatives: We estimate similarly: We use |ρ| ρ T + |Λρ| Z c ρ T δ, 0 < c ≪ 1 (7.15) to compute the first coupling term: The second coupling term is computed after an integration by parts using (7.15), the control of lower order terms (7.4) and the spherically symmetric assumption: where in the last step we used that |∂ Z ρ D | |Z|ρ D 1 Z 2 . ρ k terms. We compute: We now use the global lower bound, see properties (2.21) and (2.22) of the the profile (w, σ), to estimate using (8.17), (7.4): Next, using we estimate from (4.39): For the nonlinear term, we use (4.39), (5.26), (7.4) to estimate Next: since we are assuming that d 0 ≪ δ, and for the nonlinear term after an integration by parts: From Pohozhaev (5.26): Integrating by parts and using (8.17), (5.30): Note that the above two bounds, even though dependent on the highest order derivatives, contain no k dependence. Ψ k terms. After an integration by parts: Then and similarly, using (8.17), (7.4): where the Ψ 2 k term is controlled, with the help of the bound by using the already bounded (ρ, Ψ) km−1,σ(km−1) -norm. Then, from (6.1) and (6.9): and using (5.26): Arguing verbatim as in the proof of (5.32) produces the bound step 4 F 1 terms. We claim the bound: Source term induced by localization. From (5.36), for k m large enough: [∆ Km , H 1 ] term. We estimate from (5.37), (7.14) A k (ρ) term. From (7.7), (7.14): and (7.16) is proved for this term.
Nonlinear term. After changing indices, we need to estimate For the profile term: and therefore, recalling (5.40), (7.14): Similarly, after taking a derivative: The δJ km term above controls the case j 2 = k − 1.
Highest order term We are left with estimating the highest order term: We treat this term by integration by parts in time using (7.8): and we treat all terms in (7.18). We will systematically use the smallness (4.27). The ∂ τρk term is integrated by parts in time: and the boundary term in time is small We then estimate: Using the extra decay in Z and ∆Ψ L ∞ ≤ δ ≪ 1: Similarly, after an integration by parts: Similarly, after an integration by parts using (4.39): and similarly: step 7 Conclusion for k = k m (d, r) large enough. We now sum the collection of above bounds and obtain the differential inequality with k = k m .
We recall from (2.21), (2.22): and we now claim the pointwise coercivity of the coupled quadratic form: ∃c d,p > 0 such that ∀Z ≥ 0, which, after taking k > k * (d, p) large enough, concludes the proof of (7.6).
Proof of (7.20). The coupling term is lower order for Z large: for Z > Z(δ) large enough. On a compact set using the smallness (4.27), (7.20) is implied by: We compute the discriminant: We compute from (2.9) recalling (2.20): and hence from (2.21), (2.22) the lower bound: which together with (7.19) concludes the proof of (7.20).

Control of low Sobolev norms and proof of Theorem 1.1
Our aim in this section is to control weighted low Sobolev norms in the interior r ≤ 1 (Z ≤ Z * ). On our way we will conclude the proof of the bootstrap Proposition 4.4. Theorem 1.1 will then follow from a classical topological argument. 8.1. Exponential decay slightly beyond the light cone. We use the exponential decay estimate (3.64) for a linear problem to prove exponential decay for the nonlinear evolution in the region slightly past the light cone. We recall the notations of Section 3, in particular Z a of Lemma 3.2.

Appendix D. Behaviour of Sobolev norms
We compute Sobolev norms assuming that the leading part of the solution is given by (1.8). Computations below are formal but could be justified as a consequence of the bootstrap estimates. Dirichlet energy of the profile. We recall (1.7), (1.8) and compute: