The energy-critical nonlinear wave equation with an inverse-square potential
aa r X i v : . [ m a t h . A P ] A ug THE ENERGY-CRITICAL NONLINEAR WAVE EQUATIONWITH AN INVERSE-SQUARE POTENTIAL
CHANGXING MIAO, JASON MURPHY, AND JIQIANG ZHENG
Abstract.
We study the energy-critical nonlinear wave equation in the pres-ence of an inverse-square potential in dimensions three and four. In the de-focusing case, we prove that arbitrary initial data in the energy space lead toglobal solutions that scatter. In the focusing case, we prove scattering belowthe ground state threshold. Introduction
We consider the initial-value problem for the energy-critical nonlinear wave equa-tion (NLW) with an inverse-square potential. The underlying linear problem isgiven in terms of the operator L a := − ∆ + a | x | − . Here we restrict to dimensions d ≥ a > − ( d − ) , and we consider L a as the Friedrichs extension of the quadratic form defined on C ∞ c ( R d \{ } ) via f Z R d |∇ f ( x ) | + a | x | − | f ( x ) | dx. The lower bound on a guarantees positivity of L a ; in fact, by the sharp Hardyinequality one finds that the standard Sobolev space ˙ H is equivalent to the Sobolevspace ˙ H a defined in terms of L a (see Section 2). Furthermore, when a = 0 werecover the standard Laplacian.We consider the following nonlinear wave equation: ( ∂ t u + L a u + µ | u | d − u = 0 , ( u, ∂ t u ) | t =0 = ( u , u ) . (1.1)Here u is a real-valued function on R d with d ≥ µ ∈ {± } corresponds tothe defocusing and focusing equations, respectively. This is a Hamiltonian equation,with the conserved energy given by E a [ ~u ] = Z |∇ u | + | ∂ t u | + a | x | − | u | + µ d − d | u | dd − dx, where ~u = ( u, ∂ t u ). The notation E a [ f ] should be understood as E a [( f, L a arises often in mathematics and physics in scaling limits ofmore complicated problems, for example in combustion theory, the Dirac equationwith Coulomb potential, and the study of perturbations of space-time metrics suchas Schwarzschild and Reissner–Nordstr¨om [4, 17, 43, 44].One particularly interesting feature of the inverse-square potential is that it hasthe same scaling as the Laplacian. In particular, one cannot in general treat L a as a perturbation of − ∆, which contributes to the mathematical interest of this particular model. An additional consequence is that (1.1) has a scaling symmetry,namely, u ( t, x ) λ d − u ( λt, λx ) ,∂ t u ( t, x ) λ d ( ∂ t u )( λt, λx ) . (1.2)This rescaling leaves the energy invariant and identifies the scaling-critical spaceof initial data to be the energy space ˙ H × L . We therefore call (1.1) an energy-critical equation, and indeed when a = 0 the equation reduces to the standardenergy-critical NLW, which has been the center of a great deal of research in recentyears.On the other hand, the presence of the inverse-square potential breaks translationsymmetry, introducing new challenges into the analysis of (1.1). In particular, ourwork fits in the context of recent work on dispersive equations in the presence ofbroken symmetries, which have also attracted a great deal of interest in recent years(see e.g. [9–15, 20, 24, 25, 28, 29, 35] and in particular [16, 22, 23, 33] for the case ofnonlinear Schr¨odinger equations with an inverse-square potential).We consider the problem of global well-posedness and scattering for (1.1). Inthe defocusing case, we will prove scattering for arbitrary data in the energy space.In the focusing case, we will prove scattering below the ground state threshold.These results parallel those established for the standard energy-critical NLW (seee.g. [2, 6, 7, 18, 19, 30, 34, 37, 39]), and as in many of those works we will proceed viathe concentration-compactness/rigidity approach. Before comparing our work withthe existing literature, however, let us state our main results more precisely.Implicit in the statements below is the fact that any initial data in the energyspace leads to a unique solution that exists at least locally in time (see Proposi-tion 2.7). This local result leads to some restrictions on the parameter a , which wewill state in terms of the following constant: c d = ( d = 3 , d = 4 . (1.3)See Section 2.1 and Section 2.2 for more details.As above, given a solution u to a (linear or nonlinear) wave equation, we write ~u = ( u, ∂ t u ). We say a solution u to (1.1) scatters if there exist solutions v ± ( t ) to( ∂ t + L a ) v ± = 0 such thatlim t →±∞ k ~u ( t ) − ~v ± ( t ) k ˙ H × L = 0 . Our result in the defocusing case is the following theorem.
Theorem 1.1.
Let d ∈ { , } , a > − ( d − ) + c d , and µ = +1 . For any ( u , u ) ∈ ˙ H × L , the corresponding solution to (1.1) is global and scatters. We next turn to the focusing case. In this case, there exist global nonscatter-ing solutions, and hence we do not expect a scattering result without some sizerestrictions. Indeed, fixing a > − ( d − ) and defining β > a = ( d − ) [ β − ground state soliton for (1.1) is the static solution to (1.1)defined by W a ( x ) := [ d ( d − β ] d − (cid:2) | x | β − | x | β (cid:3) d − , which arises as an optimizer of a Sobolev embedding inequality (see [22, 38] andSection 2.3 below). Our result in the focusing case is a scattering result below the LW WITH INVERSE-SQUARE POTENTIAL 3 ground state threshold. In the following, we write a ∧ { a, } and let ˙ H a denote the Sobolev space defined in terms of √L a (see Section 2.1). Theorem 1.2.
Let d ∈ { , } , a > − ( d − ) + c d , and µ = − . Let ( u , u ) ∈ ˙ H × L satisfy E a [( u , u )] < E a ∧ [ W a ∧ ] and k u k ˙ H a < k W a ∧ k ˙ H a ∧ . (1.4) Then the corresponding solution to (1.1) is global and scatters.
As mentioned above, the restriction a > − ( d − ) guarantees that the operator L a is positive, while the further restriction a > − ( d − ) + c d arises in the devel-opment of the local theory for (1.1). Note that both results still include a rangenegative values of a , in which case the potential is attractive. This is in contrast tomany results for dispersive PDE with potentials, in which case one must considerrepulsive (or perturbative) potentials in order to obtain scattering. We restrictto dimensions d ∈ { , } to guarantee that the nonlinearity in (1.1) is algebraic(quintic and cubic, respectively), which is primarily for technical convenience andalready includes the most interesting cases. We expect that the results extend tohigher dimensions, as well. Finally, let us point out that the results above holdwithout any radial assumption on the initial data.Theorems 1.1 and 1.2 parallel the existing results for the standard energy-criticalNLW without potential (henceforth the free NLW ). In fact, as we will see, ourresults rely in an essential way on these existing results. Comparing with the resultof Kenig and Merle on the focusing NLW [19], we find that the scattering thresholdfor (1.1) is the same as that for the standard NLW in the range a > k u k ˙ H > k W k ˙ H . The existence ground state soliton then showsthat the E [ W ] is the correct energy threshold for a simple blowup/scatteringdichotomy. The situation is completely analogous when a ≤
0. On the other hand,when a > W a ; nonetheless,the condition appearing in Theorem 1.2 is sharp in terms of obtaining uniformspace-time bounds. In particular, the proof of Theorem 1.2 will show that thesolutions constructed have critical space-time norms controlled by C ( E a ∧ [ W a ∧ ] − E a [( u , u )]) for some function C ; we will show that this constant diverges as oneapproaches the threshold. Similar results have been obtained in [23, 29].We summarize the results just mentioned in the following theorem. Theorem 1.3.
Let d ∈ { , } , a > − ( d − ) + c d , and µ = − . (i) If ( u , u ) ∈ ˙ H × L satisfy E a [( u , u )] < E a ∧ [ W a ∧ ] and k u k ˙ H a > k W a ∧ k ˙ H a ∧ , then the corresponding solution to (1.1) blows up in finite time in both timedirections. (ii) If a > , then there exist a sequence of global solutions u n such that E a [ ~u n ] ր E [ W ] and k u n (0) k ˙ H a ր k W k ˙ H , with lim n →∞ k u n k L d +1) d − t,x ( R × R d ) = ∞ . C. MIAO, J. MURPHY, AND J. ZHENG
Our main focus in this paper is on the scattering results, Theorem 1.1 and 1.2.In the remainder of the introduction, we will primarily discuss the proof of theseresults. The proof of Theorem 1.3 is relatively straightforward and will be explainedin Section 7.The strategy of proof for Theorems 1.1 and 1.2 is the concentration-compactnessapproach to induction on energy, often referred to as the Kenig–Merle roadmap.In particular, supposing that either theorem is false, we show that there wouldexist a special type of solution to (1.1) (constructed as a minimal blowup solution)possessing certain compactness properties. We then preclude the possibility thatsuch solutions can exist. The precise notion of compactness is given by the following:
Definition 1.4 (Almost periodic solution) . Let ~u be a nonzero solution to (1.1)on an interval I . We call ~u almost periodic (modulo symmetries) if there exist aspatial center x : I → R d and frequency scale N : I → (0 , ∞ ) such that the set (cid:8)(cid:0) N ( t ) d − u ( t, N ( t )[ x − x ( t )]) , N ( t ) d ( ∂ t u )( t, N ( t )[ x − x ( t )]) (cid:1) : t ∈ I (cid:9) is pre-compact in ˙ H × L .The first main step of the proof, the reduction to almost periodic solutions, ap-pears as Theorem 4.3 below. The general strategy is well-established, with essen-tially two key ingredients: (i) a linear profile decomposition adapted to a Strichartzestimate (see Proposition 3.2) and (ii) a corresponding ‘nonlinear profile decompo-sition’. Both of these steps involve additional difficulties in the setting of equationswith broken symmetries. For (i), the new difficulty is related mostly to under-standing the convergence of certain linear operators that arise due to the failure oftranslation symmetry (see Lemma 3.3 and Lemma 3.4). For (ii), the key difficultyarises from the construction of (scattering) nonlinear solutions associated to profileswith a translation parameter tending to infinity. Because of the broken translationsymmetry, one cannot simply solve (1.1) and then incorporate the translation. In-stead, roughly speaking, one constructs a solution to the free NLW, incorporatesthe translation, and (because the profile lives far from the origin) shows that theresult is an approximate solution to (1.1). An application of the stability theoryfor (1.1) then yields the true solution, as desired. The construction of a suitable(scattering) solution to the free NLW relies on the full strength of [2, 19]. For moredetails, see Proposition 4.2.With the necessary ingredients in place, the reduction to almost periodic solu-tions (Theorem 4.3) follows along fairly standard lines (see Section 4.2). After this,it is useful to make some further reductions to the class of almost periodic solutionsthat we consider. Recall from Definition 1.4 above that almost periodic solutionsare described in terms of a spatial center x ( t ) and frequency scale N ( t ). Becauseof the result in Proposition 4.2, we firstly find that we must have x ( t ) ≡
0; indeed,as described above, profiles with translation parameters tending to infinity corre-spond to scattering solutions and hence do not arise in the construction of minimalblowup solutions. In Theorem 4.7, we adapt arguments of [19] to further reduce totwo scenarios, namely, the ‘forward global’ scenario and the ‘self-similar’ scenario.The preclusion of both scenarios relies heavily on certain virial/Morawetz iden-tities. For NLW with a general potential V ( x ), these identities will involve a termof the form − x · ∇ V (see Lemma 2.2, for example). For repulsive potentials (i.e.those satisfying x · ∇ V ≤ LW WITH INVERSE-SQUARE POTENTIAL 5 often restricted to the case of repulsive potentials. For the inverse square potential V ( x ) = a | x | − , one has the identity x · ∇ V = − V (a consequence of scaling). Inparticular, while the potential is repulsive only for a ≥
0, this identity ultimatelyallows us to prove suitable virial/Morawetz estimates even when a < x ( t ) ≡
0. This should be contrasted with [19], where theauthors must argue (using Lorentz boosts) that minimal blowup solutions have zeromomentum, which then allows for sufficient control over x ( t ) to run the localizedvirial argument. In the focusing case, we must also rely on the coercivity given bysharp Sobolev embedding and the fact that the minimal blowup solution is belowthe ground state in the sense of (1.4) (see Lemma 2.10).Finally, we rule out the self-similar scenario in Section 6. In this scenario, thesolution u blows up at time t = 0 and is supported at each t > B t (0), with N ( t ) = t − . To rule out such solutions, we firstly prove a virial/Morawetz estimate(Proposition 6.1), which in particular implies the vanishing of the quantity ∂ t u + x · ∇ u + d − ut along a sequence t n →
0. This particular estimate is similar to one used to studywave maps (appearing e.g. in [8, 40, 41]), and is particularly closely related to theestimate appearing in [5]. Using Proposition 6.1 and almost periodicity, we are ableto extract a true self-similar solution to (1.1), that is, a (nonzero) solution of theform v ( t, x ) = (1 + t ) − [ d − f ( x t ) . It follows that f solves a degenerate elliptic PDE in the unit ball and vanishesnear the boundary in a certain sense (see Proposition 6.2). At this point, we findourselves essentially in the same position as Kenig and Merle [19]; indeed, for thisportion of the argument the difficulty arises only at the boundary of the unit ball,and we may safely include the potential term in the nonlinearity. Following closelythe arguments of [19], we employ a change of variables to remove the degeneracy andinvoke unique continuation results to deduce that f ≡
0, yielding a contradictionand completing the proof of Theorems 1.1 and 1.2.As just described, our arguments in the self-similar scenario differ from Kenig andMerle [19] essentially only in the extraction of the elliptic solution. In particular,[19] employs a self-similar change of variables, while we utilize a virial/Morawetzestimate (actually closer to the spirit of the arguments of [5]). In fact, one observesthat setting a = 0 throughout Section 6 leads to a modified proof of the preclusionof self-similar almost periodic solutions for the free NLW.The rest of the paper is organized as follows: • In Section 2 we first set up some notation and record some useful lemmas,including the useful virial/Morawetz identity. In Section 2.1, we collectsome harmonic analysis tools adapted to L a , including some results con-cerning the equivalence of Sobolev spaces as well as Strichartz esimates. InSection 2.2 we record the basic local well-posedness and stability theory for(1.1). Finally, in Section 2.3, we record some results related to the sharpSobolev embedding and the ground state solution for (1.1). • In Section 3 we develop concentration compactness tools for (1.1). The mainresult of this section is the linear profile decomposition, Proposition 3.2.
C. MIAO, J. MURPHY, AND J. ZHENG • In Section 4, we prove the existence of minimal blowup solutions under theassumption that Theorem 1.1 or Theorem 1.2 fails (cf. Theorem 4.3). Wethen refine the class of solutions that we need to consider (see Theorem 4.7). • In Section 5 we preclude the forward global case of Theorem 4.7. • In Section 6 we preclude the self-similar case of Theorem 4.7, thus complet-ing the proof of Theorems 1.1 and 1.2. • Finally, in Section 7, we give the proof of Theorem 1.3.
Acknowledgements.
C. M. and J. Z. were supported by NSFC Grants 11831004.Part of this work was completed while J. M. was supported by the NSF postdoctoralfellowship DMS-1400706 at UC Berkeley. We are grateful to R. Killip for someuseful discussions related to Lemma 3.4 below. We are also grateful to H. Jia forhelping us to understand the works [5, 19], which played a key role in developingSection 6 of this paper. 2.
Notation and lemmas
We will employ some standard geometric notation. We let g αβ = diag( − , , . . . , R d and denote the inverse metric by g αβ . Greek indices take values in { , , . . . , d } while Roman indices take valuesin { , . . . , d } . We employ Einstein summation convention, and we raise and lowerindices with respect to the metric: ∂ α = g αβ ∂ β . For example, (1.1) may be written ∂ α ∂ α u = a | x | − u + µ | u | d − u. We write ( x α ) for space-time coordinates, with x = t . We use ∇ for the gradientin the spatial variables only; the space-time gradient will be denoted by ∇ t,x .We use the standard Lebesgue spaces L p ( R d ), as well as the mixed space-timenorms L qt L rx ( R d ), defined by k u k L qt L rx ( R d ) = (cid:13)(cid:13) k u ( t ) k L rx ( R d ) k L qt ( R ) . We write q ′ ∈ [1 , ∞ ] for the dual exponent of q ∈ [1 , ∞ ], i.e. the solution to q + q ′ = 1.We write A . B to denote A ≤ CB for some C >
0. We can similarly define A & B . We write a ± to denote a quantity of the form a ± ε for some small ε > T a that takes a pair of real-valued functions ( φ, ψ ) andreturns a single complex-valued function defined by T a ( φ, ψ ) := φ + i L − a ψ. (2.1)We apply this mapping to ~u = ( u, ∂ t u ), where u solves (1.1), that is, T a ~u = u + i L − a ∂ t u. (2.2)Thus u = Re T a ~u , and u solves (1.1) with data in ˙ H a × L if and only if thecomplex-valued function v := T a ~u solves i∂ t v − L a v − µ L − a ( | Re v | d − Re v ) = 0 (2.3)with data in ˙ H a . In these variables we have E a [ ~u ] = ˜ E a [ T a ~u ] := Z |L a T a ~u | + µ d − d | Re T a ~u | dd − dx. (2.4) LW WITH INVERSE-SQUARE POTENTIAL 7
We will need the following refinement of Fatou’s lemma in Section 3.
Lemma 2.1 (Refined Fatou, [3]) . Let ≤ p < ∞ and let { f n } be bounded in L p .If f n → f almost everywhere, then lim n →∞ Z (cid:12)(cid:12) | f n | p − | f n − f | p − | f | p (cid:12)(cid:12) dx = 0 . Finally, we record the following virial identity that will be used on a few occasionsbelow. The proof follows from direct computation and integration by parts.
Lemma 2.2 (Virial identity) . Fix a weight w : R d → R and a solution u : R d → R to the wave equation ∂ α ∂ α u = V u + G ′ ( u ) . Then ∂ t Z − ∂ t u [ ∇ u · ∇ w + u ∆ w ] dx = Z ∇ u · ∇ w ∇ u + ∆ w [ uG ′ ( u ) − G ( u )] − ∇ w · ∇ V u − ∆∆ w ( u ) dx. Harmonic analysis tools.
A harmonic analysis toolkit adapted to L a wasdeveloped in [21]. In this section, we will import several relevant results. We willalso record some Strichartz estimates adapted to the linear wave equation withinverse-square potential, which were established in [4].For r ∈ (1 , ∞ ) we write ˙ H ,ra and H ,ra for the homogeneous and inhomogeneousSobolev spaces defined in terms of L a ; these have norms k f k ˙ H ,ra = k p L a f k L r , k f k H ,ra = k p L a f k L r . When r = 2 we write ˙ H , a = ˙ H a .Let us introduce the parameter σ = d − − (cid:2) ( d − ) + a (cid:3) . One of the main results in [21] is the following result concerning the equivalence ofSobolev spaces.
Lemma 2.3 (Equivalence of Sobolev spaces, [21]) . Let d ≥ , a > − ( d − ) , and s ∈ (0 , . • If p ∈ (1 , ∞ ) satisfies s + σd < p < min { , d − σd } , then k|∇| s f k L p . kL s a f k L p . • If p ∈ (1 , ∞ ) satisfies max { sd , σd } < p < min { , d − σd } , then kL s a f k L p . k|∇| s f k L p . We will use Littlewood–Paley projections defined through the heat kernel, i.e. P aN = e −L a /N − e − L a /N , where N ∈ Z . As was shown in [31, 32], the heat kernel e − t L a ( x, y ) has upper andlower bounds of the form C (1 ∧ √ t | x | ) σ (1 ∧ √ t | y | ) σ e −| x − y | /ct . (2.5) C. MIAO, J. MURPHY, AND J. ZHENG
To state results, it will be useful to define the exponent q = ( ∞ if a ≥ , dσ if − ( d − ) < a < . We write q ′ for the dual exponent in both cases. We record the harmonic analysistools we need in the following proposition. Proposition 2.4 (Harmonic analysis tools, [21]) . Let q ′ < q ≤ r < q . • We have the following expansion: f = X N ∈ Z P aN f as elements of L r . • We have the following Bernstein estimates: – The operators P aN are bounded on L r . – The operators P aN are bounded from L q to L r with norm bounded by N dq − dr . – For any s ∈ R , N s k P aN f k L r ∼ kL s a P aN f k L r . • We have the square function estimate: (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X N ∈ Z | P aN f | (cid:19) (cid:13)(cid:13)(cid:13)(cid:13) L r ∼ k f k L r . We next turn to Strichartz estimates for the linear wave equation with inverse-square potential. Here we import results of Burq, Planchon, Stalker, and Tahvildar-Zadeh [4], specifically Theorem 5 and Theorem 9 therein. We state the estimatesin terms of the operators e ± it √L a and specialize to dimensions d ∈ { , } . Proposition 2.5 (Strichartz) . Let q, r ≥ satisfy the wave admissibility condition q + d − r ≤ d − , where in d = 3 we additionally require q, ˜ q > . Define γ via the scaling relation q + dr = d − γ. For any time interval I we have k e ± it √L a f k L qt L rx ( I × R d ) . k f k ˙ H γ ( R d ) provided the following conditions hold: • If d = 3 , then we require − min { , q a + − , q a + + 1 } < γ < min { , q a + + , q a + + 1 − q } . • If d = 4 , then we require − min { , √ a + 4 − , √ a + 1 + 1 } < γ < min { , √ a + 4 + , √ a + 1 + 1 − q } . We will also need an inhomogeneous estimate. In particular, using Proposi-tion 2.5, Lemma 2.3, and the Christ–Kiselev lemma, we have the following:
LW WITH INVERSE-SQUARE POTENTIAL 9
Corollary 2.6 (Strichartz) . Let q, r, γ be as in Proposition 2.5 and let ˜ q, ˜ r, ˜ γ bedefined similarly. Suppose q, ˜ q > . Then for any time interval I ∋ t , we have (cid:13)(cid:13)(cid:13)(cid:13)Z tt e i ( t − s ) √L a F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ( I × R d ) . k|∇| γ +˜ γ F k L ˜ q ′ t L ˜ r ′ x ( I × R d ) . Local well-posedness and stability.
We next develop the local theory for(1.1), including a stability result. As the arguments are rather standard, we willbe rather brief. As usual, the results rely primarily on Strichartz estimates, whichwere recorded in the previous section. We will construct solutions that lie locallyin L ∞ t ( ˙ H × L ) as well as the Strichartz space S ( I ) := L d +1) d − t,x ( I × R d ) ∩ L d +2 d − t L d +2 d − ) x ( I × R d ) . (2.6)Note that the scaling of S corresponds to γ = 1 in the Strichartz estimates appearingabove, and that we are able to use these spaces provided we choose a > ( − + d = 3 − d = 4 . This is the origin of the constant c d defined in (1.3) and appearing in the statementsof the main results, Theorem 1.1 and Theorem 1.2.Writing the Duhamel formulation of (1.1), namely, u ( t ) = cos( t p L a ) u + sin( t √L a ) √L a u − µ Z t sin(( t − s ) √L a ) √L a ( | u | d − u )( s ) ds, we can run a contraction mapping in the space L ∞ t ( ˙ H × L ) ∩ S by relying on thenonlinear estimate (cid:13)(cid:13)(cid:13)(cid:13)Z t sin(( t − s ) √L a ) √L a ( | u | d − u )( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L ∞ t ˙ H ∩ S . k| u | d − u k L t L x . k u k d +2 d − S . The conclusion is the following local result.
Proposition 2.7 (Local well-posedness) . Let ( u , u ) ∈ ˙ H × L , d ∈ { , } , and a > − ( d − ) + c d .There exists η such that if k cos( t p L a ) u + sin( t √L a ) √L a u k S ( I ) < η for some < η < η , then there exists a solution to (1.1) on I satisfying k u k S ( I ) . η . In particular, data in ˙ H × L leads to local-in-time solutions.The solution may be extended as long as the S -norm remains finite, and if thesolution is global with k u k S ( R ) < ∞ then the solution scatters in both time directions.Finally, given a final state ( u +0 , u +1 ) ∈ ˙ H × L , we may construct a solutionon an interval ( T, ∞ ) that scatters to ( u +0 , u +1 ) as t → ∞ . A similar result holdsbackward in time. Standard arguments relying primarily on the same Strichartz estimates as aboveyield the following stability result for (1.1).
Proposition 2.8 (Stability) . Let d, a be as in Proposition 2.7. I be a time intervaland let ˜ u satisfy ∂ t ˜ u + L a ˜ u + µ | ˜ u | d − ˜ u + e = 0 for some function e : I × R d → R . Suppose that k ˜ u k S ( I ) + k (˜ u ( t ) , ∂ t ˜ u ( t )) k ˙ H × L ≤ L for some t ∈ I and L > . There exists ε = ε ( L ) such that for < ε < ε wehave the following: if k ~u − (˜ u ( t ) , ∂ t ˜ u ( t )) k ˙ H × L + k e k L t L x ( I × R d ) < ε, then there exists a solution u to (1.1) on I with ~u ( t ) = ~u satisfying k u − ˜ u k S ( I ) . L ε and k ~u k L ∞ t ( I ; ˙ H × L ) + k u k S ( I ) . . For an introduction to these types of results, we refer the reader to [26].We remark that by applying the transformation T a introduced above, one hasequivalent local well-posedness and stability results stated in terms of the equation(2.3) with initial data in ˙ H . We will use both versions of these results below.2.3. Variational analysis.
In this section we record results related to the sharpSobolev embedding k f k L dd − ( R d ) ≤ C a k f k ˙ H a ( R d ) , (2.7)where C a denotes the sharp constant.Much of the analysis that we need was carried out in [22]; see also [38].For a > − ( d − ) , we define β > a = ( d − ) [ β − σ = d − (1 − β ). The ground state soliton is defined by W a ( x ) = [ d ( d − β ] d − (cid:2) | x | β − | x | β (cid:3) d − . We have that W a solves L a W a − | W a | d − W a = 0 , and k W a k H a = k W a k dd − L dd − = πd ( d − (cid:2) √ πβ d − Γ( d +12 ) (cid:3) d . Note that the first identity above and [22, Proposition 7.2] imply C a = k W a ∧ k − d ˙ H a ∧ . (2.8)In Section 4.1, we will need to construct scattering nonlinear solutions to (1.1)that are parametrized by a spatial center approaching infinity. To do this, we needto approximate by solutions to the nonlinear wave equation without potential; inparticular, we need to rely on the scattering result of [19]. Consequently, in thefocusing case we need to be sure that initial data lying below the threshold statedin Theorem 1.2 also lie below the appropriate threshold for the equation withoutpotential. This fact is guaranteed by the following corollary. Corollary 2.9 (Comparison of thresholds) . Let a > − ( d − ) . Then E a ∧ [ W a ∧ ] ≤ E [ W ] and k W a ∧ k ˙ H a ∧ ≤ k W k ˙ H . LW WITH INVERSE-SQUARE POTENTIAL 11
Proof.
There is nothing to prove when a ≥
0, so let us fix a <
0. We begin byobserving that k f k ˙ H a < k f k ˙ H , which implies C ≤ C a ; indeed, k f k L dd − ≤ C a k f k ˙ H a < C a k f k ˙ H . In light of (2.8), we have k W a k ˙ H a ≤ k W k ˙ H , which is one of the desired inequalities. For the remaining inequality, we again callon (2.8) and use the inequality just established to observe that E a ( W a ) = d k W a k H a ≤ d k W k H = E ( W ) . This completes the proof. (cid:3)
Finally, we record the following lemma, which is almost identical to [22, Corol-lary 7.6] and in particular follows from the same proof appearing there.
Lemma 2.10 (Coercivity) . Let d ≥ and a > − ( d − ) . Suppose u : I × R d → R is a solution to (1.1) with µ = − and initial data ~u ∈ ˙ H × L satisfying E a [ ~u ] ≤ (1 − δ ) E a ∧ [ W a ∧ ] for some δ > . If k u k ˙ H a ≤ k W a ∧ k ˙ H a ∧ , then for all t ∈ I : • k u ( t ) k ˙ H a ≤ (1 − δ ′ ) k W a ∧ k ˙ H a ∧ • R |L a u ( t, x ) | − | u ( t, x ) | dd − dx & δ k u ( t ) k H a , • E a ( ~u ) ∼ δ k ~u ( t ) k H × L ,for some δ ′ > depending on δ . Concentration compactness
A key step in the proofs of Theorem 1.1 and Theorem 1.2 is to prove thatif the result is false, then we can construct minimal blowup solutions with goodcompactness properties. This will be carried out in Section 4 (see Theorem 4.3and Theorem 4.7 below). In the present section, we will develop a key technicalingredient needed for this step, namely, a linear profile decomposition adapted toStrichartz estimates for e it √L a (see Proposition 3.2).We introduce the following notation, which helps keep track of the lack of trans-lation symmetry in L a . Definition 3.1.
Given a sequence { y n } ⊂ R d , we define L na := − ∆ + a | x + y n | and L ∞ a := ( − ∆ + a | x + y ∞ | if y n → y ∞ ∈ R d , − ∆ if | y n | → ∞ . We therefore have L a [ φ ( · − y n )] = [ L na φ ]( · − y n )]. Proposition 3.2 (Linear profile decomposition) . Let f n be a bounded sequence in ˙ H a . Passing to a subsequence, there exist J ∗ ∈ { , , . . . , ∞} , profiles { φ j } J ∗ j =1 ⊂ ˙ H a , scales { λ jn } J ∗ j =1 ⊂ (0 , ∞ ) , and space-time positions { ( t jn , x jn ) } ⊂ R d such thatfor any finite ≤ J ≤ J ∗ we have the decomposition f n = J X j =1 φ jn + r Jn , where φ jn ( x ) = ( λ jn ) − ( d − (cid:2) e it jn √ L nja φ j (cid:3) ( x − x jn λ jn ) , (3.1) where L n j a is as in Definition 3.1 with y jn = x jn λ jn . This decomposition satisfies thefollowing properties for any finite ≤ J ≤ J ∗ : • The remainder term satisfies lim J → J ∗ lim sup n →∞ k e it √L a r Jn k L d +1) d − t,x ( R × R d ) = 0 , (3.2)lim n →∞ ( λ Jn ) d − [ e − it Jn √L a r Jn ]( λ Jn x + x Jn ) ⇀ weakly in ˙ H . (3.3) • For j = k we have the orthogonality condition lim n →∞ (cid:12)(cid:12) log λ jn λ kn (cid:12)(cid:12) + | x jn − x kn | λ jn λ kn + | t jn − t kn | λ jn λ kn = ∞ . (3.4) • We also have the decouplings for each finite ≤ J ≤ J ∗ : lim n →∞ n k f n k H a − J X j =1 k φ jn k H a − k r Jn k H a o = 0 , (3.5)lim n →∞ (cid:8) k f n k dd − L dd − x − J X j =1 k φ jn k dd − L dd − x − k r Jn k dd − L dd − x (cid:9) = 0 . (3.6) Finally, we may assume that for each j either t jn ≡ or t jn → ±∞ and either x jn ≡ or ( λ jn ) − | x jn | → ∞ . The strategy for proving Proposition 3.2 is well-established: we remove onebubble at a time until the Strichartz norm is depleted. The key to isolating anindividual bubble is to first identify a scale for concentration, which can be donevia a refinement of the usual Strichartz estimate. One then finds a space-timeposition for concentration via H¨older’s inequality. The broken space translationsymmetry introduces some additional technical difficulties, related the manner inwhich we have convergence of the operators L n j a to the limiting operator L ∞ a .We begin by collecting a few lemmas related to this latter point. The first followsfrom the arguments of [22, Lemma 3.3]. Lemma 3.3 (Convergence of operators, [22]) . Let a > − ( d − ) + c d . Suppose t n → t ∈ R and y n → y ∞ ∈ R d or | y n | → ∞ . Let L na and L ∞ a be as in Definition 3.1.Then the following hold: lim n →∞ (cid:13)(cid:13)(cid:2)p L na − p L ∞ a (cid:3) ψ (cid:13)(cid:13) L = 0 for all ψ ∈ ˙ H , (3.7)lim n →∞ (cid:13)(cid:13)(cid:2)p L na − p L ∞ a (cid:3) ψ (cid:13)(cid:13) ˙ H − = 0 for all ψ ∈ L . (3.8) Furthermore, if y ∞ = 0 , then lim n →∞ (cid:13)(cid:13)(cid:2) e −L na − e −L ∞ a (cid:3) δ (cid:13)(cid:13) ˙ H − ( R d ) = 0 . (3.9)For the next result, an analogous statement appears in [22] for the case of theSchr¨odinger propagator; however, the proof relies on (endpoint) Strichartz estimatesand hence we need a new argument in our case. LW WITH INVERSE-SQUARE POTENTIAL 13
Lemma 3.4 (Convergence of operators) . Let a > − ( d − ) . Suppose y n → y ∞ ∈ R d or | y n | → ∞ . Let L na and L ∞ a be as in Definition 3.1. Then lim n →∞ k ( e it √ L na − e it √ L ∞ a ) ψ k L ∞ t L x ( R × R d ) = 0 for all ψ ∈ L . (3.10) Proof.
By approximation, it suffices to consider ψ ∈ C ∞ c ( R d \{ } ). It also sufficesto consider the case | y n | → ∞ or y n → Case 1.
Suppose y n →
0. Then L ∞ a = L a . Let us define u n ( t, x ) = [ e it √L a ψ ( · − y n )]( x ) . Then by the triangle inequality, we have k ( e it √ L na − e it √L a ) ψ k L ∞ t L x ≤ k u n ( t, x + y n ) − u n ( t, x ) k L ∞ t L x (3.11)+ k e it √L a [ ψ ( · − y n )] − e it √L a ψ k L ∞ t L x . (3.12)For (3.12), we observe(3.12) = k ψ ( · − y n ) − ψ k L x = o (1) as n → ∞ by continuity of translation in L . For (3.11), we use the fundamental theorem ofcalculus and equivalence of Sobolev spaces to bound k u n ( t, x + y n ) − u n ( t, x ) k L . | y n |k∇ u n ( t ) k L . | y n |k∇ ψ k L uniformly in t . Thus (3.11) is o (1) as well and the desired result follows. Case 2.
Suppose | y n | → ∞ . Then L ∞ a = − ∆. Define the propagator S n ( t )( f, g ) = cos( t p L na ) f + sin( t √ L na ) √ L na g, which generates solutions to ( ∂ t + L na ) u = 0. Define S ∞ ( t ) similarly. We may write e it √ L na ψ = S n ( t )( ψ,
0) + iS n ( t )(0 , p L na ψ ) , and similarly for e it √ L ∞ a ψ . Thus we have( e it √ L na − e it √ L ∞ a ) ψ = iS n ( t )(0 , ( p L na − p L ∞ a ψ ) (3.13)+ S n ( t )( ψ, − S ∞ ( t )( ψ,
0) (3.14)+ i [ S n ( t )(0 , p L ∞ a ψ ) − S ∞ ( t )(0 , p L ∞ a ψ )] . (3.15)Using (3.7), we first see k (3.13) k L ∞ t L x → V n ( x ) = L ∞ a − L na = a | x + y n | and set U n ( t ) = S n ( t )( ψ, − S ∞ ( t )( ψ, , (3.16)so that ( ∂ t + L na ) U n = V n ( x ) S ∞ ( t )( ψ, U n ( t ) = Z t t − s ) √ L na ) √ L na V n ( x ) S ∞ ( s )( ψ, ds. Using (3.16), we can firstly observe that u n is bounded in L ∞ t ˙ H − x . Thus, itsuffices to prove that u n tends to zero in L ∞ t ˙ H x . For this, we use the Duhamelformulation and use Strichartz to estimate k U n k L ∞ t ˙ H x . k V n S ∞ ( ψ, k L d +1) d +3 t,x . Recalling that ψ ∈ C ∞ c and that u ( t, x ) := S ∞ ( ψ,
0) solves the (free) linear waveequation, we are left to prove thatlim n →∞ k| x + y n | − u ( t, x ) k L d +1) d +3 t,x = 0on [0 , ∞ ) × R d (say), where u satisfies the following: • u ( t ) is supported in the ball of radius t + C for some C > • u ( t ) is uniformly bounded in L x , • u ( t ) decays like t − d − in L ∞ x .Let 0 < ε ≪
1. We first consider the contribution of 0 < t < ε | y n | . By thesupport properties of u , we have | x + y n | − . | y n | − in this region. Thus k| x + y n | − u k L d +1) d +3 t,x ( {| t | <ε | y n |}× R d ) . | y n | − kh t i dd +1 k L d +1) d +3 t ( {| t | <ε | y n |} ) k u k L ∞ t L x . | y n | − , which is acceptable.We next consider the contribution of t > ε | y n | . We split the spatial integral intothe regions where | x + y n | ≥ | x + y n | <
1, respectively. The estimates oneach region are similar, so let us consider the first case. Using the decay propertiesof u ( t ), we have k| x + y n | − u k L d +1) d +3 t,x ( {| t | >ε | y n |}×{| x + y n |≥ } ) . (cid:13)(cid:13) k| x | − k L d x ( {| x | > } ) k u ( t ) k L d ( d +1) d − d − − x (cid:13)(cid:13) L d +1) d +3 t ( {| t | >ε | y n |} ) . (cid:13)(cid:13) | t | − ( d − d +2) d ( d +1) + (cid:13)(cid:13) L d +1) d +3 t ( {| t | >ε | y n |} ) . ( ε | y n | ) − ( d − d +2) d ( d +1) + d +32( d +1) + , which is acceptable. As the modifications necessary to treat | x + y n | < | x | − in L d − ), this completes the proof. (cid:3) We now record a few corollaries that will be of use below. We first have thefollowing.
Corollary 3.5.
Let a > − ( d − ) + c d . Suppose y n → y ∞ ∈ R d or | y n | → ∞ andlet L na , L ∞ a be as in Definition 3.1. Then lim n →∞ k [ e it √ L na − e − it √ L ∞ a ] ψ k S ( R ) = 0 for all ψ ∈ ˙ H , where S ( · ) is as in (2.6) .Proof. Let us show the proof for the L d +1) d − t,x component of the S -norm. LW WITH INVERSE-SQUARE POTENTIAL 15
By Strichartz, the quantity in question is finite for ψ ∈ ˙ H . Thus, we can reduceto the case of ψ ∈ C ∞ c ( R d \{ y ∞ } ) (if y n → y ∞ ) or C ∞ c ( R d ) (if | y n | → ∞ ). For such ψ , we use H¨older to estimate k [ e it √ L na − e − it √ L ∞ a ] ψ k L d +1) d − t,x . k [ e it √ L na − e − it √ L ∞ a ] ψ k θL ∞ t L x × k [ e it √ L na − e − it √ L ∞ a ] ψ k − θL qt L rx , where θ ∈ (0 ,
1) and θ + − θr = d − d +1) , − θq = d − d +1) . Choosing θ sufficiently small so that we may apply Stirchartz and the equivalenceof Sobolev spaces holds, we can estimate k [ e it √ L na − e − it √ L ∞ a ] ψ k L qt L rx . k|∇| d − q − dr ψ k L . , so that the result follows from Lemma 3.4. (cid:3) Next, we have the following, which is completely analogous to equation (3.4)in [22].
Corollary 3.6.
Let a > − ( d − ) + c d . Suppose y n → y ∞ ∈ R d or | y n | → ∞ . Let L na and L ∞ a be as in Definition 3.1. If t n → t ∈ R , then lim n →∞ (cid:13)(cid:13) [ e − it n √ L na − e − it √ L ∞ a ] ψ k ˙ H − = 0 for all ψ ∈ ˙ H − . (3.17) Proof.
By the equivalence of Sobolev spaces, we may write ψ ∈ ˙ H − in the form p L ∞ a φ for some φ ∈ L . We write[ e it n √ L na − e it √ L ∞ a ] p L ∞ a φ = e it n √ L na (cid:2)p L ∞ a − p L na ] φ + p L na [ e it n √ L na − e it n √ L ∞ a ] φ + p L na [ e it n √ L ∞ a − e it √ L ∞ a ] φ + (cid:2)p L na − p L ∞ a (cid:3) e it √ L ∞ a φ. Applying (3.8) to the first and last terms and applying Lemma 3.4 to the secondterm, we find thatlim sup n →∞ k [ e it n √ L na − e it √ L ∞ a ] ψ k ˙ H − ≤ lim sup n →∞ k [ e it n √ L ∞ a − e it √ L ∞ a ] φ k L , which vanishes by the spectral theorem. This completes the proof. (cid:3) The next result will be important proving energy decoupling in the profile de-composition.
Corollary 3.7.
Let a > − ( d − ) + c d and ψ ∈ ˙ H . Given a sequence t n → ±∞ and any sequence { y n } ⊂ R d , we have lim n →∞ k e it n √ L na ψ k L dd − x = 0 , where L na is as in Definition 3.1. Proof.
Without loss of generality, we assume y n → y ∞ ∈ R d or | y n | → ∞ . We let L ∞ a be as in Definition 3.1.We begin by using Sobolev embedding to estimate k e it n √ L na ψ k L dd − x . k p L ∞ a (cid:2) e it n √ L na − e it n √ L ∞ a (cid:3) ψ k L x (3.18)+ k e it n √ L ∞ a ψ k L dd − x . (3.19)To estimate (3.18), we first use the triangle inequality and find k p L ∞ a (cid:2) e it n √ L na ψ − e it n √ L ∞ a ψ (cid:3) k L x . k (cid:2)p L ∞ a − p L na (cid:3) e it n √ L na ψ k L x + k e it n √ L na [ p L na − p L ∞ a ] ψ k L x + k [ e it n √ L na − e it n √ L ∞ a ] p L ∞ a ψ k L x . The second term is o (1) as n → ∞ by (3.7). The third term is o (1) as n → ∞ byLemma 3.4 (bounding an individual t n by the L ∞ norm in time). Thus we need toshow that the first term is o (1) as n → ∞ , as well. To see this, we use duality towrite k [ p L ∞ a − p L na ] e it n √ L na ψ k L x = sup (cid:12)(cid:12)(cid:10) e it n √ L na ψ, [ p L ∞ a − p L na ] g i (cid:12)(cid:12) . k [ p L ∞ a − p L na ] g k ˙ H − x k ψ k ˙ H x , where the supremum is over g ∈ L with k g k L = 1. The claim now follows from(3.8).It remains to estimate (3.19). By density, we may assume ψ ∈ C ∞ c ( R d \{ y ∞ } ) if y n → y ∞ and ψ ∈ C ∞ c if | y n | → ∞ . Writing F ( t ) = k e it √ L ∞ a ψ k L dd − x , we have by Strichartz that F ∈ L qt ( R ) for sufficiently large q < ∞ . Furthermore, F is Lipschitz; indeed, by Sobolev embedding | ∂ t F ( t ) | ≤ k ∂ t e it √ L ∞ a ψ k L dd − x . k p L ∞ a ψ k ˙ H x . . Thus F ( t n ) → n → ∞ . This completes the proof. (cid:3) We turn now to the linear profile decomposition, Proposition 3.2. As mentionedabove, the starting point is a refined Strichartz estimate for identifying a scale atwhich concentration occurs. We have the following:
Lemma 3.8 (Refined Strichartz estimate) . There exists θ ∈ (0 , so that k e − it √L a f k L d +1) d − t,x . k f k − θ ˙ H x sup N ∈ Z k e − it √L a P aN f k θL d +1) d − t,x . Proof.
Denote f N = P aN f , u ( t ) = e − it √L a f , u N = P aN u , and so on. Let us also write r = d +1) d − . By the square function estimate and Bernstein (see Proposition 2.4), LW WITH INVERSE-SQUARE POTENTIAL 17 as well as Strichartz, we have k u k rL rt,x . Z Z (cid:18)X N | u N | (cid:19) r dx dt . k (cid:0)X N | u N | (cid:1) k r − L rt,x X N ≤ N k u N k L rt L r + x k u N k L rt,x k u N k L rt,x k u N k L rt L r − x . k u k r − L rt,x (cid:2) sup N k u N k L rt,x (cid:3) X N ≤ N N k u N k L rt,x k f N k ˙ H − x . k f k r − H (cid:2) sup N k u N k L rt,x (cid:3) X N ≤ N (cid:0) N N (cid:1) k f N k ˙ H x k f N k ˙ H x . Applying Cauchy–Schwarz, the result now follows with θ = r . (cid:3) The next ingredient for the linear profile decomposition is the following inverseStrichartz estimate, which demonstrates how to remove each bubble of concentra-tion.
Proposition 3.9 (Inverse Strichartz) . Let a > − ( d − ) + c d and suppose f n ∈ ˙ H satisfy lim n →∞ k f n k ˙ H = A < ∞ and k e − it √L a f n k L d +1) d − t,x = ε > . Passing to a subsequence, there exist φ ∈ ˙ H , N n ∈ Z , and ( t n , x n ) ∈ R d suchthat g n ( x ) = N − ( d − n [ e − it n √L a f n ] (cid:0) xN n + x n (cid:1) ⇀ φ ( · ) weakly in ˙ H x , (3.20) k φ k ˙ H a & ε ( εA ) c . (3.21) Furthermore, defining φ n ( x ) = N d − n e it n √L a [ φ ( N ( x − x n ))] = N d − n [ e iN n t n √ L na φ ]( N n ( x − x n )) , where L na is as in Definition 3.1 with y n = N n x n , we have lim n →∞ (cid:8) k f n k H a − k f n − φ n k H a − k φ n k H a (cid:9) = 0 , (3.22) and lim n →∞ n k f n k dd − L dd − x − k f n − φ n k dd − L dd − x − k φ n k dd − L dd − x o = 0 , (3.23) Finally, we may assume that either N n t n → ±∞ or t n ≡ , and that either N n | x n | → ∞ or x n ≡ .Proof. Let r = d +1) d − . We use c to denote a positive constant that may changethroughout the proof. Using Lemma 3.8, there exists N n such that k e − it L a P aN n f n k L rt,x & ε ( εA ) c . Using H¨older followed by Bernstein, we have k P aN F k L rx ( | x |≤ CN − ) . C k P aN F k L rx , and hence for C sufficiently small we have k e − it L a P aN n f n k L rt,x ( R ×{| x | >CN − n } ) & ε ( εA ) c . Thus, applying H¨older, Strichartz, and Bernstein, we deduce ε ( εA ) c . k e − it √L a P aN n f n k − θL (1 − θ ) rt,x k e − it √L a P aN n f n k θL ∞ t,x . N − θ [ d − n A − θ k e − it √L a P aN n f n k θL ∞ t,x for small θ >
0. It follows that there exist ( τ n , x n ) with | x n | N n ≥ C and N − ( d − n (cid:12)(cid:12) ( e − iτ n √L a P aN n f n )( x n ) (cid:12)(cid:12) & ε ( εA ) c . (3.24)Passing to a subsequence, we may assume N n τ n → τ ∞ ∈ [ −∞ , ∞ ]. If τ ∞ is finite,define t n ≡
0; otherwise, let t n = τ n .We now let g n ( x ) := N − ( d − n [ e it n √L a f n ]( xN n + x n ) . Note that k g n k ˙ H . k f n k ˙ H . A. Therefore there exists φ ∈ ˙ H so that g n ⇀ φ weakly in ˙ H , yielding (3.20).Expanding inner products and appealing to Lemma 3.3, we can also deduce (3.22).We turn to (3.21). We now wish to define h n so that |h g n , h n i| = N − ( d − n | ( e − iτ n √L a P aN n f n )( x n ) | & ε ( εA ) c . A computation shows that we should take h n = e iN n ( τ n − t n ) √ L na P n δ where P n = e −L na − e − L na and L na is as in Definition 3.1 with y n = N n x n . Using(3.9) and (3.17), we find that h n → h ∞ := ( P ∞ δ τ ∞ ∈ {±∞} ,e − iτ ∞ √ L ∞ a P ∞ δ τ ∞ ∈ R strongly in ˙ H − , where P ∞ = e −L ∞ a − e − L ∞ a . Therefore, by strong convergence of h n and weak convergence of g n , we can conclude that ε ( εA ) c . k φ k ˙ H k h ∞ k ˙ H − . Using the heat kernel estimates (cf. (2.5)), the embedding L dd +2 ֒ → ˙ H − , and N n | x n | & c , we can show that k h ∞ k ˙ H − . , and hence (3.21) holds.Finally, we turn to (3.23). If t n ≡
0, then using Rellich–Kondrashov (to get g n → φ a.e.) and Lemma 2.1 we getlim n →∞ (cid:20) k g n k dd − L dd − x − k g n − φ k dd − L dd − x − k φ k dd − L dd − x (cid:21) = 0 , which yields (3.23) after a change of variables. In the case that t n = τ n , the resultfollows from the fact that φ n → L dd − (by Corollary 3.7).Finally, by passing to a further subsequence, we can assume that either N n | x n | →∞ or N n x n → y ∞ ∈ R d . In the latter case, we may take x n ≡ φ with φ ( · − y ∞ ). (cid:3) LW WITH INVERSE-SQUARE POTENTIAL 19
We now turn to the proof of the linear profile decomposition Proposition 3.2. Asthe proof follows along well-established lines, we will be somewhat brief.
Proof of Proposition 3.2.
The decompositon (3.1) and the decouplings (3.5) and(3.6) follow by induction. One sets r n = f n and applies Proposition 3.9 to thesequence r n to find φ n (and we set λ n = [ N n ] − ); one then applies Proposi-tion 3.9 to the sequences r Jn := r J − n − φ Jn . The process terminates at a finite J ∗ if k e − it √L a r J ∗ n k L d +1) d − t,x = 0.Defining ε J = lim n →∞ k e − it √L a r Jn k L d +1) d − t,x and A J = lim n →∞ k r Jn k ˙ H , we have that ε J → A J ≤ A ; in fact, J X j =1 ε j − (cid:0) ε j − A (cid:1) c . J X j =1 k φ jn k H a . A . By construction and (3.20), we have( λ Jn ) d − [ e − it Jn √L a r J − n ]( λ Jn x + x Jn (cid:1) ⇀ φ J for each finite J ≥ . (3.25)Recalling that r Jn = r J − n − φ J , we deduce (3.3). This will also play an importantrole in proving the orthogonality condition (3.4), to which we now turn.Putting ( j, k ) in lexicographical order, we suppose toward a contradiction that(3.4) fails for the first time at some ( j, k ) with j < k . Thus λ jn λ kn → λ , x jn − x kn √ λ jn λ kn → y , and t jn − t kn √ λ jn λ kn → t , (3.26)but (3.4) holds for every pair ( j, ℓ ) with j < ℓ < k . Now, using (3.1) to get anexpression for both r jn and r k − n , we have r jn − k − X ℓ = j +1 φ ℓn = r k − n . Therefore, using (3.25), we have( λ kn ) d − [ e − it kn √L a r jn ]( λ kn x + x kn ) − k − X ℓ = j +1 ( λ kn ) d − [ e − it kn √L a φ ℓn ]( λ kn x + x kn ) ⇀ φ k ( x ) . (3.27)To get a contradiction, we will show that both of the terms above converge weaklyto zero, contradicting that φ k is nontrivial.For the first term in (3.27), we will use (3.26). Let us introduce the notation( g jn ) − f ( x ) = ( λ jn ) d − f ( λ jn x + x kn ) . We rewrite the first term in (3.27) as( g kn ) − e i ( t jn − t kn ) √L a g jn (cid:2) ( g jn ) − e − it jn √L a r jn (cid:3) , and observe that the term inside the square brackets converges weakly to zero. Theoperator preceding the square brackets can be rewritten( g kn ) − g jn e i ( λ jn ) − ( t jn − t kn ) √ L nja , where L n j a is as in Definition 3.1 with the sequence y n = ( λ jn ) − x jn . Noting that(3.26) implies that the adjoint of ( g kn ) − g jn converges strongly and that the sequence( λ jn ) − ( t jn − t kn ) converges to some finite real number, the claim now reduces tothe following lemma. This lemma (and its proof) is completely analogous to [22,Lemma 3.8] Lemma 3.10.
Suppose f n ∈ ˙ H converges to zero weakly in ˙ H and t n → t ∞ ∈ R .Then for any y n ∈ R d , e − it n √ L na f n ⇀ weakly in ˙ H , where L na is as in Definition 3.1 with the sequence y n .Proof. Without loss of generality, we assume y n → y ∞ ∈ R d or | y n | → ∞ . We let L ∞ a be as in Definition 3.1.We claim that it suffices to prove e − it ∞ √ L na f n ⇀ H . (3.28)To see this, given ψ ∈ C ∞ c ( R d \{ y ∞ } ) (if y n → y ∞ ) or ψ ∈ C ∞ c ( R d ) (if | y n | → ∞ ),we estimate (cid:12)(cid:12)(cid:10) [ e − it n √ L na − e − it ∞ √ L na ] f n , ψ (cid:11) ˙ H x (cid:12)(cid:12) . (cid:13)(cid:13) [ e − it n √ L na − e − it ∞ √ L na ] f n (cid:13)(cid:13) L x k ∆ ψ k L x . | t n − t ∞ |k p L na f n k L x k ∆ ψ k L x , where we have used the spectral theorem and the simple inequality | e − it n √ λ − e − it ∞ √ λ | . | t n − t ∞ |√ λ for λ ≥
0. Thus the claim follows. To prove (3.28), we take ψ as above and beginby estimating |h e − it ∞ √ L na f n , ψ i ˙ H | . |h f n , [ e it ∞ √ L na − e it ∞ √ L ∞ a ]( − ∆ ψ ) i L | + |h f n , e it ∞ √ L ∞ a ( − ∆ ψ ) i L | . The first term on the right-hand side converges to zero by (3.17), using the fact that∆ ψ ∈ ˙ H − , while the second term converges to zero due to the weak convergenceof f n . This completes the proof. (cid:3) We turn now to the second term in (3.27). This time we take a similar approach,relying on the fact that (3.4) holds for each pair ( j, ℓ ) with j < ℓ < k . Omittingsome of the details, the claim boils down to the following lemma. This lemma (andits proof) is again completely analogous to [22, Lemma 3.9].
Lemma 3.11.
Let f ∈ ˙ H and let ( t n , x n ) ∈ R d and y n ∈ R d . Writing L na as inDefinition 3.1 with the sequence y n , we have [ e − it n √ L na f ]( x + x n ) ⇀ weakly in ˙ H whenever | t n | → ∞ or | x n | → ∞ .Proof. Without loss of generality, assume y n → y ∞ ∈ R d or | y n | → ∞ . Take L ∞ a as in Definition 3.1.Suppose t n → ∞ ; the case t n → −∞ is similar. We let ψ ∈ C ∞ c ( R d \{ y ∞ } ) (if y n → y ∞ ) or C ∞ c ( R d ) (if | y n | → ∞ ). Define F n ( t ) = h e − it √ L na f ]( x + x n ) , ψ i ˙ H . LW WITH INVERSE-SQUARE POTENTIAL 21
We need to show that F n ( t n ) → n → ∞ . To this end, we first compute the timederivative and observe that | ∂ t F n | . n . Thus, letting r = d +1) d − , wehave by the fundamental theorem of calculus that | F n ( t n ) | r +1 . | F n ( t ) | r +1 + k F n k rL rt ( t n ,t ) for any t > t n . In particular, it suffices to show that each F n ∈ L rt (which yields F n → t → ∞ for each fixed n ) and that lim n →∞ k F n k L rt ( t n , ∞ ) = 0 . That F n ∈ L rt follows from H¨older’s inequality and Strichartz. For the second point,we estimate by H¨older’s inequality k F n k L rt ([ t n , ∞ ]) . k [ e − it √ L na − e − it √ L ∞ a ] f k L rt,x ([ t n , ∞ ] × R d ) + k e − it √ L ∞ a f k L rt,x ([ t n , ∞ ) × R d ) . The first term converges to zero by Corollary 3.5, while the second term tends tozero as n → ∞ by Strichartz and the monotone convergence. This completes theproof in the case t n → ∞ .Finally, suppose t n is bounded (and t n → t ∞ , say) but | x n | → ∞ . In this case,we can move the translation inside appeal to Lemma 3.10, cf.[ e − it n √ L na f ]( · + x n ) = e − it √ ˜ L na [ f ( · + x n )]where ˜ L na is as in Definition 3.1 with the sequence x n + y n . This completes theproof. (cid:3) With Lemma 3.10 and Lemma 3.11 in place, we complete the proof of (3.4) andhence the proof of Proposition 3.2. (cid:3) Existence of minimal blowup solutions
In this section, we prove that if Theorem 1.1 or Theorem 1.2 fails, then we canconstruct minimal blowup solutions.We then prove the existence of scattering nonlinear profiles associated to linearprofiles with translation parameters tending to infinity (Proposition 4.2). Withthese two ingredients in place, we can then follow fairly standard arguments todeduce the existence of minimal blowup solutions (see Theorem 4.3). Finally,arguments from [19] will allow us to further reduce the class of solutions underconsideration (see Theorem 4.7).We recall the mapping T a introduced in (2.1), which takes a pair of real-valuedfunctions and returns a single complex-valued function through T a ( f, g ) = f + i L − a g. We also recall the notation ˜ E a from (2.4). Note that T ( f, g ) = f + i |∇| − g. Construction of nonlinear profiles.
We will construct nonlinear profilesvia approximation by solutions to the free nonlinear wave equation. To constructscattering solutions to the free NLW, we rely on the result of [19].
Theorem 4.1 (Scattering for the free NLW, [2, 19]) . Let ( w , w ) ∈ ˙ H × L and µ ∈ {± } . If µ = − , assume further that E [( w , w )] < E [ W ] and k w k ˙ H < k W k ˙ H . There exists a unique global solution w to ∂ t w − ∆ w + µ | w | d − w = 0 (4.1) that scatters in both time directions and obeys global L d +1) d − t,x space-time bounds.Furthermore, given ( w , w ) ∈ ˙ H × ˙ L satisfying k w k H x + k w k L x < E [ W ] and k w k ˙ H x < k W k ˙ H in the case µ = − , there exists a unique global solution to (4.1) that scatters to ( w , w ) as t → ∞ (or as t → −∞ ). We turn to the main result of this section. We assume d ∈ { , } and a > − ( d − ) + c d , as usual. In light of the application below, we will state the followingresult in terms of constructing scattering solutions to (2.3), rather than the originalequation (1.1). Proposition 4.2 (Construction of nonlinear profiles) . Suppose t n ∈ R satisfy t n ≡ or t n → ±∞ , and suppose x n ∈ R d satisfy | x n | λ − n → ∞ . Let φ ∈ ˙ H and define φ n ( x ) = λ − ( d − n [ e − it n √ L na φ ]( x − x n λ n ) , where L na is as in Definition 3.1 with the sequence y n = λ − n x n . • If µ = +1 (the defocusing case), then for n sufficiently large there exists aglobal solution v n to (2.3) with v n (0) = φ n satisfying k v n k ˙ S ( R ) . , where the implicit constant depends on k φ k ˙ H . • If µ = − (the focusing case), the same result holds provided E [ T − φ ] < E [ W ] and k Re φ k ˙ H x < k W k ˙ H , (4.2) if t n ≡ , and k φ k H x < E [ W ] and k Re φ k ˙ H x < k W k ˙ H , (4.3) if t n → ±∞ . • Furthermore, for every η > there exists N η and ψ η ∈ C ∞ c ( R d ) such thatfor n ≥ N η , k v n ( t − λ n t n , x + x n ) − λ − [ d − n ψ η ( λ − n t, λ − n x ) k L d +1) d − t,x ( R d ) < η. Proof.
Our ultimate goal is to construct solutions v n to (2.3) with data v n (0) = φ n .Equivalently, we need to construct solutions u n to (1.1) with data ~u n (0) = T − a φ n .The starting point is to appeal to Theorem 4.1 to construct a solution u associatedto the initial data (Re φ, |∇| Im φ ) . The assumptions (4.2) and (4.3) guarantee thatwe are in a position to apply Theorem 4.1.
LW WITH INVERSE-SQUARE POTENTIAL 23 If t n ≡
0, we take u to be the solution to (4.1) with initial data ~ψ := (Re φ, |∇| Im φ ) . If t n → ±∞ , we instead of u be the solution to (4.1) withlim t →±∞ k ~u ( t ) − ( S ( t ) ~ψ, ∂ t S ( t ) ~ψ ) k ˙ H × L = 0 , (4.4)where S ( t )( f, g ) = cos( t |∇| ) f + |∇| − sin( t |∇| ) g is the free linear wave propagator.In both cases, we have that u obeys global space-time bounds.We will now use u to construct approximate solutions to (1.1). For each n , welet χ n be a smooth function such that χ n ( x ) = ( | x n + λ n x | ≤ | x n | , | x n + λ n x | ≥ | x n | . In particular, χ n ( x ) → n → ∞ for each x . We further impose that χ n obeythe symbol bounds sup x | ∂ α χ n ( x ) | . [ λ − n | x n | ] −| α | for all multi-indices α .For τ >
0, we now let u n,τ ( t, x ) = λ − [ d − n ( χ n u )( λ − n t, λ − n ( x − x n )) | t | ≤ λ n τ, [ S ( t − τ λ n ) ~u n,τ ( λ n τ )]( x ) t > λ n τ, [ S ( t + τ λ n ) ~u n,τ ( − λ n τ )]( x ) t < − λ n τ, where S ( t )( f, g ) = cos( t √L a ) f + L − a sin( t √L a ) g. We claim that the u n,τ are ap-proximate solutions to (1.1) that asymptotically agree with T − a φ n , so that we mayappeal to the stability result (Proposition 2.8) to construct true solutions to (1.1)with initial data T − a φ n . To do this requires that we verify the following:lim sup τ →∞ lim sup n →∞ (cid:8) k ~u n,τ k L ∞ t ( ˙ H × L ) + k u n,τ k L d +1) d − t,x (cid:9) . , (4.5)lim sup τ →∞ lim sup n →∞ k ~u n,τ ( λ n t n ) − T − a φ n k ˙ H × L = 0 , (4.6)lim sup τ →∞ lim sup n →∞ k ( ∂ t + L a ) u n,τ + F ( u n,τ ) k L t L x = 0 , (4.7)where we have denoted F ( z ) = µ | z | d − z and ~u n,τ = ( u n,τ , ∂ t u n,τ ).We begin by estimating k ~u n,τ k L ∞ t ( ˙ H × L ) . k χ n ~u k ˙ H × L . . The space-time bound in (4.5) then follows from Strichartz and the correspondingbounds for u .We turn to (4.6). We begin with the case t n ≡
0. By construction and a changeof variables, we estimate the ˙ H component by k (1 − χ n ) Re φ k ˙ H = o (1) as n → ∞ . We turn to the L component. Again, by construction and a change of variables,we have k χ n |∇| Im φ − ( L na ) Im φ k L = o (1) as n → ∞ , where we have also made use of (3.7). We turn to the case t n → ∞ , with the case t n → −∞ being similar. Note that t n > τ for n sufficiently large. It is enough to provelim sup τ →∞ lim sup n →∞ (cid:13)(cid:13) T a ~u n,τ ( λ n t n ) − φ n (cid:13)(cid:13) ˙ H = 0 . (4.8)To this end, set u ( t, x ) := S ( t )( f, g ) . Then T a ~u ( t ) = e − it √L a T a ( f, g ) , which implies T a ~u n,τ ( λ n t n ) = e − i ( λ n t n − λ n τ ) √L a T a ~u n,τ ( λ n τ ) . Thus, performing a change of variables, we have (cid:13)(cid:13) T a ~u n,τ ( λ n t n ) − φ n (cid:13)(cid:13) ˙ H = (cid:13)(cid:13) e − i ( λ n t n − λ n τ ) √L a T a ~u n,τ ( λ n τ ) − e − it n λ n √L a g n φ (cid:13)(cid:13) ˙ H . (cid:13)(cid:13) T a ~u n,τ ( λ n τ ) − e − iλ n τ √L a g n φ (cid:13)(cid:13) ˙ H . (cid:13)(cid:13) χ n T na ~u ( τ ) − e − iτ √ L na g n φ (cid:13)(cid:13) ˙ H . (cid:13)(cid:13) T na ~u ( τ ) − e − iτ √ L na φ (cid:13)(cid:13) ˙ H + o n (1)as n → ∞ , where T na ( f, g ) := f + i ( L na ) − g. Furthermore, using (3.7), Corollary 3.5and (4.4), we derive that (cid:13)(cid:13) T a ~u n,τ ( λ n t n ) − φ n (cid:13)(cid:13) ˙ H . (cid:13)(cid:13) T na ~u ( τ ) − T ~u ( τ ) (cid:13)(cid:13) ˙ H + (cid:13)(cid:13) e − iτ √− ∆ φ − e − iτ √ L na φ (cid:13)(cid:13) ˙ H + (cid:13)(cid:13) T ~u ( τ ) − e − iτ √− ∆ φ (cid:13)(cid:13) ˙ H + o n (1) . (cid:13)(cid:13) ∂ t u ( τ ) − p L na |∇| − ∂ t u ( τ ) (cid:13)(cid:13) L + o n (1) . o n (1)as n → ∞ . Turning to (4.7), we define the errors e n,τ = ( ∂ t + L a ) u n,τ + F ( u n,τ ) , F ( z ) = µ | z | d − z, which we need to estimate in L t L x ( R d ).We first consider the contribution of times t > λ n τ , with the case t < − λ n τ being analogous. In this regime, we have e n,τ = F ( u n,τ ) . Thus, by construction and a change of variables, we have k e n,τ k L t L x ( { t>λ n τ }× R d ) . k u n,τ k d +2 d − L d +2 d − t L d +2) d − x ( { t>λ n τ }× R d ) . k S n ( t )[ χ n ~u ( τ )] k d +2 d − L d +2 d − t L d +2) d − x ((0 , ∞ ) × R d ) , where S n ( t )( f, g ) = cos( t p L na ) f + sin( t √ L na ) √ L na g. LW WITH INVERSE-SQUARE POTENTIAL 25
We claim that this term tends to zero as n, τ → ∞ , which will yield (4.7) in theregion | t | > λ n τ . Recalling the notation ~ψ from (4.4) and observing that we canreplace χ n with 1 up to errors that are o (1) as n → ∞ , we are led to estimate k S n ( t ) ~u ( τ ) k S (0 , ∞ ) . k S ( t ) ~ψ k S ( τ, ∞ ) (4.9)+ k [ S n ( t ) − S ( t )] ~u ( τ ) k S (0 , ∞ ) (4.10)+ (cid:13)(cid:13) S ( t ) (cid:2) ~u ( τ ) − (cid:0) S ( τ ) ~ψ, ∂ t S ( τ ) ~ψ (cid:1)(cid:3)(cid:13)(cid:13) S (0 , ∞ ) . (4.11)Now (4.9) is o (1) as τ → ∞ by Strichartz and monotone convergence. Next, (4.10)is o (1) for each τ by Corollary 3.5. Finally, (4.11) is o (1) as τ → ∞ by Strichartzand (4.4). This completes the proof of (4.7) in the region | t | > λ n τ .Finally, we turn to (4.7) in the region | t | ≤ λ n τ . Recalling that u is a solutionto (4.1), we compute that in this region e n,τ = λ − ( d +1) n µ [( χ n − χ d +2 d − n ) F ( u )]( λ − n t, λ − n ( x − x n )) (4.12)+ 2 λ − ( d +1) n [ ∇ χ n · ∇ u ]( λ − n t, λ − n ( x − x n )) (4.13)+ λ − ( d +1) n [∆ χ n u ]( λ − n t, λ − n ( x − x n )) (4.14)+ λ − ( d − n a | x | − [ χ n u ]( λ − n t, λ − n ( x − x n )) . (4.15)Changing variables, we estimate the contribution of (4.13) and (4.14) by τ (cid:8) k∇ χ n k L ∞ k∇ u k L + k ∆ χ n k L d k u k L dd − (cid:9) . τ λ n | x n | = o (1)as n → ∞ .For (4.12), we change variables and observe that F ( u ) ∈ L t L x (since u obeys L d +2 d − t L d +2) d − x bounds); thus the contribution of this term is o (1) as n → ∞ by thedominated convergence theorem.Finally, for (4.15) we will use Hardy’s inequality and a change of variables.Recalling the notation g n from above, first observe that in the support of g n χ n , wehave | x | & | x n | . Thus k| x | − g n [ χ n u ( λ − n t )] k L t L x ( {| t |≤ λ n τ }× R d ) . λ n | x n | k| x | − g n [ χ n u ( λ − n t )] k L ∞ t L x . λ n | x n | k∇ g n ( χ n u ( λ − n t )) k L ∞ t L x . λ n | x n | k λ − d n ∇ [ χ n u ]( λ − n t, λ − n ( x − x n )) k L ∞ t L x . λ n | x n | k∇ [ χ n u ] k L ∞ t L x = o (1)as n → ∞ . This completes the proof of (4.7).Applying Proposition 2.8, we deduce that for n sufficiently large exist true solu-tions ˜ u n to (1.1) with initial data T − a φ n . Furthermore, this solution obeys globalspace-time bounds. We now define v n = T a ~ ˜ u n to obtain the desired solutions to(2.3).Finally, the approximation result follows from the same argument in [22]. (cid:3) Reduction to almost periodic solutions.
In this section we prove thefollowing theorem.
Theorem 4.3.
Suppose Theorem 1.1 or Theorem 1.2 fails. Then there exists amaximal-lifespan solution u : I max × R → R to (1.1) that blows up in both timedirections and is almost periodic modulo symmetries with x ( t ) ≡ .In the focusing case, we have E a [ ~u (0)] < E a ∧ [ W a ∧ ] and k u (0) k ˙ H a < k W a ∧ k ˙ H a ∧ . We define L ( E ) = sup (cid:8) k u k L d − d +1 t,x ( I × R d ) (cid:9) , where the supremum is taken over all maximal-lifespan solutions u : I × R d → R to (1.1) such that E a [ ~u ] ≤ E . In the focusing case, we also restrict to solutionssatisfying k u ( t ) k ˙ H a ≤ k W a ∧ k ˙ H a ∧ for some t ∈ I . By the small-data theory, we have that L ( E ) < ∞ for E smallenough. Therefore, if Theorem 1.1 or Theorem 1.2 fails, there exists a critical E c ∈ (0 , ∞ ) (in the defocusing case) or E c ∈ (0 , E a ∧ [ W a ∧ ]) (in the focusing case)such that L ( E ) < ∞ for E < E c and L ( E ) = ∞ for E > E c . The key to establishing Theorem 4.3 is the following convergence result.
Proposition 4.4.
Suppose u n : I n × R d → R is a sequence of solutions to (1.1) with E a [ ~u n ] → E c , (4.16) and suppose t n ∈ I n are such that lim n →∞ k u n k L d +1) d − t,x ( { t>t n }× R d ) = lim n →∞ k u n k L d +1) d − t,x ( { t By time-translation symmetry, we may assume t n ≡ e − it √L a , we will generally apply the mapping T a introduced in (2.1) and work withsolutions to (2.3).We apply the linear profile decomposition (Proposition 3.2) to the sequence T a ~u n (0) = u n (0) + i L − a ∂ t u n (0) LW WITH INVERSE-SQUARE POTENTIAL 27 to write T a ~u n (0) = J X j =1 φ jn + r Jn with all of the properties stated in Proposition 3.2. We need to prove that J ∗ = 1, r n → H , x n ≡ 0, and t n ≡ n →∞ (cid:26) E a [ ~u n ] − J X j =1 ˜ E a [ φ jn ] − ˜ E a [ r Jn ] (cid:27) = 0 , where we recall the notation from (2.4).Let us first show that lim inf n →∞ ˜ E a [ φ jn ] > j. To see this, first observe that( λ jn ) − | x jn | → ∞ = ⇒ k φ jn k ˙ H a → k φ j k ˙ H > , (4.19)which is a consequence of (3.7). Thus, the claim follows from (3.5), (4.18), andLemma 2.10. Similarly, we deduce lim inf n →∞ ˜ E a [ r Jn ] ≥ J .There are now two possible cases. Case 1. Suppose sup j lim sup n →∞ ˜ E a [ φ jn ] = E c .In this case, the energy decoupling and (4.16) imply that J ∗ = 1, and hence wecan write T a ~u n (0) = φ n + r n , and in fact we can deduce that r n → H . It therefore remains to preclude λ − n | x n | → ∞ and t n → ±∞ .To this end, first suppose λ − n | x n | → ∞ . We will apply Proposition 4.2 to theprofile φ n . If t n ≡ 0, then the hypotheses of Proposition 4.2 follow from (4.19), thefact that r n → H , Lemma 3.3 and Corollary 2.9. If instead t n → ±∞ thenwe utilize Corollary 3.7, as well. Thus, by Proposition 4.2, for n large there existsa global solution v n to (2.3) with ( v n (0) , ∂ t v n (0)) = φ n satisfying global space-timebounds. Then T − a v n is a global solution to (1.1) with global space-time bounds.However, noting that k ( u n (0) , ∂ t u n (0)) − T − a φ n k ˙ H × L . k T a ~u n (0) − φ n k ˙ H → , we can therefore apply the stability result (Proposition 2.8) to deduce that the u n obey global spacetime bounds, contradicting (4.17). We conclude that x n ≡ t n → ∞ , we observe that by Strichartz, monotone convergence, r n → H , and x n ≡ 0, we have k e − it √L a T a ~u n (0) k L rt,x ( { t> }× R d ) . k e − it √L a r n k L rt,x + k e − it √L a φ k L rt,x (( t n , ∞ ) × R d ) → n → ∞ , where r = d +1) d − . By the small-data theory, this again implies global space-timebounds for the u n , yielding a contradiction. A similar argument precludes thepossibility that t n → −∞ .It therefore remains to preclude the following case: Case 2. Suppose towards a contradiction thatsup j lim sup n →∞ ˜ E a [ φ jn ] < E c − δ for some δ > . In this case, for each finite J ≤ J ∗ , we have˜ E a [ φ jn ] ≤ E c − δ for 1 ≤ j ≤ J and n large . Recalling (3.5), (4.18), and Lemma 2.10, we also have k Re φ jn k ˙ H a < (1 − δ ′ ) k W a ∧ k ˙ H a ∧ for 1 ≤ j ≤ J and n large . (4.20)We now introduce nonlinear solutions to (2.3) associated to each φ jn as follows: • If ( λ jn ) − | x jn | → ∞ then, arguing as above, the hypotheses of Proposi-tion 4.2 hold for φ j and hence we obtain a global solution v jn to (2.3) with v jn (0) = φ jn . • If x jn ≡ t jn ≡ 0, then we let v j be the maximal-lifespan solution to(2.3) with v j (0) = φ j . • If x jn ≡ t jn → ±∞ , we use Proposition 2.7 to find the maximal lifespansolution v j to (2.3) that scatters to e − it √L a φ j in ˙ H as t → ±∞ .In the latter two cases, we define v jn ( t, x ) = ( λ jn ) − ( d − v j ( tλ jn + t jn , xλ jn ) . In particular, v jn is also a solution to (2.3) with 0 in the maximal-lifespan for largeenough n and satisfying lim n →∞ k v jn (0) − φ jn k ˙ H = 0 . In particular, it follows that ˜ E a [ v jn ] ≤ E c − δ for 1 ≤ j ≤ J and n large enough. Bythe definition of E c , (4.20), and Proposition 4.2 (for those j for which ( λ jn ) − | x jn | →∞ ), we have that each v jn is global in time with uniform space-time bounds; more-over, (again using Proposition 4.2 if ( λ jn ) − | x jn | → ∞ ) for any η > ψ jη ∈ C ∞ c ( R d ) such that k v n ( t − λ n t n , x + x n ) − λ − ( d − n ψ η ( λ − n t, λ − n x ) k L d +1) d − t,x ( R d ) < η for n sufficiently large.We will now construct approximate solutions to (1.1) that asymptotically match T a ~u n (0), but which have uniform space-time bounds. Using the stability result(Proposition 2.8), this will lead to a contraction to (4.17).We define w Jn = J X j =1 v jn ( t ) + e − it √L a r Jn , (4.21)which we immediately observe satisfieslim n →∞ k w Jn (0) − T a ~u n (0) k ˙ H = 0 for all J. We claim that it remains to prove the following lemma. Lemma 4.5 (Approximate solutions) . The functions w Jn satisfy lim sup n →∞ (cid:8) k w Jn (0) k ˙ H + k w Jn k S ( R ) (cid:9) . uniformly in J, (4.22) LW WITH INVERSE-SQUARE POTENTIAL 29 and lim J → J ∗ lim sup n →∞ kL a (cid:2) ( i∂ t − L a ) w Jn − µ L − a | Re w Jn | d − Re w Jn (cid:3) k L t L x = 0 . (4.23)Indeed, with Lemma 4.5 in place, we can use Proposition 2.8 and (4.21) to deducethat the solutions T a ~u n to (2.3) inherit the uniform space-time bounds of the u Jn for large n , contradicting (4.17).The proof of Lemma 4.5 follows along standard lines, so we will be somewhatbrief. One essential ingredient is the orthogonality of parameters given in (3.4). Inparticular, (3.4) and approximation by functions that are C ∞ c in space-time implythe following: Lemma 4.6 (Orthogonality) . For j = k , we have lim n →∞ k v jn v kn k L d +1 d − t,x + k v jn v kn k L d +22( d − t L d +2 d − x = 0 . We turn to Lemma 4.5. Proof of Lemma 4.5. The ˙ H bound in (4.22) is straightforward. Using this anddecoupling, we deduce lim sup n →∞ J X j =1 k φ jn k H . J . Utilizing (4.19) for those j with ( λ jn ) − | x jn | → ∞ , this implies ∞ X j =1 k φ j k H . . Thus for J sufficiently large (depending on the small-data threshold), we can usethe small-data theory to deducesup J lim sup n →∞ J X j = J k v jn k S ( R ) . X j ≥ J k φ j k H ≪ , from which we then getlim sup n →∞ J X j =1 k v jn k S ( R ) . J. Writing r = d +1) d − , we use Lemma 4.6 to estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) J X j =1 v jn (cid:13)(cid:13)(cid:13)(cid:13) rL rt,x − J X j =1 k v jn k rL rt,x (cid:12)(cid:12)(cid:12)(cid:12) . J X j = k k v jn k r − L rt,x k v jn v kn k L r t,x → n → ∞ . As the remainder term r Jn is controlled in L rt,x uniformly, we deducethe L rt,x bound appearing in (4.22).We turn to (4.23) and set F ( z ) = µ (cid:2) | Re z | d − Re z (cid:3) , so that (using that each v jn solves (2.3)) L a ( i∂ t − L a ) w Jn − F ( w Jn ) = J X j =1 F ( v jn ) − F ( J X j =1 v jn ) (4.24)+ F ( w Jn − e − it √L a r Jn ) − F ( w Jn ) . (4.25)In particular, we need to estimate (4.24) and (4.25) in L t L x .First, by Lemma 4.6,lim n →∞ k (4.24) k L t L x . J lim n →∞ X j = k k v jn v kn k L d +22( d − t L d +2 d − x k v kn k − dd − L d +2 d − t L d +2) d − x = 0 (4.26)for all J . Thus lim J → J ∗ lim sup n →∞ k (4.24) k L t L x = 0 , as desired.We turn to (4.25). Employing the vanishing condition (3.2) (and interpolation),we findlim J → J ∗ lim n →∞ k (4.25) k L t L x . lim J → J ∗ lim n →∞ k e − it √L a r Jn k L d +2 d − t L d +2) d − x (cid:2) k w Jn k L d +2 d − t L d +2) d − x + k r Jn k ˙ H (cid:3) d − = 0 , as desired.This completes the proof of Lemma 4.5. (cid:3) With Lemma 4.5 in place, we complete the proof of Proposition 4.4 (and hencethe proof of Theorem 4.3. (cid:3) Further reductions. In this section, we perform some further reductions tothe class of solutions constructed in Theorem 4.3. We begin with the observationthat the frequency scale of an almost periodic solution obeys a local constancyproperty, namely, N ( t ) ∼ N ( t ′ ) whenever | t − t ′ | ≪ N ( t ) − . This is essentially aconsequence of the local theory (cf. [26, Lemma 5.18], for example). Using this,we may always divide the lifespan of an almost periodic solution into characteristicsubintervals J k on which N ( t ) is equal to some constant N k , with | J k | ∼ N − k .We next record a ‘non-triviality’ condition for almost periodic solutions. Notethat while the ˙ H × L -norm of ~u ( t ) is bounded away from zero, each componentindividually may spend some time near zero. Nonetheless, by adapting the argu-ments of [27, Lemma 3.4] one readily observes that almost periodicity implies thatfor any δ > 0, we have |{ t ∈ [ t , t + δN ( t ) − ] : k u ( t ) k ˙ H ≥ ε }| ≥ εN ( t ) − (4.27)for some small ε = ε ( δ, u ) > t ).We will proceed in a similar fashion to [19] prove the following. Theorem 4.7. Suppose there exist almost periodic solutions to (1.1) as in Theo-rem 4.3. Then we may find an almost periodic solution u : I max × R d → R to (1.1) conforming to one of the following two scenarios. (i) Let I = [0 , ∞ ) . Then I max ⊃ I , x ( t ) ≡ , and inf t ∈ I N ( t ) ≥ . (ii) Let I = (0 , . Then I max ⊃ I with inf I max = 0 , x ( t ) ≡ , and N ( t ) = t − . Furthermore, for each t ∈ I , ( u, ∂ t u ) is supported in B t (0) . LW WITH INVERSE-SQUARE POTENTIAL 31 In the focusing case, we have E a ( u, ∂ t u ) < E a ∧ ( W a ∧ , and sup t ∈ I k u ( t ) k ˙ H a < k W a ∧ k ˙ H a ∧ . We call scenario (i) the forward-global case and scenario (ii) the self-similar case.Proof. As mentioned above, we follow the arguments in [19]. In fact, the proof issimplified by the fact that the solutions in Theorem 4.3 have x ( t ) ≡ u : I max × R d → R as in Theorem 4.3. A standard rescaling argument showsthat we may assume N ( t ) ≥ u , say [0 , T max ).We then split into two cases, namely T max = ∞ or T max < ∞ .If T max = ∞ , then we are in scenario (i). Thus it remains to show that if T max < ∞ , then we may extract a solution conforming to scenario (ii).Suppose T max < ∞ . By time reversal and scaling, we may assume that I max ⊃ I = (0 , 1] with inf I max = 0. A standard rescaling argument relying on almostperiodicity and local well-posedness shows that we must have N ( t ) & u t − .We begin by showing that for each t ∈ I , ( u ( t ) , ∂ t u ( t )) is supported in B t (0).Using the fact that x ( t ) ≡ N ( t ) → ∞ as t → 0, we first deducelim t → + Z | x | >R |∇ u ( t, x ) | + | ∂ t u ( t, x ) | dx = 0 for any R > . Using the small-data theory and finite speed of propagation, this implieslim t → + Z | x |≥ R + | t − s | |∇ u ( s, x ) | + | ∂ s u ( s, x ) | dx = 0 for any R > , s ∈ [0 , . Now fix η > R so that R < η and let s ∈ (0 , t n → + , we havefor n sufficiently large that t n < s and {| x | ≥ s + η } ⊂ {| x | ≥ R + s − t n } . Thus, sending n → ∞ , we get Z | x |≥ s + η |∇ u ( s, x ) | + | ∂ s u ( s, x ) | dx = 0 . As η, s were arbitrary, the claim follows.We next wish to show that N ( t ) . u t − . Combining this with the upper bound,we will then be able to modify the compactness modulus by a uniformly boundedfunction and take N ( t ) = t − .To this end, we will apply the virial identity Lemma 2.2 with the weight w ( x ) = | x | . We write M ( t ) = Z − ∂ t u [ x · ∇ u + d u ] dx. Because of the support properties of ( u, ∂ t u ), we do not need to truncate the weight w . In fact, using H¨older’s inequality and Sobolev embedding, | M ( t ) | . t → t → + .With G ( u ) = µ d − d | u | dd − and V ( x ) = a | x | , we have uG ′ ( u ) − G ( u ) = µd | u | dd − and − x · ∇ V = V. Thus the virial identity becomes M ′ ( t ) = Z |L a u | + µ | u | dd − dx. In particular, using Lemma 2.10 in the focusing case, we deduce M ′ ( t ) & k u ( t ) k H . Using the fundamental theorem of calculus (cf. M ( t ) → t → M ( t ) & t .Now suppose toward a contradiction that there exists t n → + so that N ( t n ) t n →∞ . We will show that M ( t n ) = o ( t n ), contradicting the fact that M ( t n ) & u t n . Tosee this, we fix η > C ( η ) < N ( t n ) t n for n large, where C ( · ) is thecompactness modulus of u . We then write | M ( t n ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z | x |≤ C ( η ) N ( tn ) ∂ t u [ x · ∇ u + d u ] dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z C ( η ) N ( tn ) ≤| x |≤ t n ∂ t u [ x · ∇ u + d u ] dx (cid:12)(cid:12)(cid:12)(cid:12) . By almost periodicity, H¨older’s inequality, and Sobolev embedding, the second termis controlled by η · t n . The first term is controlled by C ( η ) N ( t n ) = o ( t n ). As η wasarbitrary, we conclude M ( t n ) = o ( t n ), as desired. (cid:3) To complete the proof of our main results, Theorem 1.1 and Theorem 1.2, ittherefore suffices to rule out the possibility of solutions to (1.1) as in scenarios (i)and (ii) of Theorem 4.7.5. Preclusion of the forward global case In this section we suppose that u is an almost periodic solution to (1.1) con-forming to scenario (i) in Theorem 4.7 and derive a contradiction. In particular,we have I max ⊃ [0 , ∞ ), N ( t ) ≥ 1, and x ( t ) ≡ 0. Moreover, in the focusing case, u is below the ground state threshold.We will apply the virial identity Lemma 2.2 with w ( x ) = R φ ( xR ), where φ is asmooth function satisfying φ ( x ) = ( | x | | x | ≤ | x | > . We recall that with G ( u ) = µ d − d | u | dd − and V ( x ) = a | x | , we have uG ′ ( u ) − G ( u ) = µd | u | dd − and − x · ∇ V = V. Applying Lemma 2.2 with w as above and employing the fundamental theorem ofcalculus, H¨older’s inequality, and Sobolev embedding, we deduce that Z t t Z R d |L a u | + µ | u | dd − dx dt . sup t ∈ [ t ,t ] R k∇ t,x u k L x + O (cid:18)Z t t Z R< | x | < R |∇ u | + R − | u | + | u | dd − dx dt (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12)Z t t Z | x | >R |∇ u | + a | x | | u | + | u | dd − dx dt (cid:12)(cid:12)(cid:12)(cid:12) for any 0 < t < t < ∞ .We will seek lower bounds for the left-hand side and upper bounds for the right-hand hand side that together will yield a contradiction. LW WITH INVERSE-SQUARE POTENTIAL 33 We begin with the left-hand side. Using Lemma 2.10 in the focusing case, wefirstly observe that Z R d |L a u | + µ | u | dd − dx & k u ( t ) k H . Thus, utilizing (4.27) and breaking into characteristic subintervals, we deduce that Z T Z R d |L a u | + µ | u | dd − dx & T δ, uniformly in T for some small δ = δ ( u ) > η > 0. By almost periodicity and the fact that inf t ∈ [0 , ∞ ) N ( t ) ≥ R = R ( η ) large enough thatsup t ∈ [0 , ∞ ) Z | x | >R |∇ u ( t, x ) | + a | x | | u ( t, x ) | + | u | dd − dx dt < η. Using H¨older’s inequality as well, we can take R possibly even larger to guaranteethat sup t ∈ [0 , ∞ ) Z R< | x | < R |∇ u | + R − | u | + | u | dd − dx < η. Combining the estimates above on an interval fo the form [0 , T ], we deduce that T δ . u R + T η for any T > 0. However, choosing η = η ( u, δ ) sufficiently small and then T = T ( η )sufficiently large, this leads to a contradiction. We conclude that there are nosolutions to (1.1) as in scenario (i) of Theorem 4.7.6. Preclusion of the self-similar case In this section we preclude the possibility of self-similar almost periodic solutionsas in Theorem 4.7. Recall that a self-similar almost periodic solution satisfies x ( t ) ≡ N ( t ) = t − . In particular such solutions blow up at t = 0; furthermore,at each t > B t (0).We recall the notation x β = ( t, x ). Proposition 6.1 (Virial/Morawetz estimate) . Suppose u : (0 , × R d → R is a self-similar almost periodic solution to (1.1) as in Theorem 4.7. For any t > t > ,we have Z t t Z | x | We write the equation in the form ∂ α ∂ α u = V u + G ′ ( u ) , (6.1)where V ( x ) = a | x | − and G ( u ) = d − d | u | dd − . We next introduce the function ρ = ρ ( t, x ) = (cid:2) (1 + ε ) t − | x | (cid:3) − , which satisfies x β ∂ β ρ = − ρ. (6.2) We now define the space-time region S = [ t 0) and x β n β = ± t, ( t, x ) ∈ Σ . We multiply the equation (6.1) by ρ [ x β ∂ β u + d − u ] and integrate over S . Thisyields 0 = Z S ρ [ x β ∂ β u + d − u ][ ∂ α ∂ α u − V u − G ′ ( u )]= Z S ρ [ x β ∂ α ( ∂ α u∂ β u ) − x β ∂ β ( ∂ α u∂ α u )] (6.4)+ Z S d − ρ [ ∂ α ( u∂ α u ) − ∂ α u∂ α u ] (6.5) − Z S ρ [ x β V ∂ β ( u ) + d − V u ] (6.6) − Z S ρ [ x β ∂ β G ( u ) + d − G ′ ( u ) u ] . (6.7)Integration by parts (using (6.2), (6.3), ∂ α x β = g αβ , and ∂ β x β = d + 1) yields(6.4) + (6.5) = Z S − ∂ α ρ∂ α u (cid:2) x β ∂ β u + d − u (cid:3) (6.8)+ Z Σ ρ ( ∂ α u )( x β ∂ β u ) g αγ n γ (6.9) − Z Σ ρ ( ∂ α u )( ∂ α u ) x β n β (6.10)+ Z Σ d − ρu ( ∂ α u ) g αγ n γ . (6.11)Further integration by parts (using x β ∂ β V = − V and (6.3)) yields(6.6) = − Z ∂S ρx β n β V u = − Z Σ ρx β n β V u , (6.7) = − Z ∂S ρx β n β G ( u ) = − Z Σ ρx β n β G ( u ) . LW WITH INVERSE-SQUARE POTENTIAL 35 Collecting the identities above now yields Z S ∂ α ρ∂ α u (cid:2) x β ∂ β u + d − u (cid:3) = Z Σ ρ ( ∂ α u )( x β ∂ β u ) g αγ n γ (6.12)+ Z Σ d − ρu ( ∂ α u ) g αγ n γ (6.13) − Z Σ ρ ( ∂ α u )( ∂ α u ) x β n β (6.14) − Z Σ ρx β n β [ V u + G ( u )] . (6.15)To estimate (6.12)–(6.15), we use ρ ≤ ( εt ) − on Σ and x β n β = ± t on Σ . Then,since t − ≤ | x | − on Σ , we have by ˙ H × L bounds (and Hardy’s inequality) that(6.12) + (6.13) + (6.14) + (6.15) . ε − . We now turn to the left-hand side. We wish to exhibit a coercive term andcontrol the remaining error terms.To this end, note that ∂ α ρ∂ α u = ρ [ x α ∂ α u + ε t∂ t u ] , so that the left-hand side of (6.12) is given by ∂ α ρ∂ α u [ x β ∂ β u + d − u ] = ρ ( x β ∂ β u + d − u ) (6.16)+ ρ ( x β ∂ β u + d − u )( ε t∂ t u − d − u ) . (6.17)We further expand (6.17) to write(6.17) = − d − ρ x β ∂ β ( u ) − ( d − ρ u (6.18)+ ε ρ ( x β ∂ β u + d − u ) t∂ t u. (6.19)An integration by parts shows Z S (6.18) = Z S ( d +1)( d − ρ u + d − ρ ( x β ∂ β ρ ) u − ( d − ρ u − Z Σ ρ d − x β n β u . The first term on the right-hand side is zero, and hence using ρ . ε − t − . ε − t − | x | − and | x β n β | = t on Σ , we have by Hardy’s inequality (cid:12)(cid:12)(cid:12)(cid:12)Z S (6.18) (cid:12)(cid:12)(cid:12)(cid:12) . ε − Z Σ u | x | . ε − . Collecting our estimates, we have so far established Z S ρ ( x β ∂ β u + d − u ) + ε ρ ( x β ∂ β u + d − u ) t∂ t u . ε − + ε − . We next use ε ρ ( x β ∂ β u + d − u ) t∂ t u ≤ ρ ( x β ∂ β u + d − u ) + ε ρ ( t∂ t u ) , along with the fact that ε Z t t Z | x | Suppose there exists a self-similar almost periodic solution to (1.1) as in Theorem 4.7. Then there exists a nonzero H solution f : B → R to ∆ f − x · ∇ f x − dx · ∇ f = d ( d − f + a | x | − f + µ | f | d − f (6.20) satisfying f | ∂B = 0 and Z B | f | dd − (1 − | x | ) dx + Z B | / ∇ f | (1 − | x | ) dx . , (6.21) where / ∇ denotes the angular derivative.Proof. Suppose u is a self-similar almost periodic solution. Let us first extract thestationary solution f .We first claim that Proposition 6.1 yields a sequence t n ↓ Z t n t n Z | x | 0. Then Proposition 6.1 implies that there exists j = j ( J ) such that Z j +1 − J j − J Z | x | By almost periodicity (and the fact that N ( t ) = t − ), we have (passing to afurther subsequence)(˜ u n (0) , ∂ t ˜ u n (0)) := ( t d − n u ( t n , t n · ) , t d n ( ∂ t u )( t n , t n · )) → ( v , v )strongly in ˙ H × L for some ( v , v ) ∈ ˙ H × L . Note that ( v , v ) are supportedin the unit ball. We let v : [0 , δ ) × R d → R be the solution to (1.1) with initial data( v , v ).Now observe that by scaling symmetry,˜ u n ( t, x ) = t d − n u ( t n (1 + t ) , t n x )is the solution to (1.1) with initial data (˜ u n (0) , ∂ t ˜ u n (0)). By a change of variables,(6.22) implieslim n →∞ Z Z | y | < (cid:12)(cid:12) ∂ t ˜ u n ( s, y ) + ys +1 · ∇ ˜ u n ( s, y ) + d − 22 ˜ u n ( s,y ) s +1 (cid:12)(cid:12) dx dt = 0 , from which we deduce ∂ t v + xt +1 ∇ v + d − vt +1 ≡ , δ ] × {| x | < } . By the method of characteristics, this implies that v ( t, x ) = ( t + 1) − [ d − f ( x t )for some f ∈ H supported in the unit ball. Combining this form with the factthat v solves (1.1), we immediately deduce that f solves (6.20).We turn to establishing (6.21). To begin, we collect a few properties about thesolution f .First, because the solution v belongs to L d +1) d − t,x locally in time, a change ofvariables yields f ∈ L d +1) d − x . Next, we observe that Z | f | (1 − | x | ) s dx . s ≤ . (6.23)The case s = 0 is clear, while the case s = 2 can be deduced as a consequence ofHardy’s inequality.Using (6.23) and H¨older’s inequality, we also observe that Z | f | dd − (1 − | x | ) dx . (cid:18)Z | f | (1 − | x | ) (cid:19) (cid:18)Z | f | d +2)3( d − dx (cid:19) . , (6.24)where we use d +2)3( d − < d +1) d − . This gives the first bound in (6.21).We turn to the second estimate in (6.21). Let us define the weight w ( x ) =(1 − | x | ) − . We multiply both sides of (6.20) by f w and integrate by parts. Thisyields − Z [ |∇ f | − ( x · ∇ f ) ] w + Z ( x · ∇ f ) f w + f { ( x · ∇ f )( x · ∇ w ) − ∇ f · ∇ w } dx = Z d ( d − | f | w + a | x | − | f | w + µ | f | dd − w dx. Noting that ∇ w = xw , we find that the second term on the left-hand side vanishes.Using (6.23) and (6.24), we deduce that Z (cid:12)(cid:12) |∇ f | − ( x · ∇ f ) (cid:12)(cid:12) (1 − | x | ) dx . . As |∇ f | − ( x · ∇ f ) = (1 − | x | ) |∇ f | + | x | | / ∇ f | , we deduce that (6.21) holds. (cid:3) To rule out the self-similar scenario of Theorem 4.7, it therefore suffices to pre-clude the possibility of a solution to (6.20) as in Proposition 6.2. For this we willrely on unique continuation results for elliptic PDE. In fact, at this point we are inalmost an identical situation to [19, Proposition 6.12]. Indeed the remaining issuesto address are all related to the degeneracy of (6.20) at | x | = 1; in particular, thepresence of the potential term a | x | − f plays essentially no role. For the sake ofcompleteness, however, let us not simply quote [19, Proposition 6.12] and concludethe proof. Instead let us briefly go through the argument (parallel to that in [19])to preclude the existence of a solution as in Proposition 6.2.It remains to prove the following: Proposition 6.3. Suppose f is a solution to (6.20) as in Proposition 6.2. Then f ≡ .Proof. As just mentioned, the PDE (6.20) is a degenerate elliptic PDE, with thedegeneracy occurring as | x | → 1. In particular, by standard unique continuationresults (see [1]), the result will follow if we can prove f ≡ { − δ < | x | < } for some small δ > f = f ( r, ω )and rewriting the left-hand side of the PDE (6.20) as(1 − r ) ∂ r f + ( d − r − dr ) ∂ r f + r / ∆ f, where / ∆ denotes the spherical Laplacian. We now introduce g ( s, ω ) = f ( r ( s ) , ω )for a function r ( s ) to be defined shortly. The left-hand side of (6.20) becomes − r ( r ′ ) ∂ s g + (cid:8) − (1 − r ) r ′′ ( r ′ ) + d − rr ′ − drr ′ (cid:9) ∂ s g − r / ∆ g, where r = r ( s ). If we choose r ( s ) = 1 − (1 − s ) (as in [19]), then the expression above becomes(1 + r ) ∂ s g + (1 − r ) − (cid:8) d − r − ( d − ) r + } ∂ s g − r / ∆ g. Furthermore, the domain { − δ < r < } corresponds to Ω := { − δ ′ < s < } ,where δ ′ = 2 √ δ . In particular, the PDE for g , namely(1 + r ) ∂ s g + (1 − r ) − { d − − ( d − ) r + } ∂ s g − r / ∆ g = N ( g ) , (6.25)where N ( g ) = d ( d − g + ar − g + µ | g | d − g, is nondegenerate and hence amenable to unique continuation results.From this point on, the strategy is as follows: LW WITH INVERSE-SQUARE POTENTIAL 39 (i) Collect bounds on g that show, in particular, that it is a standard solutionto (6.25).(ii) Define ˜ g ( s, ω ) = χ ( s ) g ( s, ω ), where χ is the characteristic function of (0 , g is a weak solution to (6.25) on { − δ ′ < s < } .With (i) and (ii) in place, we can (as in [19]) invoke unique continuation (cf. [1]) todeduce that g ≡ Z Ω | g | dd − + | / ∇ g | . . Similarly, Z Ω | ∂ s g | − s . Z | ∂ r f | . Z Ω | g | (1 − s ) . Z | f | (1 − r ) . , where we use (6.23) for the last bound.(ii) We turn to (ii) define ˜ g as above. In order to write down the weak formulationof (6.25), it is useful to rewrite (6.25) as ∂ s (1 + r ) ∂ s g + d − r (1 − r ) ∂ s g + (1 + r ) − r − / ∆ g = (1 + r ) − N ( g ) . Now recall that g solves (6.20) on Ω. Thus (letting φ be a test function andintegrating by parts), we find that to prove that ˜ g is a weak solution reduces toproving Z ∂ s φ (1 + r ) ∂ s ( χg ) ds dω = Z ∂ s g (1 + r ) ∂ s ( χφ ) ds dω and Z gχ∂ s { d − r (1 − r ) φ } ds dω = − Z { d − r (1 − r ) ∂ s g } χφ ds dω. Letting χ ε be smooth approximations to χ , the problem therefore reduces to provinglim ε → Z {| g∂ s φ | + | ∂ s gφ |} (1 + r ) ∂ s χ ε ds dω = 0 , (6.26)lim ε → Z g { d − r (1 − r ) φ } ∂ s χ ε ds dω = 0 . (6.27)The bounds established in (i) are well-suited for proving (6.26) and (6.27). Considerfor example, the second term in (6.26). Assuming ∂ s χ ε is of size ε − supported inan interval I ε = (1 − ε, − ε ), we get the bound ε − Z s ∈ I ε | φ | | ∂ s g | ds dω . Z s ∈ I ε | ∂ s g || − s | ds . (cid:18)Z s ∈ I ε | ∂ s g | | − s | (cid:19) [log( − ε − ε )] , which tends to zero as ε → 0. As the other terms can be treated similarly, thiscompletes the proof. (cid:3) Proof of Theorem 1.3 In this section we give the proof of Theorem 1.3, which contains two statements:(i) a blowup result below the ground state energy, and (ii) the failure of uniformspace-time bounds as one approaches the ground state threshold in the case a > Proof of Theorem 1.3 (i). Suppose ( u , u ) ∈ ˙ H × L satisfies E a [( u , u )] < E a ∧ [ W a ∧ ] and k u k ˙ H a > k W a ∧ k ˙ H a ∧ and u is the corresponding solution to (1.1). We will show that u blows up in finitetime. To this end we introduce the function y R ( t ) := Z | u ( t, x ) | ϕ ( xR ) dx, where R > ϕ is a smooth function satisfying ϕ ≡ | x | ≤ ϕ ≡ | x | > ≤ ϕ ≤ < | x | < 2. Direct computation using (1.1) yields y ′ R ( t ) = 2 Z u∂ t uϕ ( xR ) dx and y ′′ R ( t ) = 2 Z | ∂ t u | − |∇ u | − a | x | | u | + | u | dd − dx + r ( R ) , (7.1)where r ( R ) := 2 Z [1 − ϕ ( xR )] · [ | ∂ t u | − |∇ u | − a | x | | u | + | u | dd − ] dx − R Z u ∇ u · [ ∇ ϕ ]( xR ) dx. Now, using E a [ ~u ] < E a ∧ [ W a ∧ ], we can show that2 Z | u | dd − dx ≥ dd − Z (cid:2) | ∂ t u | + |∇ u | + a | x | | u | (cid:3) dx − d − k W a ∧ k H a ∧ + δ (7.2)for some δ > 0. Combining this with (7.1), we get y ′′ R ( t ) ≥ d − d − Z | ∂ t u | dx + d − (cid:0) k u k H a − k W a ∧ k H a ∧ (cid:1) + δ + r ( R ) ≥ d − d − Z | ∂ t u | dx + δ + r ( R ) . Now observe that if we additionally assume u ∈ L , then the formulas abovemake sense even as R → ∞ (in which case r ( R ) becomes identically zero). In thiscase an application of Cauchy–Schwarz leads to the lower bound y ( t ) y ′′ ( t ) ≥ d − d − [ y ′ ( t )] , (7.3)from which an ODE argument yields finite time blowup.In the general case, we need to estimate the term r ( R ). To this end note thatby finite speed of propagation, for any ε > M = M ( ε ) such that Z | x | >M + t | ∂ t u | + |∇ u | + | a || x | | u | dx < ε. Choosing ε ≪ δ and R > M , we deduce | r ( R ) | ≪ δ uniformly for t ∈ (0 , R ) . Thus we have y ′′ R ( t ) ≥ d − d − Z | ∂ t u | ϕ ( xR ) dx for t ∈ (0 , R ) , which in particular yields an estimate like (7.3) for y R on the interval (0 , R ).In particular, a similar ODE type argument (with a careful choice of parameters) LW WITH INVERSE-SQUARE POTENTIAL 41 once again yields finite-time blowup. As the complete details appear in [19] (cf.Theorem 3.7 and Theorem 7.1(ii) therein) and apply equally well in our case, weomit the details here. (cid:3) Finally, we turn to the proof of Theorem 1.3 (ii). Proof of Theorem 1.3 (ii). Recall we are in the setting of a > µ = − φ n ( x ) := (1 − ε n ) W ( x − x n ) , where ε n → | x n | → ∞ . We can show that E a [ φ n ] ր E [ W ] , and k φ n k ˙ H a ր k W k ˙ H , and hence (by Theorem 1.2) there exist global scattering solutions u n to (1.1) withdata ( φ n , n → ∞ .To this end, we define˜ u n ( t, x ) = (1 − ε n )[ χ n W ]( x − x n ) , where χ n is as in Proposition 4.2, that is, a smooth function satisfying χ n ( x ) = ( , if | x + x n | ≤ | x n | , if | x + x n | ≥ | x n | with sup x (cid:12)(cid:12) ∂ α χ n ( x ) | . | x n | −| α | for all multi-indices α. One can verify that k ˜ u n (0) − u n (0) k ˙ H = (cid:13)(cid:13)(cid:2) (1 − ǫ n ) χ n − (cid:3) W (cid:13)(cid:13) ˙ H → n → ∞ , with k ˜ u n k L d +1) d − t,x ([ − T,T ] × R d ) & T d − d +1) for T > . Using the equation − ∆ W = | W | d − W , we find e n := ( ∂ t + L a )˜ u n − | ˜ u n | d − ˜ u n = (cid:2) (1 − ǫ n ) χ n ( x − x n ) − (1 − ǫ n ) d +2 d − χ n ( x − x n ) d +2 d − (cid:3) | W | d − W (7.4)+ (1 − ǫ ) (cid:2) W ∆ χ n + 2 ∇ χ n · ∇ W (cid:3) ( x − x n ) (7.5) − a | x | (1 − ǫ n )[ χ n W ]( x − x n ) . (7.6)We now claim that for any fixed T > k e n k L t L x ([ − T,T ] × R d ) → n → ∞ . We begin with the estimate of (7.4). As W ∈ L p for any p > dd − , we have k (7.4) k L t L x ([ − T,T ] × R d ) . T (cid:13)(cid:13)(cid:2) (1 − ǫ n ) χ n ( x − x n ) − (1 − ǫ n ) d +2 d − χ n ( x − x n ) d +2 d − (cid:3) | W | d − W (cid:13)(cid:13) L ∞ t L x . T (cid:13)(cid:13)(cid:2) (1 − ǫ n ) χ n ( x − x n ) − (1 − ǫ n ) d +2 d − χ n ( x − x n ) d +2 d − (cid:3) W (cid:13)(cid:13) L ∞ t L dx × k W k d − L d ( d − d − x → n → ∞ . Next, we estimate of (7.5). We have k (7.5) k L t L x ([ − T,T ] × R d ) . 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