A Semi-linear Energy Critical Wave Equation With Applications Ruipeng Shen Department of Mathematics and Statistics McMaster University Hamilton, ON, Canada 5 1 January 5, 2015 0 2 n Abstract a In thiswork we consider a semi-linear energy critical wave equation in Rd (3≤d≤5) J 1 ∂2u−∆u=±φ(x)|u|4/(d−2)u, (x,t)∈Rd×R t with initial data (u,∂ u)| =(u ,u )∈H˙1×L2(Rd). Herethe function φ∈C(Rd;(0,1]) ] t t=0 0 1 P convergestozeroas |x|→∞. Wefollow thesame compactness-rigidity argument asKenig A and Merle applied in their paper[32] on theCauchy problem of theequation h. ∂t2u−∆u=|u|4/(d−2)u t a and obtain a similar result when φ satisfies some technical conditions. In the defocusing m case we prove that the solution scatters for any initial data in the energy space H˙1×L2. [ While in the focusing case we can determine the global behaviour of the solutions, either scattering or finite-time blow-up,according to their initial datawhen theenergy is smaller 1 than a certain threshold. v 3 2 1 Introduction 3 0 In this work we consider a semi-linear energy critical wave equation in Rd with 3 d 5: 0 ≤ ≤ 1. ∂2u ∆u=ζφ(x)upc−1u, (x,t) Rd R; 0 ut(,0−)=u H˙1(R| d|); ∈ × (CP1) 5  ∂ u·(,0)=0u∈ L2(Rd); 1  t · 1 ∈ : v Here the coefficient function φ(x) satisfies i X φ C(Rd;(0,1]), lim φ(x)=0. (1) ∈ |x|→∞ r a The exponent p = 1+ 4 is energy-critical and ζ = 1. If ζ = 1, then the equation is called c d−2 ± focusing, otherwise defocusing. Solutions to this equation satisfy an energy conservation law: E (u,∂ u)= 1 u2+ 1 ∂ u2 ζ φu2∗ dx=E (u ,u ). (2) φ t ZRd(cid:18)2|∇ | 2| t | − 2∗ | | (cid:19) φ 0 1 Here the notation 2∗ represents the constant 2d 2∗ = . d 2 − By the Sobolev embedding H˙1(Rd) ֒ L2∗(Rd), the energy E (u ,u ) is finite for any initial φ 0 1 → data (u ,u ) H˙1 L2(Rd). 0 1 ∈ × 1 1.1 Background Pure Power-type Nonlinearity Wave equations with a similar nonlinearity have been ex- tensivelystudiedinmanyworksoverafewdecades,inparticularwithapower-typenonlinearity ζ up−1u. Thereis alargegroupofsymmetriesacts onthe setofsolutionsto anequationofthis | | kind. For example, if u(x,t) is a solution to ∂2u ∆u=ζ up−1u (3) t − | | . 1 x x0 t t0 with initial data (u ,u ), then u˜(x,t)= u − , − is another solution to (3) with 0 1 λp−21 (cid:18) λ λ (cid:19) initial data 1 x x 1 x x 0 0 u − , u − (cid:18)λp−21 0(cid:18) λ (cid:19) λp−21+1 1(cid:18) λ (cid:19)(cid:19) at t = t , where λ > 0, x Rd and t R are arbitrary constants. One can check that the 0 0 0 ∈ ∈ energy defined by 1 1 ζ E(u,∂ u)= u2+ ∂ u2 up+1 dx t t ZRd(cid:18)2|∇ | 2| | − p+1| | (cid:19) is preserved under the transformations defined above, i.e. E(u,∂ u) = E(u˜,∂ u˜), if and only if . t t p=p =1+ 4 . This is the reasonwhy the exponent p is calledthe energy-criticalexponent, c d−2 c and why the equation (3) with p=p is called an energy-critical nonlinear wave equation. c PreviousResults Alargenumberofpapershavebeendevotedtothestudyofwaveequations withapower-typenonlinearity. ForinstancealmostcompleteresultsaboutStrichartzestimates, which is the basis of a local theory, can be found in [19, 31]. Local and global well-posedness has been consideredfor example in [30, 44]. In particular,there are a lotof worksregardingthe global existence and well-posedness of solutions with small initial data such as [7, 15, 16, 39]. Questions on global behaviour of larger solutions, such as scattering and blow-up, are usually consideredmoresubtle. Grillakis[21,22]andShatah-Struwe[47,48]provedtheglobalexistence and scattering of solutions with any H˙1 L2 initial data in the energy-critical, defocusing case × in 1990’s. The focusing, energy-criticalcase has been the subject of several more recent papers. This current work is motivated by one of them, F. Merle and C. Kenig’s work [32]. I would like to describe briefly its main results and ideas here. Merle and Kenig’s work Let us consider the focusing, energy-criticalwave equation ∂2u ∆u= upc−1u, (x,t) Rd R; t − | | ∈ × u(,0)=u H˙1(Rd); (CP0)  ∂ u·(,0)=0u∈ L2(Rd);  t 1 · ∈ Unlikethe defocusingcase,the solutionsto thisequationdonotnecessarilyscatter. Theground states, defined as the solutions of (CP0) independent of the time t and thus solving the elliptic equation ∆W = W pc−1W, are among the most important counterexamples. One specific − | | example of the ground states is given by the formula 1 W(x,t)=W(x)= . d−2 1+ |x|2 2 d(d−2) (cid:16) (cid:17) Kenig and Merle’s work classifies all solutions to (CP0) whose energy satisfies the inequality E(u ,u )=. 1 u 2+ 1 u 2 1 u 2∗ dx<E(W,0) 0 1 ZRd(cid:18)2|∇ 0| 2| 1| − 2∗| 0| (cid:19) into two categories: 2 (I) If u < W ,thenthe solutionuexistsgloballyintime andscatters. Theexact 0 L2 L2 k∇ k k∇ k meaning of scattering is explained in Definition 2.14 blow. (II) If u > W , then the solution blows up within finite time in both two time 0 L2 L2 k∇ k k∇ k directions. Please note that u = W can never happen if E(u ,u ) < E(W,0). Thus the 0 L2 L2 0 1 k∇ k k∇ k classification is complete under the assumption that E(u ,u ) < E(W,0). The scattering part 0 1 of this result is proved via a compactness-rigidity argument, which consists of two major steps. (I) Ifthescatteringresultwerefalse,thentherewouldexistanon-scatteringsolutionto(CP0), called a “critical element”, with a minimal energy among those non-scattering solutions, that has a compactness property up to dilations and space translations. (II) A “critical element” as described above does not exist. Solutions with a greater energy Beforeintroducingthemainresults,theauthorwouldlike to mention a few works that discuss the properties of the solutions to (CP0) with an energy E E(W,0). These works include [10, 11, 40] (Radial case) and [41] (Non-radial Case). ≥ 1.2 Main Results of this work Inthis work,we willprovethatsimilarresultsasmentionedinthe previoussubsectionstillhold for the equation (CP1), at least for those φ’s that satisfy some additional condition besides (1). Defocusing Case As in the case of the wave equation with a pure power-type nonlinearity, we expect that all solutions in the defocusing case scatter. In fact we have Theorem1.1. Let3 d 5. Assumethecoefficientfunctionφ C1(Rd)satisfiesthecondition ≤ ≤ ∈ (1) and (d 2)x φ(x) φ(x) − ·∇ >0, for any x Rd. (4) − 2(d 1) ∈ − Then the solution to the Cauchy Problem (CP1) in the defocusing case with any initial data (u ,u ) H˙1 L2(Rd) exists globally in time and scatters. 0 1 ∈ × Remark 1.2. Any positive radial C1 function satisfies the condition (4) as long as it decreases as the radius r = x grows . | | Focusing Case As in the case of a pure power-type nonlinearity, we can classify all solutions with an energy smaller than a certain positive constant. The threshold here is again the energy of the ground state W for the equation (CP0), defined by E (W,0)= 1 W 2 1 W 2∗ dx. 1 ZRd(cid:18)2|∇ | − 2∗| | (cid:19) Please note that W is no longer a ground state of (CP1) and that the energy above is not the energyE (W,0)fortheequation(CP1)asdefinedin(2). Thiscanbeexplainedbythefollowing φ fact1: If φ(x )=1, then the rescaled version of W defined by 0 1 x x 0 W (x)= W − λ,x0 λd−22 (cid:18) λ (cid:19) is “almost” a ground state for (CP1) as λ 0+ with its energy E (W ,0) E (W,0), as → φ λ,x0 → 1 shown by Lemma 3.15. 1Withoutlossofgenerality,weassumethevalueofφisequaltotheupperbound1somewhere;otherwisethe thresholdcanbeimproved,seecorollary2.23. 3 Theorem 1.3. Let 3 d 5. Assume the function φ C1(Rd) satisfies the condition (1) and ≤ ≤ ∈ 2∗(1 φ(x))+(x φ(x)) 0, for any x Rd. (5) − ·∇ ≥ ∈ Given initial data (u ,u ) H˙1 L2(Rd) with an energy E (u ,u ) < E (W,0), the global 0 1 φ 0 1 1 ∈ × behaviour, and in particular, the maximal interval of existence I =( T (u ,u ),T (u ,u )) of − 0 1 + 0 1 − thecorrespondingsolution utotheCauchy problem (CP1) inthefocusingcasecanbedetermined by: (i) If u < W , then I =R and u scatters in both time directions. 0 L2 L2 k∇ k k∇ k (ii) If u > W , then u blows up within finite time in both two directions, namely 0 L2 L2 k∇ k k∇ k T (u ,u )<+ ; T (u ,u )<+ . − 0 1 + 0 1 ∞ ∞ Remark 1.4. The function φ(x)=( |x| )σ satisfies the conditions in Theorem 1.3 as long as sinh|x| 2 σ 2∗. ≤ ≤ Remark 1.5. The compactness process works for any φ that satisfies the basic assumption (1). Thus the main theorem might still work without the assumption (4) or (5), if we could develop a successful rigidity theory for more general φ’s. 1.3 Idea of the proof In this subsection we briefly describe the idea for the scattering part of our main theorems. We focus on the focusing case, but the defocusing case, that is less difficult, can be handled in the same way. Let us first introduce (M >0) Statement 1.6 (SC(φ, M)). There exists a function β : [0,M) R+, such that if the initial data (u ,u ) H˙1 L2(Rd) satisfy → 0 1 ∈ × u < W , E (u ,u )<M; 0 L2 L2 φ 0 1 k∇ k k∇ k then the solution u to (CP1) in the focusing case with the initial data (u ,u ) exists globally in 0 1 time, scatters in both two time directions with u <β(E (u ,u )). k kLdd−+22L2(dd−+22)(R×Rd) φ 0 1 Remark 1.7. According to Remark 2.20, if u < W , then we have 0 L2 L2 k∇ k k∇ k E (u ,u ) (u ,u ) 2 0. φ 0 1 ≃k 0 1 kH˙1×L2 ≥ Therefore we have The expression β(E (u ,u )) is always meaningful. φ 0 1 • Proposition 2.12 guarantees that the statement SC(φ, M) is always true if M > 0 is • sufficiently small. Compactness Process It is clear that the statement SC(φ, E (W,0)) implies the scattering 1 part of our main theorem 1.3. If the statement above broke down at M <E (W,0), i.e. SC(φ, 0 1 M) holds for M = M but fails for any M > M , then we would find a sequence of non- 0 0 scattering solutions u ’s with initial data (u ,u ), such that E (u ,u ) M . In this n 0,n 1,n φ 0,n 1,n 0 → caseacriticalelementcanbeextractedasthelimitofsomesubsequenceof u byapplyingthe n { } profiledecomposition. ThisprocessissomewhatstandardforthewaveorSchr¨odingerequations. However, this is still some difference between our argument and that for a wave equation with a pure power-type nonlinearity. The point is that dilations and space translations are no longer contained in the symmetric group of this equation. The situation is similar when people are considering the compactness process for wave/Schr¨odinger equations on a space other than the Euclidean spaces, see [27, 42], for instance. We start by introducing the profile decomposition, before more details are discussed. 4 The profile decomposition One of the key components in the compactness process is the profile decomposition. Given a sequence (u ,u ) H˙1 L2(Rd), we can always find a 0,n 1,n ∈ × subsequence of it, still denoted by (u0,n,u1,n) n∈Z+, a sequence of free waves (solutions to the linearwaveequation),denotedby{{Vj(x,t)}j∈Z}+, andatriple (λj,n,xj,n,tj,n)∈R+×Rd×Rfor each pair (j,n), such that For each integer J >0, we have the decomposition • J (u ,u )= (V (,0),∂ V (,0))+(wJ ,wJ ). 0,n 1,n j,n · t j,n · 0,n 0,n j=1 X Here V is a modified version of V via the application of a dilation, a space translation j,n j and/or a time translation: 1 x x t t 1 x x t t j,n j,n j,n j,n (V (x,t),∂ V (x,t))= V − , − , ∂ V − , − ; j,n t j,n (cid:18)λd−22 j(cid:18) λj,n λj,n (cid:19) λd2 t j(cid:18) λj,n λj,n (cid:19)(cid:19) and(wJ ,wJ )representsaremainderthatgraduallybecomesnegligibleasJ andngrow. 0,n 1,n • Thesequences{(λj,n,xj,n,tj,n)}n∈Z+ and{(λj′,n,xj′,n,tj′,n)}n∈Z+ are“almostorthogonal” for j =j′. More precisely we have 6 λj,n λj′,n xj,n xj′,n tj,n tj′,n lim + + | − | + | − | =+ . n→∞(cid:18)λj′,n λj,n λj,n λj,n (cid:19) ∞ We can also assume λ λ [0, ) , x x Rd and t /λ j,n j j,n j j,n j,n • t R , as n → fo∈r eac∞h fix∪ed{∞j.} → ∈ ∪{∞} − → j ∈ ∪{∞ −∞} →∞ The nonlinear profile In the case of a pure power-type nonlinearity, we can approximate the solution to (CP0) with initial data (V (,0),∂ V (,0)) by a nonlinear profile U , which j,n t j,n j · · is another solution to (CP0), up to a dilation, a space translation and/or a time translation, and then add these approximations up to obtain an approximation of u , thanks to the almost n orthogonality. The fact that the equation (CP0) is invariant under dilations and space/time translations plays a crucial role in this argument. As a result, this can no longer be done for the equation (CP1). However, this problem can still be solved if we allow the use of nonlin- ear profiles that are not necessarily solutions to (CP1) but possibly solutions to other related equations instead. In fact, the solution to (CP1) with initial data (V (,0),∂ V (,0)) can be j,n t j,n · · approximatedby a nonlinear profile U as described below, up to a dilation, a space translation j and/or a time translation. I (Expanding Profile) If λ = , then the profile spreads out in the space as n . j ∞ → ∞ Eventually a given compact set won’t contain any significant part of the profile. The combinationof this fact and our assumption lim φ(x)=0 implies that the nonlinear |x|→∞ term φ(x)upc−1u is actually negligible as n . As a result, the nonlinear profile Uj in | | →∞ this case is simply a solution to the free linear wave equation. II (TravelingProfile)Ifλ < butx = ,thentheprofiletravelstotheinfinityasn . j j ∞ ∞ →∞ Againthis enables us to ignore the nonlinear term andchoose a nonlinear profile from the solutions to the linear wave equation. In fact, we can make the remainder term absorb those profiles in the cases (I) and (II). III (Stable Profile) If λ R+ and x Rd, then the profile approaches a limiting scale and j j ∈ ∈ position as n . Therefore the nonlinear profile U is a solution to (CP1). j →∞ IV (Concentrating Profile) If λ = 0 and x Rd, then the profile concentrates around a j j ∈ fixed point xj as n . The nonlinear term φ(x)upc−1u performs almost the same → ∞ | | as φ(xj)|u|pc−1u. As a result, the nonlinear profile Uj is a solution to ∂t2u − ∆u = φ(xj)upc−1u. | | 5 Extraction of a critical element After the nonlinear profiles are assigned, we can proceed step by step (I) Firstof all, we show there is atleast one non-scattering profile U, whose energy is at least M . 0 (II) By considering the estimates regarding the energy, we show that U is the only nonzero profile and its energy is exactly M . This also implies that this nonlinear profile is a 0 solution to (CP1). (III) Finally we prove that the solution U is “almost periodic”, i.e. the set (U(,t),∂ U(,t))t I t { · · | ∈ } is pre-compact in H˙1 L2(Rd), where I is the maximal lifespan of U, by considering a × new sequence of solutions derived from U via time translations and repeating the whole compactness process. A direct corollary is that the maximal lifespan I is actually R. Nonexistence ofa critical element Finallyweshowthatacriticalelementmayneverexist. In the defocusing case, we apply a Morawetz-type inequality, which gives a global integral • estimate. This contradicts with the “almost periodicity”. In the focusing case, we follow the same idea used in Kenig and Merle’s work. We show • that the derivative d d (x u)u ϕ dx+ ϕ uu dx t R R t dt(cid:20)ZRd ·∇ 2ZRd (cid:21) has a negative upper bound but the integral itself is always bounded for all time t. This givesusacontradictionwhenweconsideralongtimeinterval. Hereϕ isacut-offfunction. R 1.4 Structure of this Paper This paper is organized as follows: In Section 2 We make a brief review on some preliminary results such as the Strichartz estimates, the local theory and some results regarding the wave equation with a pure power-type nonlinearity. We then consider the linear profile decomposi- tion, define the nonlinear profiles and discuss their properties in Section 3. After finishing the preparation work, we perform the crucial compactness procedure and extract a critical element inSection4. Next weprovethatthe criticalelementcanneverexist,thus finishthe proofofthe scatteringpartofourmaintheoreminSection5. FinallyinSections6weprovetheblow-uppart of our main theorem. Section 7 is an extra, showing an application of our main theorem, about the radial solutions to the focusing, energy-critical shifted wave equation on the 3-dimensional hyperbolic space. 2 Preliminary Results 2.1 Notations Definition2.1. Throughoutthispaper thenotationF representsthefunctionF(u)=ζ upc−1u. | | The parameter ζ = 1 is determined by whether the equation in question is focusing (ζ =1) or ± defocusing (ζ = 1). − Definition 2.2 (Dilation-translation Operators). We define T to be the dilation operator λ 1 x 1 x T (u (x),u (x))= u , u ; λ 0 1 λd/2−1 0 λ λd/2 1 λ (cid:18) (cid:16) (cid:17) (cid:16) (cid:17)(cid:19) 6 and T to be the dilation-translation operator λ,x0 1 x x 1 x x T (u (x),u (x))= u − 0 , u − 0 ; λ,x0 0 1 λd/2−1 0 λ λd/2 1 λ (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) Here x is the spatial variable of the functions. Similarly we can define these operators in the same manner when both the input and output are written as column vectors. Definition 2.3. Let S (t) be the linear propagation operator, namely we define L u u(t ) S (t )(u ,u )=u(t ) S (t ) 0 = 0 L 0 0 1 0 L 0 u u (t ) 1 t 0 (cid:18) (cid:19) (cid:18) (cid:19) if u is the solution to the linear wave equation ∂2u ∆u=0; t − (u,∂ u) =(u ,u ). t t=0 0 1 (cid:26) | Similarly S (t) and S (t) represents the nonlinear propagation operator with nonlinearity F(u) 1 φ and φF(u), respectively. Definition 2.4 (The energy). Let φ : Rd [0,1] be a function and ζ = 1. We define → ± Eζ,φ(u0,u1) to be the energy of the nonlinear wave equation ∂t2u−∆u=ζφ|u|pc−1u with initial data (u ,u ) H˙1 L2(Rd). In other words, we define 0 1 ∈ × E (u ,u )= 1 u 2+ 1 u 2 ζ φu 2∗ dx. ζ,φ 0 1 ZRd(cid:18)2|∇ 0| 2| 1| − 2∗ | 0| (cid:19) We may omit ζ and use E (u ,u ) instead when the choice of ζ is obvious. φ 0 1 Definition 2.5 (Space-time Norms). Assume 1 q,r and let I be a time interval. The ≤ ≤ ∞ norm kukLqLr(I×Rd) represents ku(x,t)kLr(Rd;dx) Lq(I;dt). In particular, if 1 ≤ q,r < ∞, then we have (cid:13) (cid:13) (cid:13) (cid:13) q/r 1/q kukLqLr(I×Rd) = ZI(cid:18)ZRd|u(x,t)|rdx(cid:19) dt! Definition 2.6 (Function Spaces). Let I be a time interval. We define the following norms. kukY(I) =kukLpcL2pc(I×Rd) kukY⋆(I) =kφ1/pckY(I) =kφ1/pcukLpcL2pc(I×Rd) (u ,u ) = (u ,u ) k 0 1 kH k 0 1 kH˙1×L2(Rd) 2.2 Local Theory In this subsection we briefly discuss the local theory of the nonlinear equation (CP1). Our local theory is based on the Strichartz estimates below. Proposition 2.7 (Strichartz estimates, see [19]). There is a constant C determined solely by the dimension d 3,4,5 , such that if u is a solution to the linear wave equation ∈{ } ∂2u ∆u=F(x,t); (x,t) Rd I; t − ∈ × (u,∂ u) =(u ,u ); t t=0 0 1 (cid:26) | where I is a time interval containing 0; then we have the inequality supk(u(·,t),∂tu(·,t))kH˙1×L2(Rd)+kukY(I) ≤C k(u0,u1)kH˙1×L2(Rd)+kFkL1L2(I×Rd) . t∈I h i 7 Remark 2.8. We can substitute Y norm in Proposition 2.7 by any LqLr norm if 1 d d 2 q , 2 r < , + = 1, ≤ ≤∞ ≤ ∞ q r 2 − asshown inthepaper [19]. Thesespace-timenormsarecalled(energy-critical) Strichartznorms. Definition 2.9 (Solutions). Let (u ,u ) be initial data in H˙1 L2(Rd) and I be a time interval 0 1 × containing0. Wesayu(t)is asolution of(CP1) definedon thetimeintervalI, if(u(t),∂ u(t)) t C(I;H˙1 L2(Rd)) comes with a finite norm u for any bounded closed interval J I an∈d Y⋆(J) × k k ⊆ satisfies the integral equation t sin((t τ)√ ∆) u(t)=S(t)(u ,u )+ − − [φF(u(,τ))]dτ 0 1 √ ∆ · Z0 − holds for all time t I. ∈ Remark 2.10. We can substitute Y⋆(I) norm above with Y(I) norm to make an equivalent definition, by applying the Strichartz estimate on the integral equation above kukY⋆(I) ≤kukY(I) ≤C k(u0,u1)kH˙1×L2(Rd)+kφF(u)kL1L2(I×Rd) h i =C (u ,u ) + u pc k 0 1 kH˙1×L2(Rd) k kY⋆(I) h i Using the inequalities kφF(u)kL1L2(I×Rd) =kukpYc⋆(I); kφF(u1)−φF(u2)kL1L2(I×Rd) ≤Cku1−u2kY⋆(I) ku1kpYc⋆−(I1)+ku2kpYc⋆−1 ; (cid:16) (cid:17) the Strichartzestimate anda fixed-pointargument,we havethe followinglocaltheory. (Our ar- gumentissimilartothoseinalotofearlierpapers. Thereforeweonlygiveimportantstatements but omit most of the proof here. Please see, for instance, [44] for more details.) Proposition 2.11 (Local solution). For any initial data (u ,u ) H˙1 L2(Rd), there is a 0 1 ∈ × maximal interval ( T (u ,u ),T (u ,u )) in which the Cauchy problem (CP1) has a solution. − 0 1 + 0 1 − Proposition 2.12 (Scattering with small data). There exists δ > 0 such that if the norm of the initial data (u ,u ) < δ, then the Cauchy problem (CP1) has a global-in-time k 0 1 kH˙1×L2(Rd) solution u with kukY(R) .k(u0,u1)kH˙1×L2(Rd). Lemma 2.13 (Standard finite blow-upcriterion). Let u be a solution to (CP1) with a maximal lifespan ( T ,T ). If T < , then u = . − − + + ∞ k kY⋆([0,T+)) ∞ Definition 2.14 (Scattering). We say a solution u to (CP1) with a maximal lifespan I = ( T ,T ) scatters in the positive time direction, if T = and there exists a pair (v ,v ) − + + 0 1 H˙−1 L2(Rd), such that ∞ ∈ × u(,t) v t→li+m∞(cid:13)(cid:18)∂tu·(·,t)(cid:19)−SL(t)(cid:18)v01(cid:19)(cid:13)H˙1×L2(Rd) =0. (cid:13) (cid:13) (cid:13) (cid:13) In fact, the scattering can b(cid:13)e characterized by a mor(cid:13)e convenient but equivalent condition: kukY⋆([T′,T+)) < ∞, or still equivalently, kukY([T′,T+)) < ∞. Here T′ is an arbitrary time in I. The scattering at the negative time direction can be defined in the same manner. 8 Theorem 2.15 (Long-time perturbation theory, see also [6, 32, 33, 49]). Let M be a positive constant. There exists a constant ε = ε (M) > 0, such that if an approximation solution u˜ 0 0 defined on Rd I (0 I) and a pair of initial data (u ,u ) H˙1 L2(Rd) satisfy 0 1 × ∈ ∈ × (∂2 ∆)(u˜) φF(u˜)=e(x,t), (x,t) Rd I; t − − ∈ × u˜ <M; (u˜(0),∂ u˜(0)) < ; k kY⋆(I) k t kH˙1×L2(Rd) ∞ . ε=ke(x,t)kL1L2(I×Rd)+kS(t)(u0−u˜(0),u1−∂tu˜(0))kY⋆(I) ≤ε0; then there exists a solution u(x,t) of (CP1) defined in the interval I with the initial data (u ,u ) 0 1 and satisfying u(x,t) u˜(x,t) C(M)ε. Y⋆(I) k − k ≤ u(t) u˜(t) u u˜(0) stu∈pI (cid:13)(cid:18)∂tu(t)(cid:19)−(cid:18)∂tu˜(t)(cid:19)−SL(t)(cid:18)u10−−∂tu˜(0)(cid:19)(cid:13)H˙1×L2(Rd) ≤C(M)ε. (cid:13) (cid:13) Proof. Let us fi(cid:13)rst prove the perturbation theory when M is(cid:13)sufficiently small. Let I be the (cid:13) (cid:13) 1 maximallifespanofthesolutionu(x,t)totheequation(CP1)withthegiveninitialdata(u ,u ) 0 1 and assume [ T ,T ] I I . By the Strichartz estimates, we have 2 1 2 1 − ⊆ ∩ u˜ u k − kY⋆([−T1,T2]) ≤kSL(t)(u0−u˜(0),u1−u˜(0))kY⋆([−T1,T2])+Cke+φF(u˜)−φF(u)kL1L2([−T1,T2]×Rd) ≤ε+CkekL1L2([−T1,T2]×Rd)+CkF(φ1/pcu˜)−F(φ1/pcu)kL1L2([−T1,T2]×Rd) ε+Cε+C φ1/pc(u˜ u) ( φ1/pcu˜ pc−1 + φ1/pc(u˜ u) pc−1 ) ≤ k − kY([−T1,T2]) k kY([−T1,T2]) k − kY([−T1,T2]) Cε+C u˜ u (Mpc−1+ u˜ u pc−1 ). ≤ k − kY⋆([−T1,T2]) k − kY⋆([−T1,T2]) By a continuity argument in T and T , there exist M = M (d) > 0 and ε = ε (d) > 0, such 1 2 0 0 0 0 that if M M and ε ε , we have 0 0 ≤ ≤ u˜ u C(d)ε. k − kY⋆([−T1,T2]) ≤ Observing that this estimate does not depend on T or T , we are actually able to conclude 1 2 I I by the standard blow-up criterion and obtain 1 ⊆ u˜ u C(d)ε. Y⋆(I) k − k ≤ In addition, by the Strichartz estimate we have u(t) u˜(t) u u˜(0) stu∈pI (cid:13)(cid:18)∂tu(t)(cid:19)−(cid:18)∂tu˜(t)(cid:19)−SL(t)(cid:18)u10−−∂tu˜(0)(cid:19)(cid:13)H˙1×L2(Rd) (cid:13) (cid:13) (cid:13)(cid:13)≤CkφF(u)−φF(u˜)−ekL1L2(I×Rd) (cid:13)(cid:13) ≤C kekL1L2(I×Rd)+kF(φ1/pcu)−F(φ1/pcu˜)kL1L2(I×Rd) (cid:16) (cid:17) C ε+ u u˜ u˜ pc−1 + u u˜ pc−1 ≤ k − kY⋆(I) k kY⋆(I) k − kY⋆(I) Cεh. (cid:16) (cid:17)i ≤ This finishes the proof as M M . To deal with the general case, we can separate the time 0 ≤ intervalI intofinite numberofsubintervals I ,sothat u˜ <M ,andtheniterate { j}1≤j≤n k kY⋆(Ij) 0 our argument above. Remark2.16. IfK isacompactsubsetofthespaceH˙1 L2(Rd), thenthereexistsT =T(K)> × 0 such that for any (u ,u ) K, we have T (u ,u ) > T(K) and T (u ,u ) > T(K). This is 0 1 + 0 1 − 0 1 ∈ a direct corollary from the perturbation theory. 2The letter C in the argument represents a constant depending solely on the dimension d, it may represent differentconstants indifferentplaces. 9 2.3 Ground States and the Energy Trapping In this subsection we make a brief review on the properties of ground states for the equation (CP0)andunderstandthe“energytrapping”phenomenon. Letusfirstrecallaparticularground state 1 W(x)= . d−2 1+ |x|2 2 d(d−2) The ground state is not unique, because giv(cid:16)en any cons(cid:17)tants λ R+ and x Rd, the function 0 ∈ ∈ 1 x x 0 W (x)= W − λ,x0 λd−22 (cid:18) λ (cid:19) is also a ground state. Any ground state constructed in this way share the same H˙1 and L2∗ norms as W. The ground state W, or any other ground state we constructed above, can be characterized by the following lemma. Lemma 2.17 (Please see [56]). The function W gives the best constant C in the Sobolev d embedding H˙1(Rd)֒ L2∗(Rd). Namely, the inequality → kukL2∗ ≤Cdk∇ukL2 holds for any function u H˙1(Rd) and becomes an equality for u=W. ∈ Remark 2.18. Because the function W is a solution to ∆W = W d−42W, we also have − | | ZRd|∇W|2dx=ZRd|W|2∗dx=⇒k∇Wk2L2 =kWk2L∗2∗ =Cd2∗k∇Wk2L∗2 As a result, we have C2∗ W 2∗−2 =1 and E (W,0)=(1/d) W 2 >0. d k∇ kL2 1 k∇ kL2 Proposition 2.19 (Energy Trapping, see also [32, 33]). Let δ >0. If u is a solution to (CP1) in the focusing case with initial data (u ,u ) so that 0 1 u < W , E (u ,u )<(1 δ)E (W,0); 0 L2 L2 φ 0 1 1 k∇ k k∇ k − then for any time t in the maximal lifespan I of u we have (u(,t),∂ u(,t)) <(1 2δ/d)1/2 W (6) k · t · kH˙1×L2(Rd) − k∇ kL2 u(x,t)2 u(x,t)2∗ dx u(x,t)2dx (7) δ ZRd(cid:16)|∇ | −| | (cid:17) ≃ ZRd|∇ | u (x,t)2dx+ u(x,t)2 u(x,t)2∗ dx E (u ,u ). (8) t δ φ 0 1 ZRd| | ZRd(cid:16)|∇ | −| | (cid:17) ≃ Proof. If we have u(,t ) < W 2 for some time t I, then we have k∇ · 0 kL2 k∇ kL 0 ∈ ku(·,t0)kL2∗ ≤Cdk∇u(·,t0)kL2 <Cdk∇WkL2 =kWkL2∗. Therefore 1 (u(,t ),∂ u(,t )) 2 =E (u ,u )+ 1 φu(x,t )2∗dx 2k · 0 t · 0 kH˙1×L2(Rd) φ 0 1 2∗ ZRd | 0 | <(1 δ)E (W,0)+ 1 W 2∗dx − 1 2∗ ZRd| | 1 = W 2 δE (W,0) 1 2k∇ k − 1 δ = W 2. 2 − d k∇ k (cid:18) (cid:19) 10