SCATTERING PROFILE FOR GLOBAL SOLUTIONS OF THE ENERGY-CRITICAL WAVE EQUATION THOMAS DUYCKAERTS1, CARLOS KENIG2, AND FRANK MERLE3 6 1 Abstract. Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. 0 Weprovethatanyglobalsolutionwhichisboundedintheenergyspaceconvergesintheexterior 2 of wave cones to a radiation term which is a solution of the linear wave equation. n a J 9 1. Introduction P] In this note we consider the energy-critical nonlinear wave equation on RN, N ∈ {3,4,5}: A (cid:40)∂2u−∆u−|u|N4−2u = 0 h. (1.1) t t (u,∂tu)(cid:22)t=0 = (u0,u1) ∈ H˙1×L2, a m where H˙1 = H˙1(RN), L2 = L2(RN), and u is real-valued. If f is a function of space and time, [ we will denote f(cid:126)= (f,∂ f). It is known that the equation (1.1) is locally well posed in H˙1×L2 t 1 and that the energy v (cid:90) (cid:90) (cid:90) 07 E((cid:126)u(t)) = 12 |∇u(t,x)|2dx+ 12 |∂tu(t,x)|2dx− N2N−2 |u(t,x)|N2−N2 dx 1 2 is conserved. 0 All solutions of the defocusing equation ((1.1) with a + instead of a − sign in front of the . 1 nonlinearity) are global and scatter to a solution of the linear equation (see e.g. [27, 17, 15, 18, 0 25, 20, 26, 3, 24]). The dynamics of the focusing equation (1.1) is richer: small data solutions 6 are global and scatter, however blow-up in finite time may occur [23]. Global, non-scattering 1 : solutions also exist. Examples of such solutions are given by solutions of the elliptic equation: v Xi (1.2) −∆Q = |Q|N4−2Q, Q ∈ H˙1, r (see [7] for the existence of such solutions) and their Lorentz transforms a (cid:32)(cid:32) (cid:32) (cid:33) (cid:33) (cid:33) t 1 1 (1.3) Q (t,x) = Q − + −1 (cid:96)·x (cid:96)+x = Q (0,x−t(cid:96)) (cid:96) (cid:112)1−|(cid:96)|2 |(cid:96)|2 (cid:112)1−|(cid:96)|2 (cid:96) where (cid:96) ∈ RN, |(cid:96)| < 1. Note that Q (t,x) = Q (0,x−t(cid:96)), (cid:96) (cid:96) Date: January 12, 2016. 1LAGA, Université Paris 13, Sorbonne Paris Cité, UMR 7539. Partially supported by ERC Grant Blowdisol 291214 and French ANR Grant SchEq ANR-12-JS01-0005-01. 2University of Chicago. Partially supported by NSF Grants DMS-1265429 and DMS-1463746. 3Cergy-Pontoise (UMR 8088), IHES. Partially supported by ERC Grant Blowdisol 291214. 1 2 T.DUYCKAERTS,C.KENIG,ANDF.MERLE so that Q is a solitary wave traveling at speed |(cid:96)|. The energy of Q is given by: (cid:96) (cid:96) 1 (1.4) E(Q(cid:126) (0)) = E(Q(cid:126)(0)) −→ +∞. (cid:96) (cid:112) 1−|(cid:96)|2 |(cid:96)|→1 It is conjectured that any bounded, global solution of (1.1) is a sum of modulated, decoupled traveling waves and a scattering part. More precisely: Conjecture 1 (Soliton resolution). Let u be a solution of (1.1) on [0,+∞)×RN such that (1.5) sup(cid:107)(cid:126)u(t)(cid:107) < ∞. H˙1×L2 t≥0 Then there exist a solution vl of the linear wave equation (cid:40)(∂t2−∆)vl = 0 (1.6) (cid:126)vl(cid:22)t=0 = (v0,v1) ∈ H˙1×L2 an integer J ≥ 0, and for j ∈ {1,...,J}, a (nonzero) traveling wave Qj (|(cid:96) | < 1), and (cid:96)j j parameters x (t) ∈ RN, λ (t) ∈ RN such that j j J (cid:32) (cid:18) (cid:19) (cid:18) (cid:19)(cid:33) (1.7) t→li+m∞(cid:126)u(t)−(cid:126)vl(t)−(cid:88)j=1 λj(t1)N2−2Qj(cid:96)j 0, ·−λjx(jt()t) , λj(1t)N2 ∂tQj(cid:96)j 0, ·−λjx(jt()t) = 0 in H˙1×L2 and x (t) λ (t) j j ∀j ∈ {1,...,J}, lim = (cid:96) , lim = 0 j t→∞ t t→∞ t |x (t)−x (t)| λ (t) λ (t) j k j k ∀j,k ∈ {1,...,J}, j (cid:54)= k =⇒ lim + + = +∞. t→+∞ λj(t) λk(t) λj(t) This conjecture was proved by the authors in [11] for radial solutions in space dimension N = 3 (in this case x (t) ≡ 0 and the only stationary solutions in the expansion (1.7) are W j and −W, where 1 W(x) = . (cid:16) (cid:17)1/2 |x|2 1+ 3 In this article we extract the linear profile vl(t) which appears in the expansion (1.7). More precisely, we prove: Theorem 2. Let u be a solution of (1.1) on [0,+∞)×RN that satisfies (1.5). Then there exists a solution vl of the linear wave equation (1.6) such that, for all A ∈ R, (cid:90) (1.8) t→li+m∞ |∇(u−vl)(t,x)|2+|∂t(u−vl)(t,x)|2+ |x1|2|u(t,x)|2+|u(t,x)|N2−N2 dx = 0. |x|≥t+A WeseeTheorem2asafirststeptowardtheproofofConjecture1. Itwillbeusedinanarticle by Hao Jia and the authors to prove a weak form of the conjecture, i.e. that the expansion (1.7) holds for a sequence of times t → +∞ (see [8]). Note that Theorem 2 implies that the following n limit exists: (cid:90) lim lim 1|∇u(t,x)|2+ 1(∂tu(t,x))2− N −2|u(t,x)|N2−N2 dx = El(v0,v1), A→−∞t→+∞ |x|≥t+A 2 2 2N SCATTERING PROFILE FOR THE ENERGY-CRITICAL WAVE EQUATION 3 where (cid:90) 1 El((cid:126)vl(t)) = |∇vl(t,x)|2+(∂tvl(t,x))2dx 2 is the conserved energy for the linear wave equation. One can also prove (using for example the profile decomposition of [2] recalled in §2.4), as a consequence of Theorem 2: (1.9) S(cid:126)l(−t)(cid:126)u(t) −−−(cid:42) (v0,v1), t+∞ where S(cid:126)l denotes the linear evolution (see §2.1 below). We next mention a few related works. Theorem 2 is proved in [9] in the radial case in dimension 3 (where it is significantly simpler). A very close proof yields Theorem 2 for radial solutions of (1.1) in higher dimensions (see e.g. [5]) and of the defocusing analogue of (1.1) with an additional linear potential [19], still in the radial setting. It is proved in [6] by an adaptation of the proof of [9]. Related results for energy-critical, mass-supercritical Schrödinger equations were proved by T. Tao in [28, 29, 30]. Let us give a short outline of the paper. We first prove (Section 3), as a consequence of small data theory and finite speed of propagation, that (1.8) holds for large positive A. We then argue by contradiction, assuming that (1.8) does not hold for all A ∈ R, and defining A as the largest real number such that (1.8) does not hold. We divide (see Section 4) the elements of the sphere SN−1 between regular directions (in an angular neighbourhood of which (1.8) holds locally, for some A < A), and singular directions (other elements of SN−1), and prove, using geometrical considerations and again small data theory and finite speed of propagation, that the set of singular directions is finite. To conclude the proof, we show in Sections 5, 6 and 7 that the set of singular directions is empty, which will contradict the definition of A. The core of the proof of this fact is in Section 6, where we prove, using virial type identities, that there are no nonlinear profiles remaining close to the wave cone {t = |x|}. This is coherent with the intuition that nonlinear objects with finite energy travel at a speed strictly slower than 1, as the traveling waves (1.3) and their energies (1.4) suggest. Let us also mention that we never need to know, in all this proof, what happens inside the wave cone (that is for (t,x) such that |t|−|x| (cid:29) 1). This is of course made possible by finite speed of propagation. In Section 2, we give some preliminary results on linear and nonlinear wave equations. We introduce in particular an isometry between the initial data and the asymptotic profile of a solution of the linear wave equation that is known (see e.g. the work of Friedlander [14]) but seems to have been somehow forgotten. We construct this isometry (that we use many times in the article) in Appendices A and B for the sake of completeness. 2. Preliminaries and notations 2.1. Linear wave equation. If (v ,v ) ∈ H˙1×L2, we let 0 1 √ √ sin(t −∆) Sl(t)(v0,v1) := cos(t −∆)v0+ √ v1 −∆ the solution vl of (1.6), and (cid:18) (cid:19) ∂ S(cid:126)l(t)(v0,v1) :=(cid:126)vl(t) = Sl(t)(v0,v1), (Sl(t)(v0,v1)) . ∂t 4 T.DUYCKAERTS,C.KENIG,ANDF.MERLE The linear energy (cid:90) (cid:90) 1 1 El((cid:126)vl(t)) = |∇vl(t,x)|2dx+ (∂tvl(t,x))2dx 2 2 is conserved. We will often use radial coordinates, denoting, for x ∈ RN \{0}, r = |x| > 0 and ω = x/|x| ∈ SN−1. We also denote ∂ x 1 x ∂ = = ·∇, ∇ = ∇− ∂ . r ω r ∂ |x| r |x| r We will often use the following asymptotic property. Theorem 2.1. Assume N ≥ 3 and let vl be a solution of the linear wave equation (1.6). Then (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)1 (cid:13) (cid:13)1 (cid:13) (2.1) lim (cid:13) ∇ωvl(t)(cid:13) +(cid:13) vl(t)(cid:13) = 0 t→+∞(cid:13)r (cid:13) (cid:13)r (cid:13) L2 L2 and there exists a unique G ∈ L2(R×SN−1) such that + (cid:90) +∞(cid:90) (cid:12) (cid:12)2 (2.2) lim (cid:12)(cid:12)rN2−1∂tvl(t,rω)−G+(r−t,ω)(cid:12)(cid:12) dωdr = 0 t→+∞ 0 SN−1 (cid:90) +∞(cid:90) (cid:12) (cid:12)2 (2.3) lim (cid:12)(cid:12)rN2−1∂rvl(t,rω)+G+(r−t,ω)(cid:12)(cid:12) dωdr = 0. t→+∞ 0 SN−1 Furthermore, (cid:90) (2.4) El(v0,v1) = |G+(η,ω)|2dηdω = (cid:107)G+(cid:107)2L2 R×SN−1 and the map √ (v ,v ) (cid:55)→ 2G 0 1 + (H˙1×L2)(RN) → L2(R×SN−1) is a bijective isometry. Remark 2.2. Using Theorem 2.1 on the solution (t,x) (cid:55)→ vl(−t,x) of (1.6), we obtain that (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)1 (cid:13) (cid:13)1 (cid:13) lim (cid:13) ∇ωvl(t)(cid:13) +(cid:13) vl(t)(cid:13) = 0, t→−∞(cid:13)r (cid:13) (cid:13)r (cid:13) L2 L2 that there exists G ∈ L2(R×SN−1) such that − (cid:90) +∞(cid:90) (cid:12) (cid:12)2 lim (cid:12)(cid:12)rN2−1∂tvl(t,rω)−G−(t+r,ω)(cid:12)(cid:12) dωdr = 0 t→−∞ 0 SN−1 (cid:90) +∞(cid:90) (cid:12) (cid:12)2 lim (cid:12)(cid:12)rN2−1∂rvl(t,rω)−G−(t+r,ω)(cid:12)(cid:12) dωdr = 0. t→−∞ 0 SN−1 √ and that the map (v ,v ) (cid:55)→ 2G is a bijective isometry from (H˙1×L2)(RN) to L2(R×SN−1) 0 1 − We will call G+ (respectively G−) the radiation fields associated to vl. SCATTERING PROFILE FOR THE ENERGY-CRITICAL WAVE EQUATION 5 Theorem 2.1 is known (see in particular the works of Friedlander [13, 14]). We give a proof in Appendices A and B for the sake of completeness. Let us mention that the following identity for the conserved momentum is also available (but will not be used in this article): (cid:90) (cid:90) ∇v v = − ω|G (η,ω)|2dη. 0 1 + SN−1×R 2.2. Strichartz estimates. If Ω is a measurable subset of R ×RN, of the form Ω = (cid:83) {t}× t x t∈R Ω , Ω ⊂ RN measurable, and u a measurable function defined on Ω, we denote t t (cid:32)(cid:90) +∞(cid:18)(cid:90) (cid:19)1 (cid:33)NN−+22 2(N+2) 2 (2.5) (cid:107)u(cid:107)S(Ω) := |u| N−2 dx dt . −∞ Ωt If I ⊂ R is measurable, we will abuse notation, writing S(I) for S(I ×RN). ThespacesS(I)appearinthefollowingStrichartzestimate(see[16]): letf ∈ L1(cid:0)R,L2(RN)(cid:1), (u ,u ) ∈ (H˙1×L2)(RN), and 0 1 (cid:90) t u(t) = Sl(t)(u0,u1)+ SL(t−s)(0,f(s))ds 0 the solution of (2.6) (∂t2−∆)u = f, (cid:126)u(cid:22)t=0 = (u0,u1) ∈ H˙1×L2. Then u is well-defined, (cid:126)u ∈ C0(R,H˙1×L2), u ∈ S(R) and (cid:0) (cid:1) (cid:107)u(cid:107) +sup(cid:107)(cid:126)u(t)(cid:107) ≤ C (cid:107)f(cid:107) +(cid:107)(u ,u )(cid:107) . S(R) H˙1×L2 L1(R,L2) 0 1 H˙1×L2 t∈R We will use occasionally other Strichartz estimates: in the preceding inequality, one can replace S(R) by L2(NN−+21)(RN+1), and also by L4(R,L12) (if N = 3) or L2(R,LN2−N3) (if N ≥ 4). 2.3. Miscellaneous properties of the critical nonlinear wave equation. We recall the Cauchy theory for equation (1.1): Theorem 3. (a) Small data theory: there exists δ > 0 such that if I is an interval contain- 0 ing 0 and (u ,u ) ∈ H˙1 ×L2 is such that (cid:107)S (t)(u ,u )(cid:107) < δ , then there exists a 0 1 L 0 1 S(I) 0 unique solution of (1.1) (cid:126)u ∈ C0(I,H˙1×L2). Furthermore (cid:13) (cid:13) N+2 stu∈pI (cid:13)(cid:13)(cid:126)u(t)−S(cid:126)l(t)(u0,u1)(cid:13)(cid:13)H˙1×L2 +(cid:107)u−Sl(·)(u0,u1)(cid:107)S(I) ≤ C(cid:107)Sl(·)(u0,u1)(cid:107)SN(−I)2. (b) If (u ,u ) ∈ H˙1×L2, there exists a unique maximal solution u of (1.1). Letting (T ,T ) 0 1 − + the maximal interval of existence of u, we have the following blow-up criterion: T < ∞ =⇒ (cid:107)u(cid:107) = +∞. + S((0,T+)) (cid:0) (cid:1) (c) If u is a solution of (1.1) such that u ∈ S (0,T ) , then T = +∞ and u scatters for + + positive times: there exists a solution ul of (1.6) such that (2.7) lim (cid:107)(cid:126)u(t)−(cid:126)ul(t)(cid:107)H˙1×L2 = 0. t→+∞ (see [21]). In the theorem, a solution of (1.1) on I is by definition a solution in the Duhamel sense which is in S(J) for all J (cid:98) I. Point (c) can be seen as the consequence of the following result on the non-homogeneous linear wave equation, that we will use repeatedly in the paper: 6 T.DUYCKAERTS,C.KENIG,ANDF.MERLE Claim 2.3. Let f ∈ L1(R,L2(RN)) and u the solution of (2.6) (in the Duhamel sense). Then, there exists a solution vl of (1.6) such that (2.8) lim (cid:107)(cid:126)u(t)−(cid:126)vl(t)(cid:107)H˙1×L2 = 0. t→∞ Proof. The existence of vl follows from the formula (cid:90) t S(cid:126)l(−t)(cid:126)u(t) = (u0,u1)+ S(cid:126)l(−s)(0,f(s))ds 0 and energy estimates. (cid:3) We recall the finite speed of propagation property: if R > 0, x ∈ RN and u is a solution 0 of (2.6) such that (u ,u )(x) = 0 for |x − x | ≤ R, and f(t,x) = 0 for |x − x | ≤ R − t, 0 1 0 0 t ∈ [0,R], then u(t,x) = 0 for |x−x | ≤ R−t, t ∈ [0,R]. As a consequence, if the initial data 0 of two solutions of (1.1) coincide for |x−x | ≤ R, then the two solutions coincide for t ∈ [0,R], 0 |x−x | < R−t, if t is in the domains of existence of both solutions. A consequence of finite 0 speed of propagation and small data theory is the following claim: Claim 2.4. There exists δ > 0 with the following property. Let u be a solution of (1.1) such 1 that T (u) = +∞. Let T > T (u), and A > −T. + − (a) Assume (cid:107)u(cid:107)S({|x|≥A+t,t≥T}) = δ < δ1. Then (cid:107)Sl(·−T)(cid:126)u(T)(cid:107)S({|x|≥A+t,t≥T}) ≤ 2δ. (b) Assume (cid:107)Sl(·−T)(cid:126)u(T)(cid:107)S({|x|≥A+t,t≥T}) = δ(cid:48) < δ1. Then (cid:107)u(cid:107)S({|x|≥A+t,t≥T}) ≤ 2δ(cid:48). Proof. Let u˜ be the solution of (cid:40) ∂t2u˜−∆u˜ = |u|N4−2u11{|x|≥A+t} (u˜,∂tu˜)(cid:22)t=T =(cid:126)u(T). By Strichartz estimates, for all T > T, 1 (cid:12) (cid:12) N+2 (cid:12)(cid:12)(cid:107)u˜(cid:107)S({|x|≥A+t,T≤t≤T1})−(cid:107)Sl(·−T)(cid:126)u(T)(cid:107)S({|x|≥A+t,T≤t≤T1})(cid:12)(cid:12) ≤ C(cid:107)u(cid:107)SN(−{|2x|≥A+t,T≤t≤T1}). By finite speed of propagation, u(t,x) = u˜(t,x) if t ≥ T, |x| > A+t, and thus (cid:12) (cid:12) N+2 (cid:12)(cid:12)(cid:107)u(cid:107)S({|x|≥A+t,T≤t≤T1})−(cid:107)Sl(·−T)(cid:126)u(T)(cid:107)S({|x|≥A+t,T≤t≤T1})(cid:12)(cid:12) ≤ C(cid:107)u(cid:107)SN(−{|2x|≥A+t,T≤t≤T1}), Assuming δ small, point (a) follows immediately. An easy bootstrap argument yields point 1 (b). (cid:3) (cid:8) (cid:9) 2.4. Profile decomposition. We finally recall that any sequence (u ,u ) bounded in 0,n 1,n n H˙1×L2 has a subsequence (still denoted by (cid:8)(u ,u )(cid:9) ) that admits a profile decomposition 0,n 1,n n (cid:16) (cid:17) Ulj,{tj,n,xj,n,λj,n}n , where for all j, Ulj is a solution of the linear wave equation (1.6) and j≥1 for all j,n, λ > 0, x ∈ RN and t ∈ R, have the following properties: j,n j,n j,n λ |x −x | |t −t | j,n j,n k,n j,n k,n j (cid:54)= k =⇒ lim + + = +∞ n→∞ λk,n λj,n λj,n (pseudo-orthogonality) and lim limsup(cid:13)(cid:13)wJ(cid:13)(cid:13) = 0, n→∞ n→∞ n S(R) SCATTERING PROFILE FOR THE ENERGY-CRITICAL WAVE EQUATION 7 where J (2.9) wnJ(t) = Sl(t)(u0,n,u1,n)−(cid:88)Ulj,n(t), j=1 and (cid:18) (cid:19) 1 t−t x−x Ulj,n(t,x) = N−2Uj λ j,n, λ j,n . λ 2 j,n j,n j,n The existence of the profile decomposition was established in [2] for N = 3 (see [4] for higher dimensions). We refer to [2] for the properties of this profile decomposition (see also [12, Section 3] for a review). 3. Scattering to a linear solution outside a large wave cone We let τ → +∞ and (after extraction), n (3.1) (v0,v1) = w− lim S(cid:126)l(−τn)(cid:126)u(τn) in H˙1×L2 n→∞ (3.2) vl(t) = Sl(t)(v0,v1), where w−lim stands for the weak limit. Proposition 3.1. Let u be as in Theorem 2, and assume that the conclusion of Theorem 2 does not hold. Then there exists A ∈ R with the following properties: (3.3) ∀A > A, (cid:107)u(cid:107) < ∞ S({t>0,|x|>t+A}) (3.4) ∀A > A, tl→im∞(cid:13)(cid:13)|x|−1u(t)(cid:13)(cid:13)L2({|x|>t+A})+(cid:107)u(t)(cid:107)LN2−N2({|x|>t+A}) = 0 (3.5) ∀A > A, lim (cid:107)∇t,x(u−vl)(t)(cid:107)L2({|x|>t+A}) = 0 t→∞ (3.6) (cid:107)u(cid:107) = ∞. S({t>0,|x|>t+A}) Proof. Step 1. We note that (cid:107)u(cid:107) is finite for large A > 0. Indeed, since S{|x|>|t|+A} (cid:107)Sl(·)(u0,u1)(cid:107)S(R) < ∞, we have, for large A, (cid:107)Sl(·)(u0,u1)(cid:107)S({|x|≥|t|+A}) < δ1, where δ is given by Claim 2.4. The conclusion follows from Claim 2.4. 1 Step 2. We let A ∈ R∪{−∞} be defined by (3.7) A := inf(cid:8)A ∈ R : (cid:107)u(cid:107) < ∞(cid:9). S({t>0,|x|>t+A}) In particular (3.3) holds. We prove here that (3.5) holds. Let A > A. Let v be the solution of (3.8) (cid:40)∂t2v−∆v = |u|N4−2u11|x|>t+A (cid:126)v(cid:22)t=0 = (u0,u1). By the definition of A, we see that the right-hand side of the first equation in (3.8) is in L1([0,+∞),L2(RN)). By Claim 2.3, there exists vA, solution of (1.6), such that l (cid:90) (3.9) lim (cid:12)(cid:12)∇t,x(v−vlA)(t,x)(cid:12)(cid:12)2 dx = 0. t→+∞ 8 T.DUYCKAERTS,C.KENIG,ANDF.MERLE By finite speed of propagation, u(t,x) = v(t,x) for (t,x) such that t > 0, |x| > A+t, and thus (cid:90) (3.10) lim (cid:12)(cid:12)∇t,x(u−vlA)(t,x)(cid:12)(cid:12)2 dx = 0 t→+∞ |x|>t+A and (cid:90) lim 1 |u(t,x)|2+|u(t,x)|N2−N2 dx = 0. t→+∞ |x|2 |x|>t+A It remains to prove t→li+m∞(cid:13)(cid:13)∇t,x(cid:0)vlA(t)−vl(t)(cid:1)(cid:13)(cid:13)L2(|x|>t+A) = 0. We let G,GA ∈ L2(R×SN−1) be the radiation fields associated to vl and vlA respectively (see Theorem 2.1). We will prove (3.11) ∀ω ∈ SN−1, ∀η > A, G(η,ω) = GA(η,ω), which, in view of (3.10) and Theorem 2.1, will yield the conclusion of Step 2. Fixε > 0andΦ ∈ L2(R×SN−1)suchthatΦ(η,ω) = 0ifη ≤ A+ε. Letwl(t) = Sl(t)(w0,w1) be the solution of the linear wave equation (1.6) whose associated radiation field for t → +∞ is Φ (see Theorem 2.1). In other words, (cid:90) +∞(cid:90) (cid:12) (cid:12)2 (3.12) lim (cid:12)(cid:12)rN2−1∂twl(t,rω)−Φ(r−t,ω)(cid:12)(cid:12) dωdr = 0 t→+∞ 0 SN−1 (cid:90) +∞(cid:90) (cid:12) (cid:12)2 (3.13) lim (cid:12)(cid:12)rN2−1∂rwl(t,rω)+Φ(r−t,ω)(cid:12)(cid:12) dωdr = 0. t→+∞ 0 SN−1 On one hand, we have (cid:16) (cid:17) (cid:16) (cid:17) (3.14) (cid:126)u(τn),Sl(τn)(w0,w1) = Sl(−τn)(cid:126)u(τn),(w0,w1) H˙1×L2 H˙1×L2 −→ ((v ,v ),(w ,w )) , n→∞ 0 1 0 1 H˙1×L2 by the definition (3.1) of (v ,v ), and thus, using the isometry property of radiation fields: 0 1 (cid:90) +∞(cid:90) (3.15) nl→im∞((cid:126)u(τn),Sl(τn)(w0,w1))H˙1×L2 = 2 −∞ SN−1G(η,ω)Φ(η,ω)dωdη. On the other hand: (cid:90) ((cid:126)u(τn),Sl(τn)(w0,w1))H˙1×L2 = ∇t,xu(τn,x)·∇t,xwl(τn,x)dx (cid:90) +∞(cid:90) N−1 = r 2 (∂tu(τn,rω)−∂ru(τn,rω))Φ(r−τn,ω)dωdr+on(1) 0 SN−1 (cid:90) +∞(cid:90) = 2 GA(r−τ ,ω)Φ(r−τ ,ω)dωdr+o (1), n n n 0 SN−1 where at the last line we used that Φ(r−τ ,ω) = 0 if r−τ ≤ A, (3.10), and the definition of n n GA. Hence (cid:16) (cid:17) (cid:90) +∞(cid:90) lim (cid:126)u(τn),S(cid:126)l(τn)(w0,w1) = 2 GA(η,ω)Φ(η,ω)dωdη. n→∞ H˙1×L2 −∞ SN−1 SCATTERING PROFILE FOR THE ENERGY-CRITICAL WAVE EQUATION 9 Combiningwith(3.15), weobtainthatforallΦ ∈ L2(R×SN−1)suchthatΦ(η) = 0ifη ≤ A+ε, (cid:90) +∞ (GA−G)Φ = 0. −∞ Using this equality with Φ(η,ω) = (GA(η,ω)−G(η,ω))11 η≥A+ε yields (3.11) (since ε > 0 can be taken arbitrarily small), which concludes Step 2. Step 3. By Step 2, and since we are assuming that the conclusion of Theorem 2 does not hold, A ∈ R. In this step we prove: (3.16) (cid:107)u(cid:107) = ∞, S({t>0,|x|>A+t}) which will conclude the proof of Proposition 3.1. We argue by contradiction, assuming (3.17) (cid:107)u(cid:107) < ∞. S({t>0,|x|≥A+t}) Let δ be given by Claim 2.4 and T (cid:29) 1 such that 1 (cid:107)u(cid:107) < δ /4. S({t>T,|x|≥A+t}) 1 Then by Claim 2.4, (cid:107)Sl(t−T)(cid:126)u(T)(cid:107)S({t>T,|x|>A+t}) < δ1/2. Let ε > 0 such that (3.18) (cid:107)Sl(t−T)(cid:126)u(T)(cid:107)S({t>T,|x|>A−ε+t}) < δ1. Then, again by Claim 2.4, (cid:107)u(cid:107) < 2δ , S({t>T,|x|>A−ε+t}) 1 which contradicts the definition of A, concluding the proof. (cid:3) 4. Singular and regular directions In this section, we still assume that u is a solution of (1.1) that satisfies the assumptions of Theorem 2 and not its conclusion. We let A ∈ R be defined by Proposition 3.1 in the preceding section. Definition 4.1. The set R of regular directions is the set of ω ∈ SN−1 such that there exists ε > 0 with (4.1) (cid:107)u(cid:107)S(cid:16)(cid:110)t>0,|x|>A−ε+tand(cid:12)(cid:12)((cid:92)ω,x)(cid:12)(cid:12)<ε(cid:111)(cid:17) < ∞, (cid:12) (cid:12) (cid:92) where (ω,x) ∈ [−π,π) is the angle between ω and x. The set S of singular directions is defined as S := SN−1\R. In this section we prove: Proposition 4.2. Under the above assumptions, the set S is finite and nonempty. Proposition 4.3. There exists δ > 0 such that if ω ∈ SN−1 satisfies, for some ε > 0 2 (4.2) lTim→+in∞f(cid:107)Sl(·−T)(cid:126)u(T)(cid:107)S(cid:16)(cid:110)t>T,A−ε+t≤|x|≤A+ε+t and ((cid:92)ω,x)≤√4 (cid:111)(cid:17) < δ2, T then ω ∈ R. 10 T.DUYCKAERTS,C.KENIG,ANDF.MERLE We first prove Proposition 4.2 assuming Proposition 4.3. 4.1. Finiteness of the set of singular directions. We first prove by contradiction that S is non-empty. Assume that S is empty. Thus R = SN−1 and for all ω belonging to SN−1, there exists ε(ω) such that (4.3) (cid:107)u(cid:107)S(cid:16)(cid:110)t>0,|x|>A−ε(ω)+tand(cid:12)(cid:12)((cid:92)ω,x)(cid:12)(cid:12)<ε(ω)(cid:111)(cid:17) < ∞. (cid:12) (cid:12) By the compactness of SN−1, we can find ω ,...,ω in SN−1 such that 1 J J (cid:91) (cid:110) (cid:12) (cid:92) (cid:12) (cid:111) (4.4) SN−1 = ω ∈ SN−1 : (cid:12)(ω,ω )(cid:12) < ε(ω ) . (cid:12) j (cid:12) j j=1 Letting ε = min ε(ω ), we see that j=1...J j (cid:107)u(cid:107) < ∞, S({t>0,|x|>A−ε+t}) contradicting the definition of A. We next prove that S is finite. Let ω ,...,ω be two by two distinct elements of S. Then for 1 J large T, the sets (cid:26) (cid:12)(cid:92)(cid:12) 4 (cid:27) (t,x) : t > T, A−ε+t ≤ |x| ≤ A+ε+t and (cid:12)(ω ,x)(cid:12) ≤ √ (cid:12) j (cid:12) T are pairwise disjoint. As a consequence, for large T, J (cid:88) 2(N+1) (4.5) (cid:107)Sl(·−T)(cid:126)u(T)(cid:107) N−2 j=1 L2(NN−+21)(cid:16)(cid:110)t>T,A−ε+t≤|x|≤A+ε+tand(cid:12)(cid:12)(cid:12)((cid:92)ωj,x)(cid:12)(cid:12)(cid:12)≤√4T(cid:111)(cid:17) 2(N+1) ≤ (cid:107)S (·−T)(cid:126)u(T)(cid:107) N−2 . L 2(N+1) L N−2 ([A−ε+T,+∞)×RN) Since u(T) is bounded in H˙1 × L2, the right-hand side of (4.5) is bounded independently of T > 0. Let (cid:110) (cid:12)(cid:92)(cid:12) √ (cid:111) Ω = (t,x) : t > T, A−ε+t ≤ |x| ≤ A+ε+t and (cid:12)(ω ,x)(cid:12) ≤ 4/ T . j (cid:12) j (cid:12) We will prove that S is finite distinguishing between N = 3 and N = 4,5. If N = 3, we have 3 2 (4.6) (cid:107)Sl(·−T)(cid:126)u(T)(cid:107) ≤ (cid:107)Sl(·−T)(cid:126)u(T)(cid:107)5 (cid:107)Sl(·−T)(cid:126)u(T)(cid:107)5 S(Ωj) L4tL1x2(Ωj) L8(Ωj) Since ω ∈ S, the left-hand side of (4.6) is, according to Proposition 4.3, bounded from below j by δ2 for large n. Combining with the boundedness of (cid:126)u(t) in H˙1×L2 and Strichartz estimates, 2 we deduce that for large t, 2 δ2 ≤ C(cid:107)Sl(·−T)(cid:126)u(T)(cid:107)L58(Ωj). Hence by (4.5), Jδ220 ≤ C(cid:107)Sl(·−T)(cid:126)u(T)(cid:107)8L8([A−ε+T,+∞)×RN) ≤ C, where the right-hand side inequality is a consequence of Strichartz inequality and the boundedness of (cid:126)u in H˙1×L2. This proves that S is finite.

Scattering profile for global solutions of the energy-critical wave equation

Similar Scattering profile for global solutions of the energy-critical wave equation