Scattering to a stationary solution for the superquintic radial wave equation outside an obstacle
aa r X i v : . [ m a t h . A P ] O c t SCATTERING TO A STATIONARY SOLUTION FOR THESUPERQUINTIC RADIAL WAVE EQUATION OUTSIDE ANOBSTACLE
THOMAS DUYCKAERTS AND JIANWEI YANG Abstract.
We consider the focusing wave equation outside a ball of R , withDirichlet boundary condition and a superquintic power nonlinearity. We clas-sify all radial stationary solutions, and prove that all radial global solutionsare asymptotically the sum of a stationary solution and a radiation term. Introduction
Let K be a compact subset of R and Ω = R \ K . Consider a wave equation onΩ with Dirichlet boundary condition(1.1) ( ( ∂ t − ∆) u ( t, x ) = F ( u ) , ( t, x ) ∈ R × R ( u, ∂ t u ) | t =0 = ( u , u ) , u ↾ ∂ Ω = 0 , where the initial data ( u , u ) is assumed to be in a Sobolev space, and in par-ticular to have some decay at infinity. We will mainly be interested in a focusingsupercritical nonlinearity F ( u ) = | u | m u , where m > R . We first review known results in more general cases.The global dynamics of the linear wave equation ( F ( u ) = 0) is quite well under-stood, and depends on the geometry of the obstacle: • When K is non-trapping, for example convex, the global-in-time dispersiveproperties of the wave equation on the whole space R still hold. The localenergy of smooth, compactly supported solutions decay exponentially (see[29]. Strichartz estimates are available (see e.g. [33]). • When K is a trapping obstacle, some of the preceding properties persist,but it might be in weaker forms that depend on the geometry. In someweakly trapped geometries, the same Strichartz estimates as in R hold, asproved in [26]. In full generality, the decay of the energy is only logarithmic(see [4]) and Strichartz estimates might hold only locally.The defocusing equation F ( u ) = −| u | m u was mainly considered in the energy-critical situation m = 2 with a non-trapping obstacle. Once Strichartz estimatesare known, the proof of global well-posedness can be easily adapted to this case(see [32]). Under geometric assumptions that imply in particular that the obstacleis non-trapping, and are satisfied when K is convex, it is proved in [1] that all Date : October 3, 2019.1991
Mathematics Subject Classification. LAGA (UMR 7539), Universit´e Paris 13, Sorbonne Paris Cit´e, and Institut Universitaire deFrance. Department of Mathematics, Beijing Institute of Technology and LAGA (UMR 7539), Uni-versit´e Paris 13. Partially supported by the Labex MME-DII. AND JIANWEI YANG solutions scatter to a solution of the linear wave equation (see also [7] for Neumannboundary conditions in a radial setting). This property persists in the super-criticalcase m > F ( u ) = | u | m u , except therecent preprint of P. Bizo´n and M. Maliborski [2]. As in the case without obstacle,it is easy to construct, for any m >
0, solutions blowing up in finite time, usingblow-up solutions of the ODE y ′′ = | y | m y and finite speed of propagation.We are interested in the behaviour of global solutions. The energy-critical casewith m = 2 on the whole space R was treated in a series of work initiated in [23].The equation has an explicit stationary solution W ( x ) = (1 + | x | / − / , which isunique up to scaling and change of sign. In [10], it is proved that any radial globalsolution of the equation is asymptotically the sum of decoupled rescaled stationarysolutions and a solution of the free (linear) wave equation. When m > B ⊂ R be the unit ball centered at the origin, set Ω = R \ B and consider radial solutions of the equation (1.1) with F ( u ) = | u | m u , m > ( ( ∂ t − ∆) u ( t, x ) = | u | m u, ( t, x ) ∈ R × Ω( u, ∂ t u ) | t =0 = ( u , u ) ∈ H , u ↾ ∂ Ω = 0 , where H is the space of radial functions in ˙ H (Ω) × L (Ω) . One can prove that(1.2) is locally well-posed in H . The energy(1.3) E ( ~u ( t )) = 12 Z Ω (cid:12)(cid:12) ∇ u ( t, x ) (cid:12)(cid:12) dx + 12 Z Ω (cid:12)(cid:12) ∂ t u ( t, x ) (cid:12)(cid:12) dx − m + 1) Z Ω (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) m +1) dx is conserved by the flow. As mentioned before, the equation admits solutionsblowing-up in finite time. More interestingly, there are also stationary solutions: Proposition 1.1.
Assume m > is an integer. For any integer k ≥ , there existsa unique radial stationary solution Q k ∈ C ∞ (Ω) of (1.2) such that Q k ( x ) = 0 for x ∈ ∂ Ω , and r Q k ( r ) has exactly k zeros on (1 , ∞ ) , and is positive for large r .More precisely, there exists c k > such that (cid:12)(cid:12)(cid:12) Q k ( r ) − c k r (cid:12)(cid:12)(cid:12) . r , lim r →∞ r Q k ( r ) = − c k . Moreover, the sequence ( E ( Q k , k ∈ N is increasing. Finally the set of stationarysolutions of (1.2) is exactly { Q k , k ∈ N } ∪ {− Q k , k ∈ N } ∪ { } . Our main result is that the stationary solutions Q k are the only obstruction tolinear scattering for global solutions. Consider the linear wave equation outside Ω:(1.4) ( ( ∂ t − ∆) u ( t, x ) = 0 , ( t, x ) ∈ R × Ω( u, ∂ t u ) | t =0 = ( u , u ) ∈ H , u ↾ ∂ Ω = 0 . Theorem 1.2.
Let u be a solution of (1.2) on [0 , ∞ ) × Ω . Then there exists asolution v L of the linear wave equation (1.4) , and a stationary solution Q of (1.2) such that lim t →∞ k ~u ( t ) − ~v L ( t ) − ( Q, k ˙ H (Ω) × L (Ω) = 0 . The same statement holds true for t → −∞ . According to Proposition 1.1, Q must be 0 (and in this case the solutions scattersto a linear solution) or one of the nonzero stationary solutions ± Q k . The set ofinitial data leading to scattering is open in H . We conjecture that the set of dataleading to blow-up is open, and that the set of solutions converging locally to ± Q k isa closed submanifold of H , of codimension k + 1 in H . We will study this conjecturein a forthcoming paper. See [2] for numerical and analytical evidences toward thisconjecture in the case k = 0.Note that Theorem 1.2 implies that for any R > t →∞ Z ≤| x |≤ R |∇ ( u ( t, x ) − Q ( x )) | + ( ∂ t u ( t, x )) dx = 0 . An interesting question is the exact rate of this convergence when Q = ± Q k . Thisproblem is discussed in [2] using both theoretical and numerical methods, in thecase k = 0. Our method, based on a contradiction argument, does not give anyquantitative information of this type. Remark 1.3.
The proof of Theorem 1.2 can be adapted to prove that all solutions ofthe corresponding defocusing wave equation scatter to a linear solution (see Remarks2.20, 3.2 and 3.5). See also [6] , where a similar result is proved and used to treatnonradial perturbations of a radial solution.
The proof of Theorem 1.2 relies on the “channels of energy” method, which wasintroduced in [8], and was used in [10] to prove the analog of Theorem 1.2 for theradial energy-critical wave equation in space dimension 3. The proof for equation(1.2) is somehow simpler, since equation (1.2) does not admit any scaling invariance.The core of the proof is the rigidity result (Proposition 3.1) that states that anyradial solution of (1.2) such that X ± lim t →±∞ Z | x | > | t | |∇ t,x u ( t, x ) | dx = 0is stationary. This also implies the following one-pass theorem: Theorem 1.4.
Let ε > be small and k ∈ N . There exists δ > with thefollowing property. For all radial solution u of (1.2) such that there exists t < t with [ t , t ] ⊂ I max ( u ) and k ~u ( t ) − ( Q k , k H ≤ δ, k ~u ( t ) − ( Q k , k H ≥ ε one has ∀ t ∈ [ t , + ∞ ) ∩ I max ( u ) , ∀ Q ∈ { } ∪ [ j ∈ N {± Q j } , k ~u ( t ) − ( Q, k H > δ. THOMAS DUYCKAERTS AND JIANWEI YANG This type of result is important to study the global dynamics of (1.2) from adynamical system point of view (see e.g. [30] for application of this type of onepass theorems in the context of nonlinear dispersive equations).Our method also gives the classification of the dynamics below and at the groundstate energy, in the spirit of [23] and [13]. By definition, the ground state is the leastenergy nonzero stationary solution Q . The ground state and its opposite − Q arethe unique minimizers for the Sobolev type inequality: k f k L m +2 (Ω) . k∇ f k L (Ω) (see Proposition 2.21). As an immediate consequence of Theorem 1.2, variationalconsiderations and Proposition 3.1, we obtain the classification of the dynamicsbelow the energy of Q : Corollary 1.5.
Let ( u , u ) ∈ H such that E ( u , u ) ≤ E ( Q , , u be the corre-sponding solution of (1.2) , and ( T − , T + ) the maximal interval of existence of u . • If R Ω |∇ u | < R Ω |∇ Q | , then u is global, ∀ t ∈ R , Z Ω |∇ u ( t ) | < Z Ω |∇ Q | , and either u scatters in both time directions, or E ( u , u ) = E ( Q , andthere exists a sign ± such that u scatters as t → ∓∞ and (1.6) lim t →±∞ k ~u ( t ) − ( Q , k H = 0 . • If R Ω |∇ u | = R |∇ Q | , then u is one of the two stationary solutions ± Q . • If R Ω |∇ u | > R Ω |∇ Q | , then ∀ t ∈ ( T − , T + ) , Z Ω |∇ u ( t ) | > Z Ω |∇ Q | . Furthermore, at most one of the times T + or T − is infinite. If T ± is infinitefor one sign ± , then E ( u , u ) = E ( Q , and (1.6) is satisfied. In particular, if E ( u , u ) < E ( Q , R N . At the threshold energy, as in [12, 13],a new type of solutions arise, satisfying (1.6) for one (and only one) sign ± . Asin [13], one could prove the existence and uniqueness of such solutions, using theunique negative eigenvalue of the linearized operator at Q . We plan to treat thesequestions in a forthcoming paper.Let us mention some related works. The defocusing energy-critical wave equationwith a potential in dimension 3 is considered in [20, 18, 19]. For this equation,there is no blow-up in finite time and every solution is global and scatters to astationary solution, in the sense that the conclusion of Theorem 1.2 holds. Theset of stationary solution for this equation is not classified as in Proposition 1.1,altough it is proved that for generic potential this set is finite. We refer to [21]for the study of equivariant wave maps outside a ball. Again, there is no blow-upin finite time and every solution scatters to a stationary solution (an harmonicmap), which is uniquely determined by the equivariance map of the equation. Theunderlying space dimension in [21] is 5, which makes the proofs more technicallychallenging, however the dynamics of equation (1.2) is somehow richer, since blow-up in finite time is allowed, and there is a countable family of stationary solutions.In particular, one might contemplate solutions of (1.2) that scatter to two distinctstationary solutions as t → + ∞ and t → −∞ . The outline of the paper is as follows. In Section 2 we give some preliminarieson well-posedness (including a new profile decomposition for equation (1.2)) andstationary solutions of (1.2). In Section 3 we prove our main result, the classificationTheorem 1.2. In Section 4 we prove Corollary 1.5 and 1.4. Both proofs are short,relying on the rigidity Proposition 3.1, and, for Corollary 1.5, on Theorem 1.2.
Notations. If a and b are two positive quantities, we write a . b when there existsa constant C > a ≤ Cb . We will write a ≈ b when we have both a . b and b . a . We will write a ≪ b (resp. a ≫ b ) if there exists a sufficiently largeconstant C > Ca ≤ b (resp. a ≥ Cb ). We denote N the set of naturalnumbers.We use B to denote the unit open ball { x ∈ R : | x | < } and Ω = R \ B .The homogeneous Sobolev space ˙ H (Ω) to be used frequently is defined as theclosure of C ∞ (Ω) under the ˙ H norm. We refer to [3, 5, 28] for a systematicinvestigation on the homogeneous space ˙ H sD (Ω) associated to the Laplacian ∆ =∆ D subject to the Dirichlet boundary condition u | ∂ Ω = 0, with fractional s . Weremark that k f k ˙ H ≈ k√− ∆ f k L , where the latter norm is defined via the spectralresolution of ∆.For a radial function f depending on t and r := | x | , we let ~f := ( f, ∂ t f ) . We let L pt ( I, L qx ) be the space of measurable functions f on I × R such that k f k L pt ( I,L qx (Ω)) = Z I (cid:18)Z Ω | f ( t, x ) | q dx (cid:19) pq dt ! /p < ∞ . For q >
1, we use q ′ = qq − to mean its Lebesgue conjugate.We denote by S L ( t ) the linear propagator, i.e. S L ( t )( w , w ) := cos ( t √− ∆ D ) w + sin ( t √− ∆ D ) √− ∆ D w . Acknowledgement.
The first author would like to thank Piotr Bizo´n for intro-ducing equation (1.2) and fruitful discussions on the subject.2.
Preliminaries
Radial linear wave solutions on Ω . Consider u ( t, x ) a radial solution of(1.4). Assume that ( u , u ) ∈ C ( R ). Using that ( ∂ t − ∂ r )( ru ) = 0 and theboundary condition u ( t,
1) = 0, we deduce that(2.1) ru = ψ ( t + r ) − ψ ( t + 2 − r )for some function ψ ∈ C ( R ). One can compute ψ using the initial condition:(2.2) ψ ( σ ) = hR − σ ρu ( ρ ) dρ − (2 − σ ) u (2 − σ ) i , if σ < hR σ ρu ( ρ ) dρ + σ u ( σ ) i , if σ > . THOMAS DUYCKAERTS AND JIANWEI YANG and thus:(2.3) 2 ru ( t, r ) = R t + rt +2 − r ρu ( ρ ) dρ + ( r + t ) u ( r + t ) − ( t + 2 − r ) u ( t + 2 − r ) , r − < t, R t + rr − t ρu ( ρ ) dρ + ( r + t ) u ( r + t ) + ( r − t ) u ( r − t ) , r − > | t | R − t − rr − t ρu ( ρ ) dρ − (2 − r − t ) u (2 − r − t ) − ( r − t ) u ( r − t ) , r − < − t. We will also need the following exterior energy bound:
Lemma 2.1.
Let R ≥ , and u be a radial solution of the linear wave equation (1.4) with initial data ( u , u ) ∈ H . Then (2.4) X ± lim t →±∞ Z ∞ R + | t | ( ∂ r ( ru )) + ( ∂ t ( ru )) dr = Z + ∞ R ( ∂ r ( ru )) + r u dr. Proof.
By density, we can assume that ( u , u ) is C . By explicit computation, and(2.1), ( ∂ r ( ru ) + ( ∂ t ( ru )) = 2( ˙ ψ ( t + r ) + ˙ ψ ( t + 2 − r )) , and one can check that both sides of (2.4) equal2 Z ∞ R ˙ ψ + 2 Z − R −∞ ˙ ψ . (cid:3) Remark 2.2.
In the case R = 1 , we can check by integration by parts that Z + ∞ | t | ( ∂ r ( ru )) dr = Z + ∞ | t | ( ∂ r u ) r dr + o (1) , t → ±∞ and Z + ∞ ( ∂ r ( ru )) dr = Z + ∞ ( ∂ r u ) r dr, and the preceeding lemma is equivalent to X ± lim t →±∞ Z | x |≥ | t | |∇ u ( t ) | + ( ∂ t u ( t )) dx = Z | x |≥ |∇ u | + u . The following asymptotics follow from (2.2)
Lemma 2.3.
For all ( u , u ) ∈ H , we have, denoting by u the solution of (1.4)(2.5) lim t →±∞ Z | x | | u ( t, x ) | dx + k u ( t ) k L (Ω) ∩ L ∞ (Ω) = 0 . For both signs + and − , there exists G ± ∈ L ( R ) such that lim t →±∞ Z ∞ | r∂ r u ( t, r ) − G ± ( r ∓ t ) | dr = 0(2.6) lim t →±∞ Z ∞ | r∂ t u ( t, r ) ± G ± ( r ∓ t ) | dr = 0 . (2.7) Furthermore (2.8) Z R G ( η ) dη = Z R G − ( η ) dη = 12 Z + ∞ (cid:0) ( ∂ t u ( t, r )) + ( ∂ r u ( t, r )) (cid:1) r dr, and both maps ( u , u ) → G ± are bijective. Proof.
From the formula (2.2), we obtain (2.5), as well as (2.6) and (2.7) with G + ( σ ) = ψ ′ (2 − σ ), G − ( σ ) = ψ ′ ( σ ), that is: G + ( σ ) = 12 ( − σ u ( σ ) + u ( σ ) + σ u ′ ( σ ) if σ > − σ ) u (2 − σ ) + u (2 − σ ) + (2 − σ ) u ′ (2 − σ ) if σ < G − ( σ ) = 12 ( σ u ( σ ) + u ( σ ) + σ u ′ ( σ ) if σ > − (2 − σ ) u (2 − σ ) + u (2 − σ ) + (2 − σ ) u ′ (2 − σ ) if σ < . Note that u ∈ L (Ω), u ∈ ˙ H (Ω) and Hardy’s inequality imply G ± ∈ L ( R ) asannounced. Using (2.6), (2.7) and the conservation of the energy for equation (1.4),we obtain (2.8). It remains to prove that both maps ( u , u ) G ± are bijective.The injectivity follows immediately from (2.8).To prove the surjectivity, we let G + ∈ L ( R ) (the proof is the same for G − ), anddefine, for r > u ( r ) = 1 r Z r ( G + ( τ ) + G + (2 − τ )) dτu ( r ) = 1 r ( G + (2 − r ) − G + ( r )) . We notice that ( u , u ) ∈ ˙ H . Indeed, since G + ∈ L ( R ), we have Z + ∞ ( ru ) dr < ∞ , Z + ∞ ( ∂ r ( ru )) dr ≤ k G + k L . Furthermore, by a straightforward integration by parts, Z R u ( r ) ddr ( ru ) dr = 12 Z R u ( r ) dr + R u ( R ) , which shows by Cauchy-Schwarz that R R | u ( r ) | dr ≤ k G + k L for all R >
1, andthus R + ∞ | u ( r ) | dr < ∞ .Letting u be the solution of (1.4) with initial data ( u , u ), we see from (2.2)that u satisfies (2.6), (2.7) (with the + sign) which concludes the proof. (cid:3) An overview of the Cauchy theory in H . In this subsection, we recallthe local well-posedness theory of the problem (1.2) in the energy space with radialinitial data.Let us start by recalling the Strichartz estimate proved in [33, 5, 28].
Proposition 2.4.
Let ( q, r ) such that /q + 3 /r = 1 / and q > . Then thereexists C > such that, if u is a solution to the Cauchy-Dirichlet problem ( ∂ t − ∆) u ( t, x ) = F ( t, x ) , ( t, x ) ∈ R × Ω( u (0 , x ) , ∂ t u (0 , x )) = ( u , u ) ∈ ˙ H (Ω) × L (Ω) u ( t, x ) = 0 , x ∈ ∂ Ω . (2.9) one has (2.10) k u k L qt ( R ; L rx (Ω)) ≤ C (cid:16) k u k ˙ H (Ω) + k u k L (Ω) + k F k L t ( R ; L x (Ω)) (cid:17) . THOMAS DUYCKAERTS AND JIANWEI YANG In the radial case, one can extend the range of Strichartz exponents, using theradial Sobolev inequality(2.11) ∀ R > , | f ( R ) | . √ R k f k ˙ H . Note that (2.11) implies that for 6 < p ≤ ∞ , ˙ H (Ω) is embedded into L p (Ω) withcompact embedding. Corollary 2.5.
Assume that /q + 3 /r ≤ / and r is finite. There exists C > such that if u and F are radial solutions of (2.9) , then (2.10) holds.Proof. Assume that q + r < , and let q such that q + r = . Since r < ∞ , q >
2. By energy inequalities and the embedding H (Ω) ⊂ L r (Ω), we have k u k L ∞ t ( R ,L r (Ω)) . k u k ˙ H (Ω) + k u k L (Ω) + k F k L t ( R ,L (Ω)) . By standard Strichartz estimates, k u k L q t ( R ,L r (Ω)) . k u k ˙ H (Ω) + k u k L (Ω) + k F k L t ( R ,L (Ω)) , and (2.10) follows since q < q < ∞ . (cid:3) Note that the assumption m > q = 2 m + 1, r = 2(2 m + 1) satisfythe assumptions of Corollary 2.5.We state our main result in this subsection. Proposition 2.6.
Assume m ∈ (2 , + ∞ ) ∩ Z in (1.2) . Then for every −→ u :=( u , u ) ∈ H , there exists a unique maximal radial solution u of (1.2) defined in amaximal interval [0 , T ∗ ) with −→ u (0) = −→ u and T ∗ > C/ k−→ u k m ˙ H , for some universalconstant C > , satisfying u ∈ C (cid:16) [0 , T ∗ ) , ˙ H (Ω) (cid:17) ∩ C (cid:0) [0 , T ∗ ) , L (Ω) (cid:1) . In addition, we have the following properties: (i) either T ∗ = + ∞ , or T ∗ < + ∞ and (2.12) lim T ր T ∗ k u k L m +1 t (cid:16) [0 ,T ] , L m +1) x (Ω) (cid:17) = + ∞ . Moreover, for every T ∈ (0 , T ∗ ) , the flow map ( v , v ) ~v (where v is thesolution of (1.2) with initial data ( v , v ) ) is Lipschitz continuous from aneighborhood of ( u , u ) in H to C ([0 , T ] , H ) . (ii) If −→ u ∈ (cid:16) ˙ H (Ω) ∩ ˙ H (Ω) (cid:17) × ˙ H (Ω) then −→ u ( t ) ∈ (cid:16) ˙ H (Ω) ∩ ˙ H (Ω) (cid:17) × ˙ H (Ω) for all t ∈ (0 , T ∗ ) . (iii) E ( −→ u ( t )) = E ( −→ u ) for every t ∈ [0 , T ∗ ) . (iv) If k u k L m +1 t (cid:16) [0 ,T ∗ ) ,L m +1) x (Ω) (cid:17) < ∞ , then T ∗ = + ∞ and u scatters in theforward time direction, i.e. there exists −→ u + = ( u +0 , u +1 ) ∈ H such that (2.13) lim t → + ∞ k−→ u ( t ) − −→ S L ( t ) −→ u + k H = 0 , where −→ S L ( t ) −→ u + := ( S L ( t ) ~u + , ∂ t S L ( t ) ~u + ) . Conversely, if u scatters, then (2.14) k u k L m +1 t (cid:16) [0 , ∞ ) , L m +1) x (Ω) (cid:17) + k∇ x,t u k L ∞ t ([0 , ∞ ) , L x (Ω)) < + ∞ . (v) Let I ⊂ [0 , ∞ ) be a sub-interval such that (2.15) k S L ( t ) −→ u k L m +1 t (cid:16) I, L m +1) x (Ω) (cid:17) = δ with < δ ≪ being sufficiently small. Then u is defined on I . Inparticular, I ⊂ [0 , T ∗ ) and moreover, with C in (2.10) k u − S L ( t ) −→ u k L m +1 t (cid:16) I, L m +1) x (Ω) (cid:17) ε := C (2 δ ) m +1 The analogs of the statements (i)-(v) hold in the negative time direction aswell.
The proof follows mainly from a standard fixed point argument based on Strichartzestimates in Corollary 2.5 and we only sketch it here. By using energy estimate,Sobolev embedding and radial Sobolev inequality, it is readily to solve (1.2) on aninterval [0 , T ] with
T < ¯ C k−→ u k − m H for some constant ¯ C >
0, depending only on m, C and optimal constants in Sobolev embedding. Let T ∗ be the maximal timeof existence. Then (i) follows by using Strichartz estimate. (ii) is deduced fromstandard bootstrap argument based on the Duhamel formula. By using (ii) andstandard density argument, we obtain the conservation of energy (iii). Finally, (iv)and (v) is immediately verified by using Strichartz estimates and energy estimate.We next establish a long-time perturbation lemma for (1.2). Lemma 2.7.
Given
M > , we have ε M > and C M > with the followingproperties. Let I be an interval, t ∈ I , and u, ˜ u ∈ L m +1 (cid:16) I, L m +1)rad (Ω) (cid:17) suchthat ~u, ~ ˜ u ∈ C ( I, H ) and (2.16) k ˜ u k L m +1 t (cid:16) I,L m +1) x (Ω) (cid:17) ≤ M, (2.17) k eq( u ) k L t ( I,L x (Ω)) + k eq(˜ u ) k L t ( I,L x (Ω)) + k R L k L m +1 t (cid:16) I,L m +1) x (Ω) (cid:17) = ε, where ε ≤ ε M , eq( u ) = ( ∂ t − ∆) u − | u | m u in the sense of distribution and R L ( t ) = S L ( t − t )( ~u ( t ) − ~ ˜ u ( t )) . Then k u − ˜ u k L m +1 t ( I,L m +1) x (Ω)) + sup t ∈ I k∇ x,t ( u ( t ) − ˜ u ( t ) − R L ( t )) k L (Ω) ≤ C M ε. For the proof we will need the following Gr¨onwall-type lemma (see [16])
Lemma 2.8.
Let ≤ β < γ ≤ ∞ and define ρ ∈ [1 , ∞ ) by ρ = β − γ . Let < T ≤ ∞ , f ∈ L ρ (0 , T ) and ϕ ∈ L γ loc ([0 , T )) such that ∀ t ∈ [0 , T ) , k ϕ k L γ (0 ,t ) ≤ η + k f ϕ k L β (0 ,t ) . Then ∀ t ∈ [0 , T ) , k ϕ k L γ (0 ,t ) ≤ η Φ (cid:0) k f k L ρ (0 ,t ) (cid:1) , where Φ( s ) = 2Γ(3 + 2 s ) and Γ is the Gamma function.Proof of Lemma 2.7. Let w := u − ˜ u . Then we have( ∂ t − ∆) w = eq( u ) − eq(˜ u ) | {z } := e + | u | m u − | ˜ u | m ˜ u. We assume to fix ideas t = 0 and I = [0 , T ). By Duhamel’s formula(2.18) w ( t ) = R L ( t ) + Z t sin( t − s ) √− ∆ √− ∆ (cid:0) e + (˜ u + w ) m +1 − ˜ u m +1 (cid:1) ( s ) ds. AND JIANWEI YANG Using Strichartz (2.10) and H¨older inequalities, we deduce ∀ t ∈ [0 , T ) , k w k L m +1 ((0 ,t ) L m +1) (Ω)) ≤k R L k L m +1 (0 ,t,L m +1) (Ω)) + C k e k L ((0 ,t ) ,L (Ω)) + C Z t (cid:16) k w ( τ ) k L m +1) k ˜ u ( τ ) k mL m +1) (Ω) + k w ( τ ) k m +1 L m +1) (Ω) (cid:17) dτ. From (2.17), we obtain ∀ t ∈ [0 , T ) , k w k L m +1 ((0 ,t ) L m +1) (Ω)) ≤ (1 + 2 C ) ε + C Z t (cid:16) k w ( τ ) k L m +1) k ˜ u ( τ ) k mL m +1) (Ω) + k w ( τ ) k m +1 L m +1) (Ω) (cid:17) dτ. Let θ such that k w k L m +1 ((0 ,θ ) L m +1) (Ω)) ≤ C M ε ( C M to be specified). Then ∀ t ∈ [0 , θ ) , k w k L m +1 ((0 ,t ) L m +1) (Ω)) ≤ (1 + 2 C ) ε + C C m +1 M ε m +1 + C Z t (cid:16) k w ( τ ) k L m +1) k ˜ u ( τ ) k mL m +1) (Ω) (cid:17) dτ ≤ (2 + 2 C ) ε + C Z t (cid:16) k w ( τ ) k L m +1) (Ω) k ˜ u ( τ ) k mL m +1) (Ω) (cid:17) dτ, provided C C m +1 M ε m ≤ ε ≤ ε M = 1 /C m C m M ).Using Lemma 2.8 with ϕ ( t ) = k w ( t ) k L m +1) (Ω) , f ( t ) = k ˜ u ( t ) k mL m +1) (Ω) β = 1 , γ = 2 m + 1 , ρ = 2 m + 12 m , we obtain k w ( t ) k L m +1 ( (0 ,θ ) ,L m +1) (Ω) ) ≤ (2 + 2 C )Φ (cid:0) C M m (cid:1) ε. Choosing C M > (2 + 2 C )Φ (cid:0) C M m (cid:1) , we obtain by a simple bootstrap argument k w ( t ) k L m +1 ( (0 ,T ) ,L m +1) (Ω) ) ≤ C M ε. The bound of k∇ t,x ( u − ˜ u − R L ) k L (Ω) follows from Strichartz estimates and theequality (2.18). (cid:3) Definition 2.9.
Let Σ +rad be the set of radial functions ( u , u ) ∈ H such that if u is the solution of (1.2) with initial data ( u , u ), then u ( t, x ) exists on [0 , + ∞ ) andscatters to a linear wave. We define Σ − rad similarly for the negative time direction.The following proposition is an immediate consequence of Lemma 2.7 and thecharacterization of scattering from Proposition 2.6: Proposition 2.10. Σ +rad and Σ − rad are open. Profile decomposition.
We prove here that there exists a profile decom-position which is adapted to the Strichartz norm used in the scattering theory ofequation (1.2).
Proposition 2.11.
Let ( u n ) n be a sequence of radial solutions of the linear waveequation outside the ball (1.4) such that ( ~u n (0)) n is bounded in H . Then thereexists a subsequence of ( u n ) n (that we still denote by ( u n ) n ), and, for any integer j ≥ , a solution U jL of (1.4) and a sequence ( t j,n ) n ∈ R N satisfying j = j ′ = ⇒ lim n →∞ (cid:12)(cid:12) t j,n − t j ′ ,n (cid:12)(cid:12) = + ∞ , such that, letting, for J ≥ , w Jn ( t ) = u n ( t ) − J X j =1 U jL ( t − t j,n ) , we have, for all ( q, r ) ∈ (2 , ∞ ] × (6 , ∞ ) such that q + r < , (2.19) lim J →∞ lim sup n →∞ k w Jn k L qt L rx = 0 . Furthermore, ∀ j ≥ , ~u n ( t j,n ) −−−− ⇀ n →∞ ~U jL (0)(2.20) ∀ J ≥ , lim n →∞ k ~u n (0) k H − J X j =1 (cid:13)(cid:13)(cid:13) ~U jL (0) (cid:13)(cid:13)(cid:13) H − (cid:13)(cid:13) ~w Jn (0) (cid:13)(cid:13) H = 0 . (2.21)Proposition 2.11 is a consequence of the following Lemma: Lemma 2.12.
Let ( u n ) n be a sequence of radial solutions of the linear wave equa-tion on Ω (1.4) such that for all sequence ( t n ) n ∈ R N , ~u n ( t n ) −−−− ⇀ n →∞ (0 , in H . Then for all ( q, r ) ∈ (2 , ∞ ] × (6 , ∞ ) such that q + r < lim n →∞ k u n k L qt L rx = 0 . The fact that the Lemma implies Proposition 2.11 is by now standard (see e.g.the proof of Theorem 3.1 in [15]), and we omit it.
Proof of Lemma 2.12.
We argue by contradiction. Assume that there exists a se-quence of solutions ( u n ) n of (1.4) such that for all sequence ( t n ) n ∈ R N ,(2.22) ~u n ( t n ) −−−− ⇀ n →∞ (0 ,
0) in H . Assume that there exist ( q, r ) ∈ (2 , ∞ ] × (6 , ∞ ) with q + r < and ε > ∀ n, k u n k L qt L rx ≥ ε. Let ( q , r ) such that 1 q + 3 r = 1 q + 3 r , < q < q, and let r such that q + r = r (thus 6 < r < ∞ ). Then by H¨older’s inequality, k u n k L qt L rx ≤ k u n k q q L q t L r x k u n k − q q L ∞ t L r x . AND JIANWEI YANG Since by Strichartz estimates k u n k L q t L r x is bounded from above (see Corollary 2.5),we deduce that there existe ε > ∀ n, k u n k L ∞ t L r x ≥ ε . We thus can choose a sequence ( t n ) n such that ∀ n, k u n ( t n ) k L r ≥ ε . This contradicts (2.22) and the compactness of the embedding ˙ H (Ω) ⊂ L r (Ω).The proof is complete. (cid:3) We will need to consider solutions to the wave equation (1.2) outside wave cones.For this, it is convenient to multiply the nonlinearity by a characteristic function
Definition 2.13.
If ( u , u ) ∈ ˙ H (Ω) × L (Ω) and R ≥ the solution of (1.2) on {| x | > R + | t |} , with initial data ( u , u ), is by definition the restriction to {| x | > R + | t |} of the solution u of the following wave equation,(2.24) ( ( ∂ t − ∆) u ( t, x ) = F ( t, x )11 {| x | >R + | t |} , ( t, x ) ∈ R × R ( u, ∂ t u ) | t =0 = ( u , u ) , u | ∂ Ω = 0where F = ι | u | m u with ι = ± m > for equation (2.24). In particular, letting T ∗ R bethe maximal time of existence for (2.24), we have the blow-up criterion T ∗ R < ∞ = ⇒ (cid:13)(cid:13) u {| x | >R + | t |} (cid:13)(cid:13) L m +1 t (cid:16) [0 ,T ∗ R ) , L m +1) x (cid:17) = ∞ , as well as the following scattering criterion. If u {| x | >R + | t |} ∈ L m +1 t (cid:16) [0 , + ∞ ) , L m +1) x (cid:17) , then u scatters for positive times: there exists a solution u L of the linear waveequation on Ω such thatlim t → + ∞ (cid:13)(cid:13) {| x | >R + | t |} |∇ t,x u L ( t ) − ∇ t,x u ( t ) | (cid:13)(cid:13) ˙ H (Ω) × L (Ω) = 0 . Also, there exists ε > R >
1) such that if for some T ∈ (0 , ∞ ], (cid:13)(cid:13) S L ( t )( u , u )11 {| x | >R + | t |} (cid:13)(cid:13) L m +1 t (cid:16) [0 ,T ) , L m +1) x (cid:17) = ε ≤ ε then T ∗ R ≥ T andsup t ∈ [0 ,T ] (cid:13)(cid:13) |∇ t,x ( u ( t ) − S L ( t )( u , u )) | {| x | >R + | t |} (cid:13)(cid:13) L ≤ ε m +1 . We note also that if T ∗ is the maximal (positive) time of existence for the equation(1.2) with the same initial data, then T ∗ ≤ T ∗ R and the two solutions coincide on { ( t, x ) , ≤ t < T ∗ , | x | > R + | t |} . Note however that since we have truncated the nonlinearity with a nonsmooth function, thepersistence of regularity does not hold anymore Let ( u Ln ) n be a sequence of radial solutions of the linear wave equation (1.4)outside the ball. Assume that ( ~u n (0)) n is bounded in ˙ H (Ω) × L (Ω) and has a pro-file decomposition { U jL , ( t j,n ) } j ≥ as in Proposition 2.11. Extracting subsequences,reordering and time translating the profiles, we might assume(2.25) ∀ n, t ,n = 0 , ∀ j ≥ , lim n →∞ t j,n ∈ {±∞} . We define the nonlinear profile U associated to U L as the solution of the nonlinearwave equation (1.2) with initial data ~U L (0). If R ≥ U thesolution of (1.2) on {| x | > R + | t |} with the same initial data. Proposition 2.14.
Let u Ln be as above, and R ≥ . Assume that the nonlinearprofile U is well-defined for { t ≥ , | x | ≥ R + | t |} , and that {| x | >R + | t |} U ∈ L m +1 (cid:16) (0 , ∞ ) , L m +1) (cid:17) . Let u n be the solution of the nonlinear wave equation (1.2) on {| x | > R + | t |} . Thenfor large n , u n is global for positive time, and, letting ǫ Jn ( t, x ) = u n ( t, x ) − U ( t, x ) − J X j =1 U jL ( t − t j,n , x ) − w Jn ( t, x ) , one has lim J → + ∞ lim sup n →∞ sup t ≥ Z | x | >R + | t | (cid:12)(cid:12) ∇ t,x ǫ Jn ( t, x ) (cid:12)(cid:12) dx = 0 . Proof.
By Lemma 2.7 (or rather its version adapted to solutions on {| x | > R + | t |} ),it is sufficient to provelim J →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16) J X j =2 U jL ( · − t j,n ) − w Jn (cid:17) {| x | >R + | t |} (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L m +1 t (cid:16) (0 , ∞ ) ,L m +1) x (cid:17) = 0 . Using that lim J →∞ lim sup n →∞ (cid:13)(cid:13) w Jn (cid:13)(cid:13) L m +1 ( (0 , ∞ ) ,L m +1) ) = 0 , we see that it is sufficient to prove: J ≥ ⇒ lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J X j =2 U jL ( · − t j,n )11 {| x | >R + | t |} (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L m +1 t (cid:16) (0 , ∞ ) ,L m +1) x (cid:17) = 0 . Since lim n →∞ t j,n ∈ {±∞} , this last property follows from the dominated conver-gence theorem, concluding the proof. (cid:3) Zeros of stationary solutions.
In this subsection, we state several proper-ties on a class of singular stationary solutions involved in [11, 14, 15].
Proposition 2.15.
Let m > , m ∈ N , and ℓ ∈ R \ { } . Then there exists a radial, C solution Z ℓ ( x ) = Z ℓ ( | x | ) of (2.26) ∆ Z ℓ + Z m +1 ℓ = 0 on R \ { } , such that (2.27) ∀ r ≥ , (cid:12)(cid:12) r Z ℓ ( r ) − ℓ (cid:12)(cid:12) ≤ Cr
24 THOMAS DUYCKAERTS AND JIANWEI YANG (2.28) lim r →∞ r dZ ℓ dr = − ℓ . Furthermore, Z ℓ L m , where m is the critical Sobolev exponent corresponding to s m = 32 − m . In particular, Z ℓ ˙ H s m . Moreover, the zeros of Z ℓ are given by asequence { r j } ∞ j =0 such that r > r > · · · > r j > · · · −→ , j → ∞ . Remark 2.16.
The existence of such a solution Z ℓ with properties (2.27)(2.28) and Z ℓ L m had been demonstrated in [11] . It remains to show that Z ℓ ( r ) oscillatesinfinitely often towards . This provides a more precise characterization on thebehavior of Z ℓ ( r ) as r approaches the origin. The proof of the oscillatiory property of Z ℓ in Proposition 2.15 relies on thefollowing classical result due to Fowler. Lemma 2.17.
Let θ ( x ) be a solution of (2.29) d θdx + x − θ n = 0 , x ∈ (0 , + ∞ ) , where n > is an odd integer. Then θ is one of the following three distinct types (i) Special solutions (2.30) θ ( x ) = ± (cid:18) n − n − (cid:19) n − x n − ;(ii) Emden’s solutions with one arbitrary constant C (2.31) θ ( x ) = C − α ( x ) C n x , lim x →∞ α ( x ) = 1 , (iii) θ ( x ) oscillates about θ = 0 with the asymptotic forms | θ ( X n ) | ≈ AX n +3 n x n +1 − x n ≈ A n − (cid:18) n + 1 (cid:19) Γ (cid:0) (cid:1) Γ (cid:16) n +1 (cid:17) Γ (cid:16) + n +1 (cid:17) X n +3 n , (2.32) where A is a constant of integration, { X n } is the sequence of zeros of θ ′ ( x ) ,and { x n } is the sequence of zeros of θ ( x ) , that satisfy lim n x n = + ∞ .Proof. Please see p. 281–282 of [17]. (cid:3)
Remark 2.18.
The equation (2.29) along with its general form θ ′′ + x σ θ λ = 0 isusually referred as the Emden-Fowler equation. When λ > is not an integer, onemay find in [24] a similar classification on the solutions of Emden-Fowler equationsin a more general setting.Proof of the oscillation of Z ℓ . We may assume ℓ > Z − ℓ = − Z ℓ .By scaling invariance and the uniqueness of the fixed point argument, it suffices toconsider ℓ = 1 (see Remark 2.5 in [14]) and we denote by Z ( r ) = Z ( r ) for brevity. Rewrite (2.26) fulfilled by Z as the following ordinary differential equations (inthe r variables)(2.33) Z ′′ ( r ) + 2 r Z ′ ( r ) + Z ( r ) m +1 = 0 . Let h ( s ) = Z (1 /s ), s ∈ (0 , ∞ ). Then h is a C solution of(2.34) h ′′ ( s ) + s − h ( s ) m +1 = 0 , s > , which satisfies(2.35) lim s → h ( s ) s = 1 , lim s → h ′ ( s ) = 1 . We are reduced to showing that the zeros of h form a sequence { s j } ∞ j =0 such that0 < s < s < s < · · · < s j < · · · −→ ∞ . In view of Lemma 2.17, it suffices to show that h ( s ) is of type (iii). Invoking that Z ( r ) is not bounded at the origin, we see that h ( s ) can not be of the form (2.31).By (2.35), h ( s ) is not a function given by the formula (2.30). Hence h ( s ) oscillatesinfinitely often and behaves asymptotically according to formula (2.32). (cid:3) Radial stationary solutions outside the unit ball.
Let Z ( x ) be theradial solution of equation (2.26) corresponding to ℓ = 1. As we have seen in thelast subsection, the zeros of Z form a sequence { r j } ∞ j =0 with the following property(2.36) r > r > · · · > r j > · · · −→ , j → ∞ . Let Q j ( r ) = r /mj Z ( r j r ). Then Q j ( | x | ) is the radial solution of the following ellipticequation outside the unit ball Ω = R \ B with the Dirichlet boundary condition(2.37) − ∆ Q = | Q | m Q, Q | ∂ Ω = 0 , x ∈ Ω , m > , m ∈ N , where ∆ = ∆ D is the Dirichlet-Laplacian, and Q belongs to ˙ H (Ω).Notice that Q j ( r ) has exactly j zeros in (1 , + ∞ ) for each j ∈ N and Q j (1) = 0.Define the energy functional E ( Q ) = 12 Z Ω |∇ Q ( x ) | dx − m + 1) Z Ω | Q ( x ) | m +1) dx. Then(2.38) E ( Q j ) = m m + 1) r − m − m j Z + ∞ r j | Z ( r ) | m +1) r dr. This formula with (2.36) clearly yields E ( Q j ) −→ + ∞ monotonically as j → ∞ .The following Lemma shows that there are no other stationary solutions for equa-tion (1.2). Lemma 2.19.
Let Q ∈ ˙ H (Ω) , radial, such that − ∆ Q = | Q | m Q . Then Q ≡ ,or there exists a sign ± and α > such that Q ( r ) = ± α m Z ( αr ) . In particular, if Q (1) = 0 , then Q ≡ or Q = ± Q j for some j ≥ .Proof. We first prove that there exists ℓ ∈ R such that(2.39) (cid:12)(cid:12)(cid:12)(cid:12) Q ( r ) − ℓr (cid:12)(cid:12)(cid:12)(cid:12) . r m − , r ≫ . AND JIANWEI YANG Indeed, we have d dr ( rQ ) = rQ m +1 ( r ). Since by the radial Sobolev inequality(2.11), | Q ( r ) | . /r / , we obtain that ddr ( rQ ) has a limit as r → ∞ . Using that R ∞ (cid:12)(cid:12) ddr ( rQ ) (cid:12)(cid:12) dr is finite, we see that this limit is 0. Thus(2.40) ddr ( rQ ) = − Z ∞ r σQ m +1 ( σ ) dσ. Combining with the radial Sobolev inequality, we obtain (cid:12)(cid:12) ddr ( rQ ) (cid:12)(cid:12) . R ∞ r σ m − dσ . r m − . Since m ≥
3, we deduce that rQ has a limit ℓ . Plugging the estimate | Q ( r ) | . /r into (2.40) and integrating between r and ∞ , we obtain (2.39).If ℓ = 0, we let Y ( r ) = 0 for r >
1. If ℓ = 0, we let α = | ℓ | m − m , ι be the sign of ℓ , and Y ( r ) = ια m Z ( αr )One can check lim r →∞ rY ( r ) = ℓ. We will prove that Q ≡ Y . Indeed, for large r (cid:12)(cid:12)(cid:12)(cid:12) d dr ( rQ − rY ) (cid:12)(cid:12)(cid:12)(cid:12) = r (cid:12)(cid:12) Q m +1 − Y m +1 (cid:12)(cid:12) . r m − | Q ( r ) − Y ( r ) | . Integrating twice, we deduce | rQ ( r ) − rY ( r ) | . Z + ∞ r Z + ∞ σ ρ − m ρ | Q ( ρ ) − Y ( ρ ) | dρ dσ . r m − sup ρ>r | ρ ( Q ( ρ ) − Y ( ρ )) | . Taking the supremum over all r > R , where R ≫ Q ( r ) = Y ( r ) for large r . By classical ODE theory, we deduce that Y ( r ) = Q ( r ) forall r > (cid:3) Remark 2.20.
One can prove that the only stationary solution of the defocusinganalog of (1.2) (that is, with a minus sign in front of the nonlinearity) is . Moreprecisely, similarly to Proposition 2.15 there is, for all ℓ ∈ R \ { } , a solution Z ℓ ofthe elliptic wave equation defined for large r behaving as ℓ/r at infinity. However inthis case, the solution Z ℓ has a constant sign and is defined only for r ∈ ( R ℓ , + ∞ ) ,for some minimal radius of existence R ℓ > that satisfies lim r → R ℓ | Z ( r ) | = ∞ (see [14, Proposition 2.3] ). Proposition 2.21.
For any radial f ∈ ˙ H (Ω) , we have (2.41) k f k L m +1) (Ω) k∇ Q k L (Ω) ≤ k Q k L m +1) (Ω) k∇ f k L (Ω) Furthermore, the equality is achieved in (2.41) if and only if there exists σ ∈ R suchthat f = σQ .Proof. It suffices to show that if we set J ( f ) = k∇ f k m +1) L (Ω) / k f k m +1) L m +1) (Ω) , and a = inf { J ( f ) : f ∈ ˙ H (Ω) \ { } , f radial } , then a = J ( Q ). Notice that from radial Sobolev inequality, we have 0 < a < + ∞ and hence the above two quantities are well-defined. The argument is reminiscent of [34]. Take a minimizing sequence f ν ∈ ˙ H (Ω)which are radial such that J ( f ν ) → a as ν → + ∞ . Since f ν is real valued, we mayassume (replacing f ν by | f ν | if necessary), that f ν is nonnegative. Setting ϕ ν = f ν / k f ν k ˙ H (Ω) , we have J ( ϕ ν ) = J ( f ν ) and k∇ ϕ ν k L (Ω) = 1. Hence there exists asubsequence ϕ ν k converges weakly in ˙ H to ϕ ∗ as k → + ∞ with k ϕ ∗ k ˙ H (Ω) ≤ ϕ ν k converges to ϕ ∗ strongly in L m +1) (Ω). As a consequence, ϕ ∗ = 0since otherwise we would have J ( ϕ ν k ) → + ∞ by the strong convergence. It followsfrom the above discussion that a ≤ J ( ϕ ∗ ) ≤ k ϕ ∗ k m +1) L m +1) (Ω) = lim k → + ∞ k ϕ ν k k m +1) L m +1) (Ω) = a . Thus J ( ϕ ∗ ) = a and k∇ ϕ ∗ k L (Ω) = 1, which along with the weak convergenceimplies ϕ ν k → ϕ ∗ in ˙ H (Ω) strongly as k → + ∞ .It follows from the above facts that ϕ ∗ is the minimizer of the function J andsatisfies the Euler-Lagrange equation: ddε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 J ( ϕ ∗ + εη ) = 0 , ∀ η ∈ C ∞ (Ω) . Taking k∇ ϕ ∗ k L (Ω) = 1 into account, we have − ∆ ϕ ∗ = 1 k ϕ ∗ k m +1)2( m +1) | ϕ ∗ | m ϕ ∗ . Let ϕ ∗ ( x ) = k ϕ ∗ k ( m +1) /mL m +1) Q ( x ). Then we have − ∆ Q = | Q | p − Q on Ω and Q | ∂ Ω = 0, Q ( x ) ≥ x ∈ Ω. By uniqueness of the solution for the problem (2.37) (Lemma2.19), we have Q ( x ) = Q ( x ).Note that the last part of the argument above shows that any minimizer for J is proportional to Q , which concludes the proof of the proposition. (cid:3) Classification of global solutions
Rigidity.
We prove here the following rigidity result:
Proposition 3.1.
Let ρ > and u be a solution of the nonlinear wave equation (1.2) on {| x | > ρ + | t |} . Assume (3.1) X ± lim t →±∞ Z {| x |≥| t | + ρ } |∇ t,x u ( t, x ) | dx = 0 . Then ( u , u )( r ) = 0 for almost all r > ρ , or there exists ℓ ∈ R \ { } , ι ∈ {±} such that ( u , u )( r ) = ( ιZ ℓ ( r ) , for all r > ρ , where Z ℓ is defined in Proposition2.15. Remark 3.2.
Let us mention that the analog of Proposition 3.1, with the sameproof, is also valid for the defocusing equation corresponding to (1.2) . In this case,in view of Remark 2.20, the conclusion is that the solution u is identically .Proof. The proof follows the line of the analogous result for the energy-critical waveequation on R (see [10, Section 2]), with some of the arguments simplified. AND JIANWEI YANG Step 1: channels of energy.
We fix a small ε >
0, and let R ≥ ρ such that(3.2) Z | x |≥ R |∇ u | + u dx ≤ ε, and prove, letting v ( r ) = ru ( r ), v ( r ) = ru ( r ),(3.3) Z + ∞ R ( ∂ r v ) + v dr . R m +1 v m +1)0 ( R ) . Let u L be the solution of the linear wave equation with initial data ( u , u ). Wehave (see Lemma 2.1): Z + ∞ R ( ∂ r ( ru )) + ( ru ) dr ≤ X ± lim t →±∞ Z + ∞ R + | t | (cid:0) ∂ r ( ru L ( t, r )) (cid:1) + r (cid:0) ∂ t u L ( t, r ) (cid:1) dr. Furthermore, by the small data theory,sup t ∈ R (cid:13)(cid:13) {| x | > | t | + R } ( ∇ t,x u ( t ) − ∇ t,x u L ( t )) (cid:13)(cid:13) L . (cid:13)(cid:13) ( ∇ u , u )11 {| x | >R } (cid:13)(cid:13) m +1 L . By a straightforward integration by parts, we have, for any f ∈ ˙ H (Ω), and(3.4) Z + ∞ A ( ∂ r ( rf )) dr = Z + ∞ A ( ∂ r f ) r dr − Af ( A ) , which yields, using assumption (3.1),lim t →±∞ Z + ∞ R + | t | (cid:0) ∂ r ( ru ( t, r )) (cid:1) + r (cid:0) ∂ t u ( t, r ) (cid:1) dr = 0 . Combining, we obtain(3.5) Z + ∞ R (cid:0) ∂ r ( ru ) (cid:1) + ( ru ) dr . (cid:18)Z + ∞ R (cid:16) ( ∂ r u ) + u (cid:17) r dr (cid:19) m +1 . Using the formula (3.4) again, and the smallness assumption (3.2), we deduce(3.6) Z + ∞ R ( ∂ r ( ru )) + ( ru ) dr . R m +1 u m +1)0 ( R ) , hence (3.3). Step 2: limit of r u . In this step we prove that v ( R ) has a limit ℓ as R → ∞ and that there exists a constant K (depending on v ), such that(3.7) | v ( R ) − ℓ | ≤ KR m , Z + ∞ R v ( r ) dr ≤ KR m +1 . Until the end of the proof, we will always denote by K a large constant dependingon v , that may change from line to line.We first fix R, R ′ such that ρ < R < R ′ < R and the smallness assumption(3.2) is satisfied. Then | v ( R ) − v ( R ′ ) | . Z R ′ R | ∂ r v ( r ) | dr . √ R sZ + ∞ R ( ∂ r v ( r )) dr. Using Step 1, we deduce(3.8) | v ( R ) − v ( R ′ ) | . R m | v ( R ) | m +1 . By (3.2) and the integration by parts formula (3.4), we have √ R | v ( R ) | ≤ √ ε , andthus(3.9) | v ( R ) − v ( R ′ ) | . ε m | v ( R ) | . By an easy induction, we deduce that for all k ≥ | v (2 k ρ ) | . (1 + Cε m ) k | v ( ρ ) | ≤ K (1 + Cε m ) k . Going back to (3.8), we obtain | v (2 k ρ ) − v (2 k +1 ρ ) | ≤ K − km (1 + Cε m ) k (2 m +1) . Taking ε > P k ≥ | v (2 k ρ ) − v (2 k +1 ρ ) | converges, and thus that there exists ℓ ∈ R such thatlim k →∞ v (2 k ρ ) = ℓ. This implies that v (2 k ρ ) is bounded. Using (3.8) again we obtain | v (2 k ρ ) − v (2 k +1 ρ ) | ≤ − km K, and summing up: (cid:12)(cid:12) v (2 k ρ ) − ℓ (cid:12)(cid:12) ≤ K − km . By (3.8), if 2 k ρ ≤ r ≤ k +1 ρ , (cid:12)(cid:12) v (2 k ρ ) − v ( r ) (cid:12)(cid:12) ≤ K − km , which concludes the proof of the first bound in (3.7). The second bound followsfrom (3.3) Step 3. Compact support of the difference with a stationay solution. If ℓ = 0, welet Z ℓ be the radial solution of − ∆ Z ℓ = Z m +1 ℓ such that(3.10) (cid:12)(cid:12)(cid:12)(cid:12) Z ℓ ( r ) − ℓr (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kr . (see Proposition 2.15). We define Z as the zero function, so that (3.10) is alsosatisfied in the case ℓ = 0. Our goal is to prove that ( u , u ) = ( Z ℓ ,
0) for almostevery r > ρ . In this step, we prove that this equality holds for large r .We let h ( r ) = u − Z ℓ , so that the following equation is satisfied for r > ρ + | t | (3.11) ∂ t h − ∆ h = ( Z ℓ + h ) m +1 − Z m +1 ℓ . We let ( h , h )( r ) = ~h (0 , r ), and h L be the solution of the linear wave equation on {| x | > ρ + | t |} with initial data ( h , h ) at t = 0.Let R > ρ such that(3.12) sZ + ∞ R (( ∂ r h ) + h ) r dr + (cid:13)(cid:13) Z ℓ { r ≥ R + | t |} (cid:13)(cid:13) L m +1 t L m +1) x ≤ ε, where the small constant ε > ε > R . By the equation (3.11), finite speed of propagationand Strichartz/energy estimates, for all interval I containing 0,(3.13) sup t ∈ I (cid:13)(cid:13) ( ∇ t,x h ( t ) − ∇ t,x h L ( t )) 11 {| x | >R + | t |} (cid:13)(cid:13) L x + (cid:13)(cid:13) ( h − h L )11 {| x>R + | t |} (cid:13)(cid:13) L m +1 t ( I,L m +1) ) . (cid:13)(cid:13) {| x>R + | t |} | Z ℓ | m | h | (cid:13)(cid:13) L t ( I,L x ) + (cid:13)(cid:13) {| x>R + | t |} | h | m +1 (cid:13)(cid:13) L t ( I,L x ) , AND JIANWEI YANG and thus, by H¨older’s inequality, and the bound of the norm of Z ℓ in (3.12), wededuce(3.14) sup t ∈ I (cid:13)(cid:13) ( ∇ t,x h ( t ) − ∇ t,x h L ( t )) 11 {| x | >R + | t |} (cid:13)(cid:13) L x + (cid:13)(cid:13) ( h − h L )11 {| x>R + | t |} (cid:13)(cid:13) L m +1 t ( I,L m +1) ) . ε m (cid:13)(cid:13) {| x>R + | t |} h (cid:13)(cid:13) L m +1 t (cid:16) I,L m +1) x (cid:17) + (cid:13)(cid:13) {| x>R + | t |} h (cid:13)(cid:13) m +1 L m +1 t (cid:16) I,L m +1) x (cid:17) . Combining with the smallness assumption on h in (3.12), we deducesup t ∈ I (cid:13)(cid:13) ( ∇ t,x h ( t ) − ∇ t,x h L ( t )) 11 {| x | >R + | t |} (cid:13)(cid:13) L x . ε m k ( ∇ h , h )11 {| x | >R } k L x . By the same argument as in Step 1, we obtain(3.15) Z + ∞ R (cid:16) ( ∂ r ( rh )) + r h ( r ) (cid:17) dr . ε m Rh ( R ) . Arguing as in Step 2, we deduce that for
R < R ′ < R , if (3.12) holds, one has | g ( R ) − g ( R ′ ) | . ε m | g ( R ) | , where g ( R ) = Rh ( R ). By a straightforward induction argument, we deduce(3.16) | g ( R ) | . − Cε m ) k (cid:12)(cid:12) g (cid:0) k R (cid:1)(cid:12)(cid:12) . However, by Step 2 and (3.10), there exists a constant K such that | g (2 k R ) | ≤ K (2 k R ) . Taking ε small, so that 1 − Cε m > , we deduce from (3.16) that Rh ( R ) = g ( R ) = 0, if (3.12) is satisfied, that is for large R . Going back to (3.15) weobtain that h ( R ) = 0 for almost all large R . This concludes this step noting that( h , h ) = ( u , u ) − ( Z ℓ , Step 4. End of the proof.
We next prove that ( u , u ) = ( Z ℓ ,
0) for almost every r > ρ . We let ρ = inf (cid:26) ρ > ρ (cid:12)(cid:12)(cid:12) Z + ∞ ρ (cid:0) ( ∂ r h ) + h (cid:1) r dr = 0 (cid:27) . We must prove that ρ = ρ . We argue by contradiction, assuming that ρ > ρ .We thus can choose R such that ρ < R < ρ and(3.17) Z + ∞ R (cid:0) ( ∂ r h ) + h (cid:1) r dr + (cid:13)(cid:13) Z ℓ { R + | t |≤ r ≤ ρ + | t |} (cid:13)(cid:13) L m +1 t L m +1) x ≤ ε. By finite speed of propagation and the definition of ρ , r ≤ ρ + | t | on the supportof h . As a consequence, we see that the argument of Step 3 is still valid, replacing11 { r ≥ R + | t |} by 11 { R + | t |≤ r ≤ ρ + | t |} in (3.13). In particular, h ( R ) = 0, and (3.15) holdsfor this choice of R . This implies Z + ∞ R (cid:0) ( ∂ r h ) + h (cid:1) r dr = 0 , contradicting the definition of ρ . (cid:3) Boundedness along a sequence of times.Lemma 3.3.
Let u be a solution of (1.2) such that T + ( u ) = + ∞ . Then lim inf t → + ∞ Z Ω |∇ u | + ( ∂ t u ) dx ≤ m + 1)2 m E ( u , u ) . In particular, E ( u , u ) > or ( u , u ) = 0 .Proof. The proof is very close to the one of the analogous result in the energy-critical case without obstacle (see [10, Prop 3.4]). It uses a monotonicity formulathat goes back to the work of Levine [27]. We argue by contradiction, assumingthat E ( u , u ) <
0, or that there exists t > t ≥ t ,(3.18) (1 − ε ) (cid:16) k∇ u ( t ) k L (Ω) + k ∂ t u ( t ) k L (Ω) (cid:17) ≥ (cid:18) m + 1)2 m (cid:19) E ( u , u ) + ε . We let ϕ ∈ C ∞ ( R ) be a radial function such that ϕ ( r ) = 1 if r ≤ ϕ ( r ) = 0if r ≥
3. We let y ( t ) = Z Ω ϕ (cid:16) xt (cid:17) u ( t, x ) dx. We will prove that there exists γ > t ≥ t such that ∀ t ≥ t , γy ′ ( t ) ≤ y ( t ) y ′′ ( t )(3.19) ∀ t ≥ t , y ′ ( t ) > , (3.20)yielding a contradiction by a standard ODE argument (see e.g. the end of the proofof Theorem 3.7 in [23] for the details).Using the small data theory and finite speed of propagation, we obtain thatlim t →∞ Z {| x | > | t |} |∇ t,x u | + 1 | x | | u | + | u | m +2 dx = 0 . As a consequence, using also equation (1.2) and integration by parts, we obtain, as t → ∞ : y ′ ( t ) = 2 Z { ≤| x |≤ t } u∂ t u dx + o ( t )(3.21) y ′′ ( t ) = 2 Z Ω ( ∂ t u ) − Z Ω |∇ u | + 2 Z Ω | u | m +2 + o (1) . (3.22)We can rewrite (3.22):(3.23) y ′′ ( t ) = 2 m Z Ω |∇ u | + (2 m + 4) Z Ω ( ∂ t u ) − m + 1) E ( u , u ) + o (1) . Using that E ( u , u ) < ε > t , y ′′ ( t ) ≥ ε . This yieldslim inf t →∞ t y ′ ( t ) ≥ ε . In particular (3.20) holds. More precisely, for large t , R { ≤| x |≤ t } u∂ t u ≥ ε t , and(3.21) implies y ′ ( t ) ≤ (2 + o (1)) Z { ≤| x |≤ t } u∂ t u. AND JIANWEI YANG By (3.23) and the fact that E ( u , u ) is negative or that (3.18) holds for large t , weobtain that for large t , y ′′ ( t ) ≥ Z ( ∂ t u ( t, x )) dx. Using Cauchy-Schwarz inequality, (3.22) and the definition of y ( t ), we deduce(3.19), which concludes the proof. (cid:3) Existence of a radiation term.
We next prove:
Proposition 3.4.
Let u be a radial solution of (1.2) such that T + ( u ) = + ∞ . Thenthere exists a solution v L of the linear wave equation (1.4) such that (3.24) ∀ A ∈ R , lim t → + ∞ Z | x |≥ A + | t | |∇ t,x ( u − v L ) | dx = 0 . (see [10, Lemma 3.7] for the analog for radial solutions of the energy criticalequation on R ). Proof. Step 1.
We prove:(3.25) ∀ A ∈ R , (cid:13)(cid:13) {| x | >A + | t |} u (cid:13)(cid:13) L m +1 ( [0 , + ∞ ) ,L m +1) (Ω) ) < ∞ . Let ( t n ) n be a sequence given by Lemma 3.3 such that(3.26) lim n →∞ t n = + ∞ , lim sup n →∞ k ~u ( t n ) k ˙ H < ∞ . By the small data theory outside wave cones and finite speed of propagation, it issufficient to prove that for large n ,(3.27) (cid:13)(cid:13) {| x |≥ A + | t |} S L ( t − t n ) ~u ( t n ) (cid:13)(cid:13) L m +1 ([ t n , + ∞ ) ,L m +1) ) ≤ ε , where ε > (cid:16) U jL , ( t j,n ) n (cid:17) be a profile decomposition for the sequence ~u ( t n ). Without loss of generality, wecan assume(3.28) ∀ n, t ,n = 0 and ∀ j ≥ , lim n →∞ t j,n ∈ {±∞} . Let B ≥ (cid:13)(cid:13) {| x |≥ B + | t |} U L (cid:13)(cid:13) L m +1 ( [0 , ∞ ) ,L m +1) (Ω) ) ≤ ε / . By dominated convergence, using (3.28), we have for j ≥ (cid:13)(cid:13)(cid:13) {| x |≥ B + | t |} U jL ( · − t j,n ) (cid:13)(cid:13)(cid:13) L m +1 ( [0 , ∞ ) ,L m +1) (Ω) )= (cid:13)(cid:13)(cid:13) {| x |≥ B + | t + t j,n |} U jL (cid:13)(cid:13)(cid:13) L m +1 ( [ − t j,n , ∞ ) ,L m +1) (Ω) ) −→ n →∞ . This implies that for large n (cid:13)(cid:13) S L ( t ) ~u ( t n )11 {| x |≥ B + | t |} (cid:13)(cid:13) L m +1 ([0 , ∞ ) ,L m +1) (Ω) ≤ ε / , which yields (3.27) by the small data theory. Step 2.
We prove that for all A ∈ R , there exists a solution v AL of the linearwave equation (1.4) such that(3.29) lim t → + ∞ Z | x |≥ A + | t | (cid:12)(cid:12) ∇ t,x ( u − v AL ) (cid:12)(cid:12) dx = 0 . Indeed, this follows immediately from Step 1, noticing that u coincide, for | x | ≥ A + t ( t ≥ u A of(3.30) ( ∂ t u A − ∆ u A = ( u A ) m +1 {| x |≥ A + | t |} , ( t, x ) ∈ [0 , ∞ ) × Ω ~u A ↾ t =0 = ( u , u ) , u ↾ ∂ Ω = 0 . Since by Step 1 the right-hand side of the equation is in L (cid:0) (0 , ∞ ) , L (Ω) (cid:1) , weobtain the existence of v AL satisfying (3.29). Step 3.
In this step we conclude the proof, proving that v AL can be taken inde-pendent of A . We let G A be the unique element of L ( R ) such thatlim t → + ∞ Z + ∞ (cid:12)(cid:12) r∂ r v AL − G A ( r − t ) (cid:12)(cid:12) dr = 0lim t → + ∞ Z + ∞ (cid:12)(cid:12) r∂ t v AL + G A ( r − t ) (cid:12)(cid:12) dr = 0(see Lemma 2.3). By Lemma 3.3, there exists a constant C m such that(3.31) k G A k L ≤ C m E ( u , u ) . By the construction of v AL in Step 2, we havelim t →∞ Z | x |≥ B + | t | (cid:12)(cid:12) ∇ t,x ( v AL − v BL ) (cid:12)(cid:12) dx = 0if A ≤ B . This proves that G A ( η ) = G B ( η ) if η ≥ B = max( A, B ). We define G by G ( η ) = G η − ( η ) , so that if η ≥ A , G ( η ) = G A ( η ). We note in particular that by (3.31), G ∈ L ( R ).Let v L be the solution of (1.4), given by Lemma 2.3, such thatlim t → + ∞ Z + ∞ | r∂ r v L − G ( r − t ) | dr = 0lim t → + ∞ Z + ∞ | r∂ t v L + G ( r − t ) | dr = 0 . Using (3.29) and the definition of G and v L , we obtain that v L satisfies the desiredestimate (3.24). (cid:3) Proof of the soliton resolution.
In this subsection we conclude the proofof Theorem 1.2. We consider a solution u of (1.2). We assume that u is well definedfor t ≥
0, and we let v L be its dispersive component, given by Proposition 3.4. Step 1.
We prove that for all sequence t n → + ∞ such that ~u ( t n ) is bounded in H (Ω), there exists an subsequence of ( t n ) n (still denoted by ( t n ) n ), and a stationarysolution Q such that(3.32) lim n →∞ k ~u ( t n ) − ~v L ( t n ) − ( Q, k H (Ω) = 0 . Let t n be such a sequence. According to Proposition 2.11, we can assume (extractingsubsequences if necessary), that the sequence S L ( t )( ~u ( t n ) − ~v L ( t n )) has a profiledecomposition n U jL , ( t j,n ) n o j ≥ . We assume as usual ∀ j ≥ , lim n →∞ t j,n ∈ {±∞} and ∀ n, t ,n = 0 . AND JIANWEI YANG We note that the solution sequence ~S L ( − t n )( ~u ( t n )) converges weakly to ~v L (0).Denoting by U L = v L , t n = − t n , we see that n U jL , ( t j,n ) n o j ≥ is a profile decom-position for S L ( t )( ~u ( t n )). In particular, ∀ j ≥ , lim n →∞ | t n − t jn | = + ∞ . We prove by contradiction(3.33) ∀ j ≥ , U j ⇒ lim n →∞ t n − t jn = + ∞ . Assume on the contrary that there exists j ≥ U j n →∞ t jn − t n = + ∞ . Recall(3.35) ~S L (cid:0) t jn (cid:1) ~u ( t n ) −−−− ⇀ n →∞ ~U jL (0) , weakly in H . Let a n ( t ) = S L ( t ) ~u ( t n ). By the strong Huygens principle (see the first line of (2.3))(3.36) Z ≤| x |≤ M |∇ t,x a n ( t jn , x ) | dx ≤ Z t jn − M ≤| x |≤ t jn + M |∇ t,x u ( t n , x ) | dx. Since by (3.33), lim n →∞ Z | x |≥ t jn − M − t n |∇ t,x u (0 , x ) | dx = 0 , we obtain (by finite speed of propagation again)lim n →∞ Z | x |≥ t jn − M |∇ t,x u ( t n , x ) | dx = 0 , and thus (3.36) implieslim n →∞ Z ≤| x |≤ M |∇ t,x a n ( t jn , x ) | dx = 0 . By (3.35), U L is identically 0 , contradicting (3.34).As usual, we denote by U the solution of (1.2) with initial data U (0). Wenext prove that U is a stationary solution. If not, by Proposition 3.1, there exists R ≥ U is well-defined for {| x | > R + | t |} , and(3.37) X ±∞ lim t →±∞ Z {| x | >R + | t |} (cid:12)(cid:12) ∇ t,x U ( t, x ) (cid:12)(cid:12) dx > . We let w Jn ( t ) = u L ( t + t n ) − v L ( t + t n ) − J X j =1 U jL ( t − t j,n ) , and(3.38) ǫ Jn ( t ) = u ( t + t n ) − v L ( t + t n ) − U ( t ) − J X j =2 U jL ( t − t j,n ) − w Jn ( t ) . By Proposition 2.14, u ( t n + t ) is well defined for {| x | > R + | t |} , andlim J →∞ lim sup n →∞ (cid:18) sup t ∈ R (cid:13)(cid:13) {| x | >R + | t |} ∇ t,x ǫ Jn ( t ) (cid:13)(cid:13) L (cid:19) = 0 . We first consider the case where(3.39) lim t → + ∞ Z {| x | >R + | t |} (cid:12)(cid:12) ∇ t,x U ( t, x ) (cid:12)(cid:12) dx = η + > . By (3.38), for all t ≥ Z | x | >R + | t | ( ∇ t,x ( u − v L )( t + t n )) · ∇ t,x U ( t ) dx = Z | x | >R + | t | |∇ t,x U ( t ) | dx + J X j =2 Z | x | >R + | t | ∇ t,x U jL ( t − t j,n ) · ∇ t,x U ( t ) dx + Z | x | >R + | t | ∇ t,x w Jn ( t ) · ∇ t,x U ( t ) dx + o n (1) , where o n (1) goes to 0 as n goes to infinity, uniformly with respect to t ≥
0. Usingthat lim n →∞ | t jn − t kn | = + ∞ for j = k and the property (2.19) of w Jn , it is easy toprove that lines 2 and 3 of (3.40) go to 0 as n → ∞ (see e.g. Claim 3.2 in [10]),and thus, by (3.39), for large n ,lim t →∞ Z | x | >R + | t | |∇ t,x ( u − v L )( t + t n , x ) | dx ≥ η + / . In other words, for large n ,lim t →∞ Z | x | >R + t − t n |∇ t,x ( u − v L )( t, x ) | dx ≥ η + / , which contradicts the definition of v L given by Proposition 3.4.We next assume(3.41) lim t →−∞ Z {| x | >R + | t |} (cid:12)(cid:12) ∇ t,x U ( t, x ) (cid:12)(cid:12) dx = η − > . Arguing as before, we obtain that for large n , using the analog of (3.40) with t = − t n Z | x | >R + t n |∇ t,x u (0 , x ) | dx ≥ η − / . Since t n is arbitrarily large, we obtain a contradiction, proving that U is a station-ary solution Q . Note that the case Q ≡ {| x | > | t |} Q ∈ L m +1 (cid:16) R , L m +1) (cid:17) , so that the assumptions of Proposition 2.14 (and its analog in the past) are satisfiedwith R = 1. As a consequence, letting ǫ Jn ( t, x ) = u ( t + t n , x ) − v L ( t + t n , x ) − Q ( x ) − J X j =2 U jL ( t − t j,n , x ) − w Jn ( t, x ) , we have(3.42) lim J →∞ lim sup n →∞ sup t ∈ R Z | x | > | t | +1 (cid:12)(cid:12) ∇ t,x ǫ Jn ( t, x ) (cid:12)(cid:12) dx = 0 . AND JIANWEI YANG We next prove by contradiction that U jL ≡ j ≥
2. Assume that there exists j ≥ U jL is not zero. Then by Lemma 2.3, we have, for large A ,(3.43) lim t →±∞ Z | t |− A< | x | < | t | + A (cid:12)(cid:12)(cid:12) ∇ U jL ( t, x ) (cid:12)(cid:12)(cid:12) dx = η ± > . First assume lim n →∞ t j,n = −∞ . Combining (3.42), (3.43) and the pseudo-orthogonality of the time sequences ( t j,n ) n ,we can obtain that for a large fixed n ,lim t → + ∞ Z t − t n − t j,n − A ≤| x |≤ t − t n − t j,n + A |∇ t,x ( u − v L ) | dx ≥ η + / . This contradicts the definition of v L in Proposition 3.4 Next assumelim n →∞ t j,n = + ∞ . Using that by (3.42),lim J → + ∞ lim sup n →∞ sup t ∈ R Z | x | > | t n | +1 (cid:12)(cid:12) ∇ t,x ǫ Jn ( t − t n , x ) (cid:12)(cid:12) dx = 0 , we obtain that for all large n ,(3.44) Z t n + t j,n − A< | x | 2, we see that w Jn and ǫ Jn do not depend on J ≥ 2. We willdenote w n = w Jn and ε n = ε Jn . We are left with provinglim n →∞ k ~w n (0) k H (Ω) = 0 . Since by Lemma 2.1,(3.45) X ± lim t →±∞ Z | x |≥| t | +1 |∇ t,x w n | dx ≥ k ~w n (0) k H (Ω) , we can deduce, with the same arguments as before,lim t → + ∞ Z | x |≥ t + t n |∇ t,x ( u − v L ) | dx ≥ k ~w n (0) k H , if (3.45) holds for large n with a sign +, and Z | x | >t n |∇ t,x u (0) | dx ≥ k ~w n (0) k H , if (3.45) holds for large n with a sign − . This yields, in both cases, a contradiction,concluding this step. Step 2. Conclusion of the proof. Let t n → + ∞ be as in the preceding step. In viewof (3.32), we must prove(3.46) lim t →∞ k ~u ( t ) − ~v L ( t ) − ( Q, k H (Ω) = 0 . We assume that (3.46) does not hold, and fix a small ε > 0, such thatlim sup t →∞ k ~u ( t ) − ~v L ( t ) − ( Q, k H (Ω) > ε. Let t ′ n = min n t > t n s.t. k ~u ( t ) − ~v L ( t ) − ( Q, k H (Ω) > ε o , so that t n < t ′ n and(3.47) k ~u ( t ′ n ) − ~v L ( t ′ n ) − ( Q, k H (Ω) = ε. By Step 1, there exists a stationary solution Q ′ such that(3.48) lim n →∞ k ~u ( t ′ n ) − ~v L ( t ′ n ) − ( Q ′ , k H (Ω) = 0 . By the triangle inequality, (3.47) and (3.48),(3.49) k Q − Q ′ k H (Ω) ≤ ε. By (3.32), and the conservation of the linear and the nonlinear energy: E ( Q, 0) + 12 k ~v (0) k H (Ω) = E ( u , u ) . Similarly, by (3.48), E ( Q ′ , 0) + 12 k ~v (0) k H (Ω) = E ( u , u ) . This proves that E ( Q, 0) = E ( Q ′ , . By the classification of the radial stationary solutions in Subsection 2.5, we obtainthat Q = Q ′ , or Q = 0 and Q = − Q ′ . The first case contradicts (3.47) or (3.48). Inthe second case k Q − Q ′ k H = 2 k Q k H ≥ k Q k H , where Q is the ground state (seeSubsection 2.5). This contradicts (3.49) if ε is small enough. The proof is complete. Remark 3.5. Proposition 3.4 (exitence of a radiation term v L ) is still valid withthe same proof, for the defocusing analog of (1.2) . If u is a solution of the defocus-ing analog of (1.2) , then Remark 3.2, and Step 1 of the preceding proof yield theexistence of a sequence t n → + ∞ such that lim n →∞ k ~u ( t n ) − ~v L ( t n ) k H (Ω) = 0 . This implies, by the small data well-posedness theory that k u k L m +1 ( L m +1) (Ω) ) < ∞ for large n , and thus that u scatters. Further elements on the dynamics Dynamics below the energy threshold. In this section we prove Corollary1.5.Let ( u , u ) ∈ H with E ( u , u ) ≤ E ( Q , T − , T + ) its maximalinterval of existence.We start by variational considerations. Using the Sobolev inequality of Proposi-tion 2.21, the fact that R Ω |∇ Q | = R Ω Q m +20 , and the conservation of the energywe obtain(4.1) E ( Q , ≥ E ( u , u ) ≥ f (cid:0) |∇ u ( t ) | (cid:1) + 12 k ∂ t u ( t ) k L , AND JIANWEI YANG where f ( σ ) = σ − m +2 σ m +1 ( R Ω |∇ Q | ) m . The function f is increasing on (cid:0) , R Ω |∇ Q | (cid:1) ,decreasing on (cid:0)R Ω |∇ Q | , + ∞ (cid:1) and satisfies f (cid:0)R Ω |∇ Q | (cid:1) = E ( Q , E ( Q , 0) is the maximum of f and it is attained at σ = R |∇ Q | . We deducefrom (4.1) that for all t ∈ ( T − , T + ) Z Ω |∇ u ( t ) | = Z Ω |∇ Q | = ⇒ Z Ω ( ∂ t u ( t )) = 0 and E ( ~u ( t )) = E ( Q , . Thus if R Ω |∇ u ( t ) | = R Ω |∇ Q | , for one t ∈ ( T − , T + ), we must have R Ω | u ( t ) | m +2 = R Ω | Q | m +2 , and the uniqueness in Proposition 2.21 shows that ~u ( t ) = ± ( Q , u is a stationary solution. By the intermediate value theorem, Z Ω |∇ u | < Z Ω |∇ Q | = ⇒ ∀ t ∈ ( T − , T + ) , Z Ω |∇ u ( t ) | < Z Ω |∇ Q | (4.2) Z Ω |∇ u | > Z Ω |∇ Q | = ⇒ ∀ t ∈ ( T − , T + ) , Z Ω |∇ u ( t ) | > Z Ω |∇ Q | . (4.3) Case 1: global existence. Assume that we are in the case where the left-hand sideof (4.2) is satisfied. We see that u is bounded in ˙ H (Ω), and thus, by conservationof the energy, that ~u is bounded in H . Thus u is global.Furthermore, Theorem 1.2 and the condition E ( u , u ) ≤ E ( Q , 0) implies thatif u does not scatter forward (respectively backward) in time to a linear solution,then(4.4) lim t → + ∞ k ~u ( t ) − ( Q , k H = 0(respectively lim t →−∞ . . . ). However we see by Proposition 3.1 that both propertiescannot occur simultaneously, i.e. that u must scatter in at least one time direction. Case 2: finite time blow-up. Next, we assume that we are in the case where theleft-hand side of (4.3) is satisfied. Note that if u is global and scatters to a linearsolution, say forward in time, then we must havelim t → + ∞ Z Ω |∇ x,t u ( t ) | = E ( u , u ) ≤ E ( Q , < Z |∇ Q | . Thus (4.3) implies that u cannot scatter to a linear solution in any time direction.As a consequence, if T + = + ∞ , then by Theorem 1.2 (4.4) must be satisfied andsimilarly for negative times. Again, Proposition 3.1 implies that both propertiescannot occur simultaneously, which concludes the proof. (cid:3) One-pass theorem. In this subsection we prove Theorem 1.4. Denote byΣ = { } ∪ S k { Q k } ∪ S k {− Q k } the set of stationary solutions. We argue bycontradiction, assuming there there exist ε > 0, and, for all n ≥ s n < t ′ n < t n , asolution u n of (1.2) defined on [ s n , t n ] and such thatlim n →∞ (cid:18) k ~u n ( s n ) − ( Q k , k H + min Q ∈ Σ k ~u n ( t n ) − ( Q, k H (cid:19) = 0(4.5) ∀ n, k ~u n ( t ′ n ) − ( Q k , k H ≥ ε. (4.6)By the intermediate value theorem, we can replace the inequality in (4.6) by anequality. Translating in time, we can assume t ′ n = 0. Furthermore, by energy conservation, we can replace the minimum in (4.5) by k ~u n ( t n ) − ι ( Q k , k H forsome sign ι ∈ {± } . Thus we can replace (4.5) and (4.6) bylim n →∞ (cid:16) k ~u n ( s n ) − ( Q k , k H + k ~u n ( t n ) − ι ( Q k , k H (cid:17) = 0(4.7) ∀ n, k ~u n (0) − ( Q k , k H = ε, (4.8)where s n < < t n . Extracting subsequences if necessary, we consider a profiledecomposition n U jL , ( t j,n ) n o j ≥ of ~u n (0). As in Subsection 2.3, we assume ∀ n, t ,n = 0 , j ≥ ⇒ lim n →∞ t j,n ∈ {±∞} . By (4.8) and the Pythagorean expansion of the H norm, we have(4.9) (cid:13)(cid:13)(cid:13) ~U L (0) − ( Q k , (cid:13)(cid:13)(cid:13) H ≤ ε. We distinguish two cases.If ~U L (0) = ( Q k , n →∞ E ( ~u n (0)) > E ( Q k , , a contradiction with (4.7).If ~U L (0) = ( Q k , ε is small, we see that ~U L (0) is not a stationary solution.By (4.9), we also now (using again that ε is small) that the solution U of (1.2) withinitial data ~U L (0) is well-defined on { r > | t | + 1 } . As a consequence, by Proposition3.1, U satisfies: X ± lim t →±∞ Z | x | > | t | +1 |∇ U ( t, x ) | dx > t ≥ Z | x |≥| t | +1 |∇ U ( t, x ) | dx + inf t ≤ Z | x |≥| t | +1 |∇ U ( t, x ) | dx > . Thus there is a small η > n :(4.11) Z | x | > | σ n | +1 |∇ u n ( σ n , x ) | dx > η, where σ n = s n if the infimum for t ≤ σ n = t n if theinfimum for t ≥ n : Z | x | > | σ n | +1 |∇ u n ( σ n , x ) | dx > η . Combining with (4.7) we deduce that for large n Z | x | >R + | σ n | |∇ Q k ( x ) | dx > η . This is a contradiction since by (4.7) and (4.8) and the continuity of the flow forequation (1.2), we must have lim n →∞ | σ n | = + ∞ . AND JIANWEI YANG References [1] Abou Shakra, F. Asymptotics of the critical nonlinear wave equation for a class of non-star-shaped obstacles. J. Hyperbolic Differ. Equ. 10 , 03 (2013), 495–522.[2] Bizo´n, P., and Maliborski, M. Dynamics at the threshold for blowup for supercritical waveequations outside a ball. ArXiv preprint:1909.01626, 2019.[3] Blair, M. D., Smith, H. F., and Sogge, C. D. Strichartz estimates for the wave equationon manifolds with boundary. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 26 , 5 (2009), 1817–1829.[4] Burq, N. D´ecroissance de l’´energie locale de l’´equation des ondes pour le probl`eme ext´erieuret absence de r´esonance au voisinage du r´eel. Acta Math. 180 , 1 (1998), 1–29.[5] Burq, N. Global Strichartz estimates for nontrapping geometries: about an article by H.Smith and C. Sogge. Comm. Partial Differential Equations 28 , 9-10 (2003), 1675–1683.[6] D’Ancona, P. On the supercritical defocusing NLW outside a ball. In preparation, 2019.[7] Duyckaerts, T., and David, L. Scattering for critical radial Neumann waves outside a ball.2019.[8] Duyckaerts, T., Kenig, C., and Merle, F. Universality of blow-up profile for small radialtype II blow-up solutions of the energy-critical wave equation. J. Eur. Math. Soc. (JEMS)13 , 3 (2011), 533–599.[9] Duyckaerts, T., Kenig, C., and Merle, F. Scattering for radial, bounded solutions offocusing supercritical wave equations. IMRN (2012).[10] Duyckaerts, T., Kenig, C., and Merle, F. Classification of radial solutions of the focusing,energy-critical wave equation. Cambridge Journal of Mathematics 1 , 1 (2013), 75–144.[11] Duyckaerts, T., Kenig, C., and Merle, F. Scattering for radial, bounded solutions offocusing supercritical wave equations. Int. Math. Res. Not. IMRN 2014 , 1 (2014), 224–258.[12] Duyckaerts, T., and Merle, F. Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18 , 6 (2009), 1787–1840.[13] Duyckaerts, T., and Merle, F. Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation. Indiana Univ. Math. J. 58 , 4 (2009), 1971–2001.[14] Duyckaerts, T., and Roy, T. Blow-up of the critical Sobolev norm for nonscattering radialsolutions of supercritical wave equations on R . Bull. Soc. Math. France 145 , 3 (2017), 503–573.[15] Duyckaerts, T., and Yang, J. Blow-up of a critical Sobolev norm for energy-subcriticaland energy-supercritical wave equations. Anal. PDE 11 , 4 (2018), 983–1028.[16] Fang, D., Xie, J., and Cazenave, T. Scattering for the focusing energy-subcritical nonlinearSchr¨odinger equation. Sci. China Math. 54 , 10 (2011), 2037–2062.[17] Fowler, R. H. Further studies of Emden’s and similar differential equations. Quart. J. Math.os-2 , 1 (1931), 259–288.[18] Jia, H., Liu, B., Schlag, W., and Xu, G. Generic and non-generic behavior of solutionsto defocusing energy critical wave equation with potential in the radial case. Int. Math. Res.Not. IMRN , 19 (2017), 5977–6035.[19] Jia, H., Liu, B., Schlag, W., and Xu, G. Global center stable manifold for the defocusingenergy critical wave equation with potential. ArXiv preprint:1706.09284, 2017.[20] Jia, H., Liu, B., and Xu, G. Long time dynamics of defocusing energy critical 3+1 dimen-sional wave equation with potential in the radial case. Comm. Math. Phys. 339 , 2 (Oct 2015),353–384.[21] Kenig, C. E., Lawrie, A., and Schlag, W. Relaxation of wave maps exterior to a ball toharmonic maps for all data. Geometric and Functional Analysis 24 , 2 (2014), 610–647.[22] Kenig, C. E., and Merle, F. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schr¨odinger equation in the radial case. Invent. Math. 166 , 3(2006), 645–675.[23] Kenig, C. E., and Merle, F. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201 , 2 (2008), 147–212.[24] Kiguradze, I., and Chanturia, T. Asymptotic properties of solutions of nonautonomousordinary differential equations , vol. 89 of Mathematics and its Applications . Springer, Berlin,1993.[25] Krieger, J., and Schlag, W. Large global solutions for energy supercritical nonlinear waveequations on 3+ 1. J. Anal. Math. 133 , 1 (2017), 91–131. [26] Lafontaine, D. Strichartz estimates without loss outside many strictly convex obstacles.ArXiv preprint:1811.12357, 2018.[27] Levine, H. A. Instability and nonexistence of global solutions to nonlinear wave equationsof the form P u tt = − Au + F ( u ). Trans. Amer. Math. Soc. 192 (1974), 1–21.[28] Metcalfe, J. Global strichartz estimates for solutions to the wave equation exterior to aconvex obstacle. Transactions of the American Mathematical Society 356 , 12 (2004), 4839–4855.[29] Morawetz, C. S., Ralston, J. V., and Strauss, W. A. Decay of solutions of the waveequation outside nontrapping obstacles. Comm. Pure Appl. Math. 30 , 4 (1977), 447–508.[30] Nakanishi, K., and Schlag, W. Invariant manifolds and dispersive Hamiltonian evolutionequations . European Mathematical Society, 2011.[31] Shen, R. On the energy subcritical, nonlinear wave equation in R with radial data. Anal.PDE 6 , 8 (2014), 1929–1987.[32] Smith, H. F., and Sogge, C. D. On the critical semilinear wave equation outside convexobstacles. J. Amer. Math. Soc. 8 , 4 (1995), 879–916.[33] Smith, H. F., and Sogge, C. D. Global Strichartz estimates for nontrapping perturba-tions of the Laplacian: Estimates for nontrapping perturbations. Comm. Partial DifferentialEquations 25 , 11-12 (2000), 2171–2183.[34] Weinstein, M. I. Nonlinear Schr¨odinger equations and sharp interpolation estimates. Comm.Math. Phys. 87 , 4 (1982/83), 567–576. Thomas Duyckaerts, LAGA, Institut Galil´ee, Universit´e Paris 13 99, avenue Jean-Baptiste Cl´ement,n 93430 - Villetaneuse, France E-mail address : [email protected] Jianwei Yang, Department of Mathematics, Beijing Institute of Technology, Beijing100081, P. R. China E-mail address ::