Nonexistence of multi-bubble radial solutions to the 3D energy critical wave equation

Nonexistence of m ulti-bubble radial solutions to the 3D energy critical w a v e equation Ruip eng Shen Cen tre for Applied Mathematics Tianjin Univ ersity Tianjin, China Marc h 24, 2026 Abstract In this work w e consider the fo cusing, energy-critical wa ve equation in 3D radial case. It has b een v eriﬁed that an y global or type I I blo w-up solution decomposes in to a sup erposition of several decoupled grounds states, a free w av e and a small error, as time tends to inﬁnit y or the blow-up time. This is usually called soliton resolution. How ev er, all known examples of soliton resolution in the 3D radial case come with no more than one soliton. In this w ork w e prov e the nonexistence of any global or type II blo w up solution with tw o or more solitons, thus give a complete classiﬁcation of asymptotic behaviours of radial solutions. 1 In tro duction 1.1 Bac kground In this w ork w e consider radial solutions to the follo wing focusing, energy-critical wa ve equation in 3-dimensional space  ∂ 2 t u − ∆ u = + | u | 4 u, ( x, t ) ∈ R 3 × R ; ( u, u t ) | t =0 = ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 . (CP1) The solutions satisfy the energy conserv ation la w: E = Z R 3  1 2 |∇ u ( x, t ) | 2 + 1 2 | u t ( x, t ) | 2 − 1 6 | u ( x, t ) | 6  d x = Const . The equation is inv arian t under the dilation transformation. More precisely , if u is a solution to (CP1), then u λ = 1 λ 1 / 2 u  x λ , t λ  , λ ∈ R + is also a solution to (CP1) with exactly the same energy . The lo cal well-posedness of (CP1) follows from a combination of the standard ﬁxed-p oint argumen t and suitable Strichartz estimates. This argument do es not dep end on the sign of the nonlinearit y , thus applies to the defo cusing wa ve equation ∂ 2 t u − ∆ u = −| u | 4 u as well. More details can b e found in Kapitanski [26] and Lindblad-Sogge [32], for examples. The global b eha viour in the fo cusing case, how ever, is muc h more complicated than the defo cusing case. In short, all ﬁnite-energy solutions to the defocusing equation are globally deﬁned for all t and scatter in b oth time directions. Readers may refer to [16, 35, 36, 37, 38, 42], for instance. Although solutions to (CP1) with small initial data still scatter, large solutions ma y blow up in 1 ﬁnite time or exhibit more interesting global b ehaviour. Before w e start to giv e a brief review of the global/asymptotic b eha viour, we ﬁrst give a typical example of global non-scattering solution, i.e. the T alenti-Aubin solution, also called a ground state. W ( x ) =  1 3 + | x | 2  − 1 / 2 . Indeed, W ( x ) solves the elliptic equation − ∆ u = | u | 4 u . It is clear that all solutions to this elliptic equation are also solutions to (CP1) indep endent of time. They are called the stationary solutions to (CP1). Among all stationary solutions (not necessarily radially symmetric) to (CP1), W ( x ) comes with the smallest energy . This is why we call W a ground state. In the radial case, all non-trivial ﬁnite-energy radial stationary solutions can b e giv en by the dilations of W , up to a sign, i.e. {± W λ : λ ∈ R + } ; W λ . = 1 λ 1 / 2 W  x λ  . All these dilations come with the same energy . Thus they are all called ground states. Finite time blow-up A classical lo cal theory implies that if u blo ws up at time T + ∈ R + , then ∥ u ∥ L 5 L 10 ([0 ,T + ) × R 3 ) = + ∞ . Indeed there are tw o types of ﬁnite time blow-up solutions. Type I blo w-up solutions satisfy lim t → T + ∥ ( u, u t ) ∥ ˙ H 1 × L 2 = + ∞ . W e may construct suc h a solution in the following wa y: we start by an explicit solution of (CP1) u ( x, t ) =  3 4  1 / 4 ( T + − t ) − 1 / 2 , whic h blo ws up as t → T + . A com bination of cut-oﬀ tec hniques and the ﬁnite sp eed of propaga- tion then gives a ﬁnite-energy type I blow-up solution. Soliton resolution In the contrast, a t yp e I I blow-up solution satisﬁes lim sup t → T + ∥ ( u, u t ) ∥ ˙ H 1 × L 2 < + ∞ . The asymptotic b ehaviour of type I I blow-up solutions and global solutions can b e describ ed b y the follo wing soliton resolution conjecture: As time tends to the blow-up time or inﬁnit y , a solution asymptotically decomposes into a sum of decoupled solitary wa v es, a free wa ve and a small error term. Here solitary wa ves are Loren tz transformations of stationary solutions to (CP1). In the radial case, the solitary wa ves are simply ground states thus we may write the soliton resolution in the following form  u ( t ) = J X j =1 ζ j ( W λ j ( t ) , 0) +  u L ( t ) + o (1) , t → T + or + ∞ . Here  u = ( u, u t ); ζ j ∈ { +1 , − 1 } are signs; u L is a free w av e; o (1) is a error term, whose norm in the energy space ˙ H 1 × L 2 v anishes as t tends to T + or ∞ . The scale functions λ j ( t ) for type II blo w-up solutions satisfy lim t → T + λ j +1 ( t ) λ j ( t ) = 0 , j = 1 , 2 , · · · , J − 1; lim t → T + λ 1 ( t ) T + − t = 0 . 2 Similarly scale functions for global solutions satisfy lim t → + ∞ λ j +1 ( t ) λ j ( t ) = 0 , j = 1 , 2 , · · · , J − 1; lim t → + ∞ λ 1 ( t ) t = 0 . Although soliton resolution conjecture is still an op en problem in the non-radial case (see Duyc k aerts-Jia-Kenig [6] for a partial result), it has b een veriﬁed in the radial case in the past ﬁfteen y ears. Duyck aerts-Kenig-Merle [9] gav e the ﬁrst pro of of the 3-dimensional case via a com- bination of proﬁle decomp osition and channel of energy metho d. Then Duyc k aerts-Kenig-Merle [12], Duyck aerts-Kenig-Martel-Merle [7], Collot-Duyck aerts-Kenig-Merle [1] and Jendrej-Lawrie [22] veriﬁed the soliton resolution conjecture in higher dimensions for radial data. Recently the author [41] giv es another pro of of this conjecture in 3D radial case and discusses some further quan titative prop erties of the soliton resolution. In addition, soliton resolution for co-rotational w av e maps has also b een veriﬁed by Jendrej-La wrie [23]. F or the con venience of the readers, we cop y b elo w the soliton resolution theorem in the 3D radial case giv en b y Duyc k aerts-Kenig-Merle [9]. Theorem 1.1. L et u b e a r adial solution of (CP1) and T + = T + ( u ) b e the right endp oint of this maximal interval of existenc e. Then one the fol lowing holds: • T yp e I blow-up : T + < ∞ and lim t → T + ∥ ( u ( t ) , u t ( t )) ∥ ˙ H 1 × L 2 = + ∞ . • T yp e I I blo w-up : T + < ∞ and ther e exist ( v 0 , v 1 ) ∈ ˙ H 1 × L 2 , an inte ger J ≥ 1 , and for al l j ∈ { 1 , 2 , · · · , J } , a sign ζ j ∈ {± 1 } , and a p ositive function λ j ( t ) deﬁne d for t close to T + such that λ 1 ( t ) ≪ λ 2 ( t ) ≪ · · · ≪ λ J ( t ) ≪ T + − t as t → T + ; lim t → T +       ( u ( t ) , ∂ t u ( t )) −   v 0 + J X j =1 ζ j λ j ( t ) W  x λ j ( t )  , v 1         ˙ H 1 × L 2 = 0 . • Global solution : T + = + ∞ and ther e exist a solution v L of the line ar wave e quation, an inte ger J ≥ 0 , and for al l j ∈ { 1 , 2 , · · · , J } , a sign ζ j ∈ {± 1 } , and a p ositive function λ j ( t ) deﬁne d for lar ge t such that λ 1 ( t ) ≪ λ 2 ( t ) ≪ · · · ≪ λ J ( t ) ≪ t as t → + ∞ ; lim t → + ∞       ( u ( t ) , ∂ t u ( t )) −   v L ( t ) + J X j =1 ζ j λ j ( t ) W  x λ j ( t )  , ∂ t v L ( t )         ˙ H 1 × L 2 = 0 . Num b er of bubbles If the soliton resolution of a solution comes with J solitary wa ves, then w e call it a J -bubble solution. A scattering solution can b e view ed as a 0-bubble solution as time tends to inﬁnity . 1.2 Main topic and result Although the soliton resolution conjecture has been veriﬁed in the radial case, a natural question still remains to b e answ ered, i.e. can we ﬁnd an example of soliton resolution for each combination of bubble num b er and/or signs? T o b e more precise, given a p ositiv e integer J and a sequence of 3 signs ζ 1 , ζ 2 , · · · , ζ J , do es it exist a radial global solution (or type I I blow-up solution) to (CP1), suc h that the following soliton resolution holds as time tends to inﬁnit y (or blo w-up time)?  u ( t ) ≈ J X j =1 ζ j ( W λ j ( t ) , 0) +  u L ( t ) + o (1) . Let us make a brief review on relev ant results giv en in previous w orks. T yp e I I blow-up solutions The ﬁrst type II blo w-up solution was constructed by Krieger- Sc hlag-T ataru [29], then b y Krieger-Sc hlag [30] and Donninger-Huang-Krieger-Schlag [4]. All these examples come with a single soliton, but with diﬀerent c hoices of scale functions λ 1 ( t ). Please note that similar t yp e I I blo w-up solutions can also b e constructed in higher dimensions. Please see Hillairet-Rapha¨ el [17] and Jendrej [19], for examples. Global solutions The ground states are clearly non-scattering global solutions to (CP1). In addition, Donninger-Krieger [5] pro ved that one-bubble global solution exists with a scale function b eha ving like λ 1 ( t ) ≃ t µ for any suﬃciently small parameter | µ | ≪ 1. Main result In summary only one-bubble examples are previously known in the 3D radial case. In this w ork we prov e that this is the general rule, i.e. soliton resolution with tw o or more bubbles do es not exist at all in the 3D radial case. No w we in tro duce the main result of this w ork: Theorem 1.2. Ther e do es not exist any r adial glob al solution or typ e II blow-up solution to (CP1) with two or mor e bubbles in its soliton r esolution. In other wor ds, if u is a r adial solution deﬁne d for al l time t ≥ 0 , then exactly one of the fol lowing holds • Scattering: ther e exists a fr e e wave u L , such that lim t → + ∞ ∥  u ( t ) −  u L ( t ) ∥ ˙ H 1 × L 2 = 0 . • One-bubble global solution: ther e exists a ﬁnite-ener gy fr e e wave u L , a sign ζ ∈ { +1 , − 1 } and a sc ale function λ ( t ) > 0 such that lim t → + ∞      u ( t ) −  u L ( t ) − ζ λ ( t ) 1 / 2  W  x λ ( t )  , 0      ˙ H 1 × L 2 ( R 3 ) = 0; lim t → + ∞ λ ( t ) t = 0 . Similarly if a r adial solution u to (CP1) blows up at a ﬁnite time T + > 0 , then exactly one of the fol lowing holds • T yp e I blow-up: the solution u blows up in the manner of typ e I, i.e. lim t → T + ∥  u ( t ) ∥ ˙ H 1 × L 2 ( R 3 ) = + ∞ . • One-bubble t yp e I I blo w-up: ther e exists a ﬁnite-ener gy fr e e wave u L , a sign ζ ∈ { +1 , − 1 } and a sc ale function λ ( t ) > 0 such that lim t → T +      u ( t ) −  u L ( t ) − ζ λ ( t ) 1 / 2  W  x λ ( t )  , 0      ˙ H 1 × L 2 ( R 3 ) = 0; lim t → T + λ ( t ) T + − t = 0 . Please note that w e substitute ( v 0 , v 1 ) ∈ ˙ H 1 × L 2 (as given in Theorem 1.1) by a linear free w av e u L here in the type I I blow-up case, for the reason of consistence. It clearly do es not make an y diﬀerence since  u L ( t ) →  u L ( T + ) in the energy space ˙ H 1 × L 2 as t → T + . 4 Remark 1.3. Examples of al l four c ases in The or em 1.2 ar e pr eviously known to exist. As a r esult, The or em 1.2 ﬁnal ly gives a c omplete classiﬁc ation of the asymptotic b ehaviour of any r adial solution to (CP1). This is the ﬁrst c omplete classiﬁc ation r esult in the ar e a of soliton r esolution for the fo cusing, ener gy-critic al wave e quation, as far as the author knows. Remark 1.4. Multiple bubble solutions to (CP1) do exist in non-r adial c ase. Inde e d, if the r adial ly symmetric assumption is r emove d, then one may c onstruct a typ e II blow-up solution with any numb er of solitary waves by c ombining sever al typ e II blow-up solutions with diﬀer ent blow- up p oints in the sp ac e but the same blow-up time, thanks to the ﬁnite sp e e d of wave pr op agation. F urthermor e, typ e II blow-up solution with multiple bubbles shrinking to a single blow-up p oint but along diﬀer ent dir e ctions has also b e en c onstructe d r e c ently by Kadar [24]. In 5-dimensional c ase, non-r adial glob al solutions with two or mor e bubbles have also b e en c onstructe d by Martel- Merle [33, 34] and Y uan [43]. Remark 1.5. R adial multiple bubble solutions stil l exist if we c onsider the ener gy critic al wave e quation □ u = | u | 4 d − 2 in a high-dimensional sp ac e R d . F or example, tw o-bubble r adial solutions have b e en c onstructe d by Jendr ej [21] for d = 6 . However, the author c onje ctur es that given a dimension d ≥ 3 , ther e exists a p ositive inte ger N = N ( d ) , such that the soliton r esolution of r adial solutions to the ener gy-critic al wave e quation □ u = | u | 4 d − 2 c an never c ome with mor e than N solitons. Our main the or em veriﬁes that N (3) = 1 . Remark 1.6. If we c onsider a sp e cial c ase with zer o r adiation u L = 0 in the soliton r esolution, then it has b e en pr ove d by Jendr ej [20] that solutions with two bubbles of diﬀer ent signs do not exist in the r adial c ase for any dimension d ≥ 3 . Remark 1.7. R e c ently soliton r esolution with any numb er of bubbles was c onstructe d for the c o-r otational wave maps by Krie ger-Palacios [25] and Hwang-Kim [18]. Ple ase note that the bubbles c ome with alternative signs in the soliton r esolution given by b oth these two works. 1.3 General idea No w we describ e the general idea of this w ork. Generally sp eaking, we inv estigate the rela- tionship b et ween the bubbles and radiation of a solution. This idea dates back to Duyck aerts- Kenig-Merle’s proof of the s oliton resolution conjecture, and the channel of energy method they deplo yed. In fact, their pro of given in [9] utilized an imp ortan t fact that an y radial solution of (CP1) other than zero or ground states must come with nonzero radiation outside the main light cone | x | = | t | , i.e. X ± lim t →±∞ Z | x | > | t | |∇ t,x u ( x, t ) | 2 d x > 0 . Here ∇ t,x = ( ∂ t , ∇ x ). Radiation ﬁelds The theory of radiation ﬁelds ma y help us further inv estigate the asymptotic b eha viour of solutions to the wa ve equations as time tends to inﬁnity . The classic theory of radiation ﬁelds applies to the free wa v es. F or simplicit y w e fo cus on the 3D radially symmetric case. Given an y ﬁnite-energy radial free w av e u , there exist tw o functions G ± ∈ L 2 ( R ) such that lim t →±∞ Z ∞ 0 | r u t ( r , t ) − G ± ( r ∓ t ) | 2 d r = 0; lim t →±∞ Z ∞ 0 | r u r ( r , t ) ± G ± ( r ∓ t ) | 2 d r = 0 . This giv es go o d appro ximation of the gradient ( u t , ∇ u ) in the energy space. The author calls these functions G ± the radiation proﬁles. In many situations, the asymptotic b ehaviour of a 5 solution to the nonlinear wa ve equation is similar to that of a free wa ve, either in the whole space for a given time direction, or outside some light cone. As a result, similar limits to the ones giv en ab ov e hold for suitable radiation proﬁles G ± . In other words, we ma y also describ e the asymptotic b eha viour of a nonlinear solution b y sp ecifying its corresp onding radiation proﬁles. In particular, the radiation strength of a suitable solution u in the “energy channel” { x, t : | t | + r 1 < | x | < | t | + r 2 } can b e measured by the limits lim t →±∞ Z | t | + r 1 < | x | < | t | + r 2 |∇ t,x u ( x, t ) | 2 d x or equiv alen tly , the integrals Z r 2 r 1 | G ± ( s ) | 2 d s. Radiation concentration Let us assume that the following soliton resolution holds ( J ≥ 2)  u ( t ) ≈ J X j =1 ζ j  W λ j ( t ) , 0  +  u L ( t ) + o (1) . W e fo cus on the in teraction of the smallest t wo bubbles ζ J W λ J ( t ) and ζ J − 1 W λ J − 1 ( t ) , and show that a signiﬁcant radiation concentration has to happen for at least one of the radiation proﬁles G ± asso ciated to the solution u . More precisely , we hav e sup 0 κ ; as long as t is suﬃciently large (or suﬃciently close to T + ). Here κ > 0 is a constant determined solely b y the bubble n um b er J . This, together with the classic theory of maximal functions, giv es a con tradiction. Intuitiv ely strong concentration can not alwa ys happ en as w e make t → + ∞ (or t → T + in the type I I blow-up case). Bubble in teraction with no disp ersion No w let us brieﬂy explain wh y a strong radiation concen tration has to happen as describ ed abov e. Indeed, if u w ere a J -bubble solution with almost no radiation in the c hannel Ψ = { ( x, t ) : | t | < | x | < | t | + λ J − 1 ( t ) } , then we migh t linearize the wa v e equation (CP1) near the approximated solution (giv en by the soliton resolution) S ∗ = J X j =1 ζ j W λ j ( x ) + v L , where v L is the asymptotically equiv alen t free wa ve of u outside the main ligh t cone, as deﬁned in Subsection 2.3; and deduce that the error w ∗ = u − S ∗ satisﬁes the wa v e equation □ w ∗ = F ( u ) − J X j =1 ζ j F ( W λ j ) = 5 W 4 λ J  w ∗ + ζ J − 1 W λ J − 1  + low er order terms = 5 W 4 λ J  w ∗ + √ 3 ζ J − 1 λ − 1 / 2 J − 1  + low er order terms . Neglecting the low er order terms and solving the wa v e equation □ w ∗ = 5 W 4 λ J  w ∗ + √ 3 ζ J − 1 λ − 1 / 2 J − 1  , (1) w e may giv e a more precise approximation u ≈ J X j =1 ζ j W λ j ( x ) + v L + √ 3 ζ J − 1 λ − 1 / 2 J − 1 φ ( x/λ J ) . 6 Here φ ( x ) is a well-c hosen solution to the linear elliptic equation − ∆ φ = 5 W 4 φ + 5 W 4 , th us √ 3 ζ J − 1 λ − 1 / 2 J − 1 φ ( x/λ J ) is exactly a solution of (1). Please note that we make φ indep endent of the time b ecause φ is exp ected to send no radiation. This precise appro ximation ﬁnally giv es a contradiction if w e consider the b ehaviour of u near the origin b ecause we ma y prov e that φ ( x ) comes with a strong singularity near the origin. Ma jor tools This w ork utilize tw o ma jor to ols. The ﬁrst one is a family of estimates regarding the soliton resolution in term of the Strichartz norm of the asymptotically equiv alent free wa v e v L . Roughly sp eaking, the following estimates hold for suitable J -bubble solution u deﬁned in the exterior region Ω 0 = { ( x, t ) : | x | > | t |}        u (0) − J X j =1 ζ j  W λ j , 0  −  v L (0)       ˙ H 1 × L 2 ≲ J ∥ χ 0 v L ∥ L 5 L 10 ( R × R 3 ) ; λ j +1 λ j ≲ j ∥ χ 0 v L ∥ L 5 L 10 ( R × R 3 ) , j = 1 , 2 , · · · , J − 1 . Here χ 0 is the characteristic function of Ω 0 ; ζ j and λ j are signs/scales in the soliton resolution. More details can b e found in Section 3. Another important to ol is a family of reﬁned Strichartz estimates for free w av es whose ra- diation proﬁles/initial data are supp orted aw ay from the origin, which are given in Subsection 2.4. These reﬁned Stric hartz estimates imply that the inﬂuence of larger bubbles and the radia- tion part can b e neglected when w e consider the interaction of tw o smallest bubbles in suitable situations. 1.4 Structure of this w ork This work is organized as follows: In Section 2 w e ﬁrst introduce some notations, basic con- ceptions and preliminary results, including the exterior solutions, radiation ﬁelds and theory , asymptotically equiv alen t solutions, as w ell as several reﬁned Stric hartz estimates. W e then mak e a review of the soliton resolution estimates giv en b y [41] in Section 3. Next w e discuss the linear elliptic equation mentioned ab ov e in Section 4 and give a few approximations of J -bubble exterior solutions with fairly weak radiation concentration in Section 5, if this kind of solution existed. Finally in Section 6 we pro v e the radiation concentration property of J -bubble solutions and ﬁnish the pro of of the main theorem. 2 Preliminary results In this section we make a brief review of several previously known theories and results, whic h will b e used in subsequent sections. Let us start by a few notations. Implicit constants In this article we use the notation A ≲ B if there exists a constant c suc h that the inequality A ≤ cB holds. In addition, we may add subscript(s) to indicate that the implicit constant c mentioned ab o ve dep ends on the subscript(s) but nothing else. In particular, the notation ≲ 1 implies that the constan t c is actually an absolute constant. The notations ≳ and ≃ can b e understo o d in the same w ay . Similarly w e use the notation c ( · ), where the dot represents one or more parameter(s), to represent a pos itiv e constan t determined by the parameter(s) listed but nothing else. In particular, c (1) means an absolute constant. Please note that the same notation c ( · ) may represen t diﬀeren t constants at diﬀerent places, ev en if the parameters are exactly the sam e. 7 Bo x notation F or conv enience we use the notation □ = ∂ 2 t − ∆ when necessary in this work. Nonlinearit y and radial functions W e use the notation F ( u ) = | u | 4 u throughout this w ork, unless sp eciﬁed otherwise. If u is a radial solution, then we use the notation u ( r , t ) to represent the v alue u ( x, t ) with | x | = r . Space norms W e need to consider the restriction of radial ˙ H 1 functions outside a ball of radius R > 0 when we discuss the conception of exterior solutions. F or conv enience w e let H ( R ) b e the space of the restrictions of radial functions ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 to the region { x : | x | > R } , equipp ed with the norm ∥ ( u 0 , u 1 ) ∥ H ( R ) = Z | x | >R  |∇ u 0 ( x ) | 2 + | u 1 ( x ) | 2  d x ! 1 / 2 . In particular, H = H (0) is exactly the Hilb ert space of radial ˙ H 1 × L 2 functions. Channel-lik e regions Throughout this w ork w e use the follo wing notations for the channel- lik e regions Ω R =  ( x, t ) ∈ R 3 × R : | x | > | t | + R  , R ≥ 0; Ω R 1 ,r 2 =  ( x, t ) ∈ R 3 × R : | t | + R 1 < | x | < | t | + R 2  , 0 ≤ R 1 < R 2 ; and let χ R , χ R 1 ,R 2 b e their corresponding c haracteristic functions deﬁned in R 3 × R . In addition, if Ψ ⊂ R 3 × R , then the notation χ Ψ represen ts the c haracteristic function of Ψ. Stric hartz norms W e deﬁne Y norm to be the L 5 L 10 Stric hartz norm. F or example, if J is a time interv al, then ∥ u ∥ Y ( J ) = ∥ u ∥ L 5 L 10 ( J × R 3 ) = Z J  Z R 3 | u ( x, t ) | 10 d x  1 / 2 d t ! 1 / 5 . W e ma y also com bine Y norm with the characteristic function χ R deﬁned ab o ve and write ∥ χ R u ∥ Y ( J ) =   Z J Z | x | > | t | + R | u ( x, t ) | 10 d x ! 1 / 2 d t   1 / 5 . Please note that ∥ χ R u ∥ Y ( J ) is meaningful ev en if u is only deﬁned in the exterior region Ω R but not necessarily deﬁned in the whole space-time R 3 × R . 2.1 Exterior solutions In order to fo cus on the radiation property of u in some exterior region Ω R , and to a void (p ossibly) complicated behaviour of solutions inside the light cone | x | = | t | + R , w e shall adopt the conception of exterior solutions giv en b y Duyck aerts-Kenig-Merle [11]. W e start by discussing exterior solutions to the linear wa ve equations. Let u, F b e functions b oth deﬁned in the region Ω = { ( x, t ) : | x | > | t | + R, t ∈ ( − T 1 , T 2 ) } ⊆ Ω R , T 1 , T 2 ∈ R + ∪ { + ∞} . W e deﬁne the exterior solution u to the following linear w av e equation  ∂ 2 t u − ∆ u = F ( x, t ) , ( x, t ) ∈ Ω; ( u, u t ) | t =0 = ( u 0 , u 1 ) ∈ H ; 8 where F satisﬁes ∥ χ R F ∥ L 1 L 2 ( J × R 3 ) < + ∞ for an y b ounded closed time interv al J ⊂ ( − T 1 , T 2 ), b y u = S L ( u 0 , u 1 ) + Z t 0 sin( t − t ′ ) √ − ∆ √ − ∆ [ χ R ( · , t ′ ) F ( · , t ′ )]d t ′ , ( x, t ) ∈ Ω . (2) Here S L ( u 0 , u 1 ) is the linear propagation of initial data ( u 0 , u 1 ), i.e. the solution to the homoge- neous linear wa v e equation with initial data ( u 0 , u 1 ). In other words, u is exactly the restriction of the solution ˜ u , whic h solves the follo wing classic linear wa ve equation, to the exterior region Ω  ∂ 2 t ˜ u − ∆ ˜ u = χ R ( x, t ) F ( x, t ) , ( x, t ) ∈ R × ( − T 1 , T 2 ); ( ˜ u, ˜ u t ) | t =0 = ( u 0 , u 1 ) ∈ H . Please note that the ﬁnite sp eed of wa ve propagation implies that the v alues of initial data in the ball { x : | x | ≤ R } are actually irrelev ant, thus it suﬃces to specify the initial data ( u 0 , u 1 ) in the space H ( R ). W e may deﬁne an exterior solution u to nonlinear w av e equations in a similar w ay . F or instance, we sa y that a function u deﬁned in Ω is a solution to  ∂ 2 t u − ∆ u = F ( u ) , ( x, t ) ∈ Ω; ( u, u t ) | t =0 = ( u 0 , u 1 ) if and only if the inequality ∥ χ R u ∥ Y ( J ) < + ∞ holds for any b ounded closed time interv al J ⊂ ( − T 1 , T 2 ), which also implies that ∥ χ R F ( u ) ∥ L 1 L 2 ( J × R 3 ) < + ∞ , and the iden tity (2) holds with F ( x, t ) = F ( u ( x, t )). Lo cal theory A combination of the Strichartz estimates (see [15] for example) sup t ∥  u ( t ) ∥ H + ∥ u ∥ L 5 L 10 ≲ 1 ∥  u (0) ∥ H + ∥ □ u ∥ L 1 L 2 , ﬁnite sp eed of w av e propagation and a ﬁxed-point argument immediately leads to the lo cal well- p osedness theory , small data scattering theory and p erturbation theory(contin uous dep endence of solutions on the initial data) of exterior solutions. The argumen t is almost the same as the corresp onding argumen t in the whole space R 3 . More details of this argumen t in the whole space can b e found in [26, 32] for lo cal well-posedness and in [28, 39] for p erturbation theory . 2.2 Radiation ﬁelds The theory of radiation ﬁelds plays an imp ortan t role in the discussion of the asymptotic b e- ha viour of solutions to wa ve equations. It dates back to F riedlander’s w orks [13, 14] more than half a cen tury ago. The following version of statement comes from Duyck aerts-Kenig-Merle [10]. Theorem 2.1 (Radiation ﬁeld) . Assume t hat d ≥ 3 and let u b e a solution to the fr e e wave e quation ∂ 2 t u − ∆ u = 0 with initial data ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 ( R d ) . Then lim t →±∞ Z R d  |∇ u ( x, t ) | 2 − | u r ( x, t ) | 2 + | u ( x, t ) | 2 | x | 2  d x = 0 and ther e exist two functions G ± ∈ L 2 ( R × S d − 1 ) such that lim t →±∞ Z ∞ 0 Z S d − 1    r d − 1 2 ∂ t u ( r θ, t ) − G ± ( r ∓ t, θ )    2 d θ d r = 0; lim t →±∞ Z ∞ 0 Z S d − 1    r d − 1 2 ∂ r u ( r θ, t ) ± G ± ( r ∓ t, θ )    2 d θ d r = 0 . In addition, the maps ( u 0 , u 1 ) → √ 2 G ± ar e bije ctive isometries fr om ˙ H 1 × L 2 ( R d ) to L 2 ( R × S d − 1 ) . 9 In this work the author calls the functions G ± the radiation proﬁles of the free w av e u . In addition, the radiation proﬁles of ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 are deﬁned to b e the radiation proﬁles of the corresp onding free w av e with initial data ( u 0 , u 1 ). It is not diﬃcult to see that u is radial if and only if the radiation proﬁle is indep endent of the angle θ . In this w ork w e frequently utilize the following explicit formula in the 3D radial case, which give the free wa ve u and the radiation proﬁle G + in the positive time direction in term of the radiation proﬁle G − in the negativ e time direction. u ( r , t ) = 1 r Z t + r t − r G − ( s )d s ; G + ( s ) = − G − ( − s ) , s ∈ R . (3) Similar formula for other dimensions and non-radial case can be found in [2, 31]. The symmetry b et ween G ± giv en abov e also implies that an arbitrary combination of G ± ∈ L 2 ( R + ) uniquely determine a radial free w av e. W e ma y also write initial data ( u 0 , u 1 ) in term of the initial data ( u 0 , u 1 ) u 0 ( r ) = 1 r Z r − r G − ( s )d s ; u 1 ( r ) = G − ( r ) − G − ( − r ) r . (4) An integration by parts then gives us another useful formula ∥ ( u 0 , u 1 ) ∥ 2 H ( R ) = 8 π ∥ G − ∥ 2 L 2 ( { s : | s | >R } ) + 4 π R | u 0 ( R ) | 2 . (5) Next we consider the radiation proﬁle of initial data ( v 0 , v 1 ) ∈ H ( R ) for some radius R > 0. It is natural to deﬁne its radiation proﬁle G ± in the following wa y: we pick up radial initial data ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 suc h that the restriction of ( u 0 , u 1 ) in the exterior region { x : | x | > R } is exactly ( v 0 , v 1 ) and deﬁne G ± to b e the corresp onding radiation proﬁle of ( u 0 , u 1 ). Although the choice of ( u 0 , u 1 ) is NOT unique, we may uniquely determine the v alue of G ± ( s ) for | s | > R , b y the ﬁnite sp eed of propagation, as well as the v alue of Z R − R G − ( s )d s, b y the explicit formula (4). Conv ersely , if tw o radiation proﬁles G − ( s ) and ˜ G − ( s ) satisfy G − ( s ) = ˜ G − ( s ) , | s | > R ; Z R − R G − ( s )d s = Z R − R ˜ G − ( s )d s ; then the corresp onding free w av e coincide in the exterior region Ω R . In summary , the map from initial data ( u 0 , u 1 ) ∈ H ( R ) to the corresponding radiation proﬁles ( u 0 , u 1 ) − → √ 8 π G ± ( s ) , 2 √ π R 1 / 2 Z R − R G ± ( s )d s ! is an isometric homeomorphism from H ( R ) to L 2 ( { s : | s | > R } ) ⊕ R . The isometric prop erty follo ws from the identities (4) and (5). Finally w e may also consider radiation ﬁelds and proﬁles for suitable solutions to inhomoge- neous/nonlinear wa v e equations. Lemma 2.2 (Radiation ﬁelds of inhomogeneous equation) . Assume that R ≥ 0 . L et u b e a r adial exterior solution to the wave e quation  ∂ 2 t u − ∆ u = F ( t, x ); ( x, t ) ∈ Ω R ; ( u, u t ) | t =0 = ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 . 10 If F satisﬁes ∥ χ R F ∥ L 1 L 2 ( R × R 3 ) < + ∞ , then ther e exist unique r adiation pr oﬁles G ± ∈ L 2 ([ R, + ∞ )) such that lim t → + ∞ Z ∞ R + t  | G + ( r − t ) − r u t ( r , t ) | 2 + | G + ( r − t ) + r u r ( r , t ) | 2  d r = 0; lim t →−∞ Z ∞ R − t  | G − ( r + t ) − r u t ( r , t ) | 2 + | G − ( r + t ) − r u r ( r , t ) | 2  d r = 0 . In addition, the fol lowing estimates hold for G ± given ab ove and the c orr esp onding r adiation pr oﬁles G 0 , ± of the initial data ( u 0 , u 1 ) : 4 √ π ∥ G − − G 0 , − ∥ L 2 ([ R,R ′ ]) ≤ ∥ χ R,R ′ F ∥ L 1 L 2 (( −∞ , 0] × R 3 ) , R ′ > R ; 4 √ π ∥ G + − G 0 , + ∥ L 2 ([ R,R ′ ]) ≤ ∥ χ R,R ′ F ∥ L 1 L 2 ([0 , + ∞ ) × R 3 ) , R ′ > R . Please refer to Section 2 (Lemma 2.5 and Remark 2.6) of the author’s previous work [41] for the pro of of this lemma. In fact w e may giv e an explicit formula G + ( s ) − G 0 , + ( s ) = 1 2 Z ∞ 0 ( s + t ) F ( s + t, t )d t. Nonlinear equations Please note that Lemma 2.2 applies to exterior solutions u to (CP1) deﬁned in Ω R , as long as the inequalit y ∥ χ R u ∥ Y ( R ) < + ∞ holds, b ecause this assumption guaran tees that the inequality ∥ χ R F ( u ) ∥ L 1 L 2 ( R × R 3 ) < + ∞ holds. In this case the corresp onding radiation proﬁles G ± ∈ L 2 ([ R, + ∞ )) given in Lemma 2.2 is called the (nonlinear) radiation proﬁle of u . Remark 2.3. Whenever we talk ab out a r adiation pr oﬁle in subse quent se ctions without sp e c- ifying whether it is the r adiation pr oﬁle in the p ositive or ne gative dir e ction, we r efer to the r adiation pr oﬁle in the ne gative time dir e ction. 2.3 Asymptotically equiv alen t solutions Assume that u, v ∈ C ( R ; ˙ H 1 × L 2 ) and R ≥ 0. W e say that u and v are R -w eakly asymptotically equiv alent if and only if the follo wing limit holds lim t →±∞ Z | x | >R + | t | |∇ t,x ( u − v ) | 2 d x = 0 . In particular, w e sa y that u and v are asymptotically equiv alent to eac h other if R = 0. Because the integral ab ov e only inv olv es the v alues of u, v in the exterior region Ω R , the deﬁnition ab o ve also applies to suitable exterior solutions u and v deﬁned in Ω R only . Radiation part If a free w av e v L is asymptotically equiv alent to a radial exterior solution u of (CP1), then w e call v L the radiation part of u (outside the ligh t cone). It w as prov e in [40] that an exterior solution u is asymptotically equiv alent to some free wa v e in Ω 0 if and only if ∥ χ 0 u ∥ Y ( R ) < + ∞ . The suﬃciency of this condition is a direct consequence of Lemma 2.2. Indeed, if ∥ χ 0 u ∥ Y ( R ) < + ∞ , then w e may determine its (nonlinear) radiation proﬁle G ± ∈ L 2 ( R + ) by Lemma 2.2, and then construct a free w av e with the same radiation proﬁles for s > 0, which is the desired asymptotically equiv alen t free wa ve. T o see why the condition ∥ χ 0 u ∥ Y ( R ) < + ∞ is also necessary , please refer to [40]. Please note that this conception of radiation part is diﬀeren t from the radiation part u L in a soliton resolution at the blow-up time T + or + ∞ , as describ ed in the introduction section of this article. 11 Non-radiativ e solutions A ( R -w eakly) non-radiative solution is a solution u to the free wa ve equation, or the nonlinear wa ve equation (CP1), or any other related wa v e equation such that lim t →±∞ Z | x | > | t | + R |∇ t,x u ( x, t ) | 2 d x = 0 . In other words, a solution u is ( R -w eakly) non-radiative if and only if it is ( R -weakly) asymptot- ically equiv alent to zero. Non-radiative solution is one of most imp ortant topics in the channel of energy metho d (see [3, 8, 27] for example), which plays an imp ortant role in the study of nonlinear wa v e equations in recen t years. F or an example, the ground states W λ ( x ) are all non-radiative solutions to (CP1). Th us if a free wa v e v L is asymptotically equiv alen t to a solution u to (CP1), then S ∗ ( x, t ) = J X j =1 ζ j W λ j ( x ) + v L ( x, t ) is also asymptotically equiv alent to u , for an y given p ositiv e in teger J , signs ζ j ∈ { +1 , − 1 } and scales λ j > 0. Remark 2.4. The standar d gr ound state W ( x ) in this article is a dilation of (thus slightly diﬀer ent fr om) the one (1 + | x | 2 / 3) − 1 / 2 use d in most r elate d works. This choic e eliminates the addition c onstant in the asymptotic b ehaviour W λ ( r ) ≈ λ 1 / 2 r − 1 for lar ge r . 2.4 Sev eral Stric hartz estimates In this subsection w e prov e sev eral reﬁned Stric hartz estimates for further use. Most of them can b e v eriﬁed by a straight-forw ard calculation. The ﬁrst few lemmata are concerning the Stric hartz norm of the ground state in several channel-lik e regions. Lemma 2.5. L et 0 ≤ r 1 < r 2 ≤ R b e p ositive c onstants, R ≥ 1 and Ψ = { ( x, t ) : r 1 + | t | < | x | < r 2 + | t | , | x | + | t | > R } . Then we have ∥ χ Ψ W ∥ Y ( R ) ≲ 1 ( r 2 − r 1 ) 1 / 10 R − 3 / 5 . In addition, if 0 ≤ r 1 < r 2 , then ∥ χ r 1 ,r 2 W ∥ Y ( R ) ≲ 1 ( r 2 − r 1 ) 1 / 10 min n r − 3 / 5 1 , 1 o . Pr o of. The pro of follows a straigh t-forward calculation. ∥ χ Ψ W ∥ 5 Y ( R ) ≲ 1 Z ∞ R − r 2 2 Z t + r 2 max { t + r 1 ,R − t }  1 3 + r 2  − 5 r 2 d r ! 1 / 2 d t ≲ 1 Z R R − r 2 2  R − 8 ( r 2 − r 1 )  1 / 2 d t + Z ∞ R  r 2 − r 1 ( t + r 1 ) 8  1 / 2 d t ≲ 1 ( r 2 − r 1 ) 1 / 2 R − 3 . Here we only consider the p ositive time direction by time symmetry and use the inequalit y max { t + r 1 , R − t } ≥ R / 2. The second inequalities can b e prov ed in the same manner. On one 12 hand, we hav e ∥ χ r 1 ,r 2 W ∥ 5 Y ( R ) ≲ 1 Z ∞ 0 Z t + r 2 t + r 1  1 3 + r 2  − 5 r 2 d r ! 1 / 2 d t ≲ 1 Z ∞ 0  r 2 − r 1 ( t + r 1 ) 8  1 / 2 d t ≲ 1 ( r 2 − r 1 ) 1 / 2 r − 3 1 . On the other hand, since  1 3 + r 2  − 5 r 2 ≲ 1 min  1 , r − 8  , w e also ha ve ∥ χ r 1 ,r 2 W ∥ 5 Y ( R ) ≲ 1 Z ∞ 0 Z t + r 2 t + r 1  1 3 + r 2  − 5 r 2 d r ! 1 / 2 d t ≲ 1 Z 1 0 ( r 2 − r 1 ) 1 / 2 d t + Z ∞ 1  r 2 − r 1 ( t + r 1 ) 8  1 / 2 d t ≲ 1 ( r 2 − r 1 ) 1 / 2 . Com bining these t wo upper bounds, w e ﬁnish the pro of. Lemma 2.6. L et λ > 1 b e a r adius • If 1 / 2 ≤ r 1 < r 2 , then   χ r 1 ,r 2 W 4 W λ   L 1 L 2 ≲ 1 λ − 1 / 2 ( r 2 − r 1 ) 1 / 2 r − 2 1 ;   χ r 1 ,r 2 W 3 W 2 λ   L 1 L 2 ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 r − 1 1 ; • If 0 ≤ r 1 < r 2 ≤ 1 , then   χ r 1 ,r 2 W 4 W λ   L 1 L 2 ≲ 1 λ − 1 / 2 ( r 2 − r 1 ) 1 / 2 ;   χ r 1 ,r 2 W 3 W 2 λ   L 1 L 2 ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 . • If 0 ≤ r 1 < r 2 ≤ λ , then   χ r 1 ,r 2 W W 4 λ   L 1 L 2 ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 . Please note that all the L 1 L 2 norms are the abbreviation of L 1 L 2 ( R × R 3 ) in this work, unless sp eciﬁed otherwise. Pr o of. The proof is simply a straigh t forward calculation. W e ﬁrst assume that 1 / 2 ≤ r 1 < r 2 . Then we hav e   χ r 1 ,r 2 W 4 W λ   L 1 L 2 ≲ 1 Z ∞ 0 Z r 2 + t r 1 + t  1 3 + r 2  − 4 1 λ  1 3 + r 2 λ 2  − 1 r 2 d r ! 1 / 2 d t ≲ 1 Z ∞ 0  Z r 2 + t r 1 + t 1 λr 6 d r  1 / 2 d t ≲ 1 Z ∞ 0 ( r 2 − r 1 ) 1 / 2 λ 1 / 2 ( r 1 + t ) 3 d t ≲ 1 λ − 1 / 2 ( r 2 − r 1 ) 1 / 2 r − 2 1 . 13   χ r 1 ,r 2 W 3 W 2 λ   L 1 L 2 ≲ 1 Z ∞ 0 Z r 2 + t r 1 + t  1 3 + r 2  − 3 1 λ 2  1 3 + r 2 λ 2  − 2 r 2 d r ! 1 / 2 d t ≲ 1 Z ∞ 0  Z r 2 + t r 1 + t 1 λ 2 r 4 d r  1 / 2 d t ≲ 1 Z ∞ 0 ( r 2 − r 1 ) 1 / 2 λ ( r 1 + t ) 2 d t ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 r − 1 1 . On the other hand, if 0 ≤ r 1 < r 2 ≤ 1, then   χ r 1 ,r 2 W 4 W λ   L 1 L 2 ≲ 1 Z ∞ 0 Z r 2 + t r 1 + t  1 3 + r 2  − 4 1 λ  1 3 + r 2 λ 2  − 1 r 2 d r ! 1 / 2 d t ≲ 1 Z 1 0  Z r 2 + t r 1 + t 1 λ d r  1 / 2 d t + Z ∞ 1  Z r 2 + t r 1 + t 1 λr 6 d r  1 / 2 d t ≲ 1 Z 1 0 ( r 2 − r 1 ) 1 / 2 λ 1 / 2 d t + Z ∞ 1 ( r 2 − r 1 ) 1 / 2 λ 1 / 2 ( r 1 + t ) 3 d t ≲ 1 λ − 1 / 2 ( r 2 − r 1 ) 1 / 2 .   χ r 1 ,r 2 W 3 W 2 λ   L 1 L 2 ≲ 1 Z ∞ 0 Z r 2 + t r 1 + t  1 3 + r 2  − 3 1 λ 2  1 3 + r 2 λ 2  − 2 r 2 d r ! 1 / 2 d t ≲ 1 Z 1 0  Z r 2 + t r 1 + t 1 λ 2 d r  1 / 2 d t + Z ∞ 1  Z r 2 + t r 1 + t 1 λ 2 r 4 d r  1 / 2 d t ≲ 1 Z 1 0 ( r 2 − r 1 ) 1 / 2 λ d t + Z ∞ 1 ( r 2 − r 1 ) 1 / 2 λ ( r 1 + t ) 2 d t ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 . Finally , for 0 ≤ r 1 < r 2 ≤ λ , w e hav e   χ r 1 ,r 2 W W 4 λ   L 1 L 2 ≲ 1 Z ∞ 0 Z r 2 + t r 1 + t  1 3 + r 2  − 1 1 λ 4  1 3 + r 2 λ 2  − 4 r 2 d r ! 1 / 2 d t ≲ 1 Z λ 0  Z r 2 + t r 1 + t 1 λ 4 d r  1 / 2 d t + Z ∞ λ  Z r 2 + t r 1 + t λ 4 r 8 d r  1 / 2 d t ≲ 1 Z λ 0 ( r 2 − r 1 ) 1 / 2 λ 2 d t + Z ∞ λ λ 2 ( r 2 − r 1 ) 1 / 2 ( r 1 + t ) 4 d t ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 . Corollary 2.7. L et λ > 1 b e a r adius. Then   W 4 W λ   L 1 L 2 +   W W 4 λ   L 1 L 2 ≲ 1 λ − 1 / 2 . 14 Pr o of. The estimate of W 4 W λ is a direct consequence of Lemma 2.6.   W 4 W λ   L 1 L 2 ≤   χ 0 , 1 W 4 W λ   L 1 L 2 + ∞ X k =0   χ 2 k , 2 k +1 W 4 W λ   L 1 L 2 ≲ 1 λ − 1 / 2 + ∞ X k =0 λ − 1 / 2 2 − 3 k/ 2 ≲ 1 λ − 1 / 2 . Next we conduct a direct calculation   χ λ W W 4 λ   L 1 L 2 ≲ 1 Z ∞ 0 Z ∞ λ + t  1 3 + r 2  − 1 1 λ 4  1 3 + r 2 λ 2  − 4 r 2 d r ! 1 / 2 d t ≲ 1 Z ∞ 0  Z ∞ λ + t λ 4 r 8 d r  1 / 2 d t ≲ 1 Z ∞ 0 λ 2 ( λ + t ) 7 / 2 d t ≲ 1 λ − 1 / 2 . Com bining this with Lemma 2.6, we obtain   W W 4 λ   L 1 L 2 ≤   χ 0 ,λ W W 4 λ   L 1 L 2 +   χ λ W W 4 λ   L 1 L 2 ≲ 1 λ − 1 / 2 . This completes the pro of. Lemma 2.8. Assume that λ ≥ 2 and 0 ≤ r 1 < r 2 ≤ λ . Then    χ r 1 ,r 2 W 4 ( W λ − √ 3 λ − 1 / 2 )    L 1 L 2 ≲ 1 ( r 2 − r 1 ) 1 / 2 λ − 5 / 2 ln λ. Pr o of. w e observe that    W λ − √ 3 λ − 1 / 2    ≲ 1 min n λ − 5 / 2 r 2 , λ − 1 / 2 o . Th us LHS ≲ 1 Z ∞ 0 Z r 2 + t r 1 + t  1 3 + r 2  − 4    W λ − √ 3 λ − 1 / 2    2 r 2 d r ! 1 / 2 d t ≲ 1 Z λ 0 Z r 2 + t r 1 + t λ − 5  1 3 + r 2  − 1 d r ! 1 / 2 d t + Z ∞ λ  Z r 2 + t r 1 + t λ − 1 r − 6 d r  1 / 2 d t ≲ 1 Z λ 0 ( r 2 − r 1 ) 1 / 2 λ 5 / 2  1 3 + ( r 1 + t ) 2  − 1 / 2 d t + Z ∞ λ ( r 2 − r 1 ) 1 / 2 λ 1 / 2 ( r 1 + t ) 3 d t ≲ 1 ( r 2 − r 1 ) 1 / 2 λ − 5 / 2 ln λ. The follo wing results are concerning the L 5 L 10 norm of free w av es in the c hannel-lik e regions. Lemma 2.9. L et v b e a r adial fr e e wave with a r adiation pr oﬁle G and 0 ≤ r 1 < r 2 ≤ R b e r adii. If G ( s ) = 0 for | s | < R , then we have ∥ χ r 1 ,r 2 v ∥ Y ( R ) ≲ 1  r 2 − r 1 R  1 10 ∥ G ∥ L 2 ( R ) . 15 Pr o of. W e recall the explicit formula v ( r, t ) = 1 r Z t + r t − r G ( s )d s, whic h immediately giv es the point-wise estimate | v ( r, t ) | ≲ 1 r − 1 / 2 ∥ G ∥ L 2 ( R ) . In addition, if | t | < r < R/ 2, then the supp ort of G guarantees that v ( r, t ) = 0. Th us | v ( r, t ) | ≲ 1 min n r − 1 / 2 , R − 1 / 2 o ∥ G ∥ L 2 , r > | t | . A direct calculation then gives ∥ χ r 1 ,r 2 v ∥ 5 Y ( R ) ≲ 1 Z R Z | t | + r 2 | t | + r 1 min { r − 5 , R − 5 }∥ G ∥ 10 L 2 · r 2 d r ! 1 / 2 d t ≲ 1 Z R − R Z | t | + r 2 | t | + r 1 R − 5 r 2 ∥ G ∥ 10 L 2 d r ! 1 / 2 d t + Z | t | >R Z | t | + r 2 | t | + r 1 r − 3 ∥ G ∥ 10 L 2 d r ! 1 / 2 d t ≲ 1 Z R − R  R − 3 ∥ G ∥ 10 L 2 ( r 2 − r 1 )  1 / 2 d t + Z | t | >R  ( r 2 − r 1 ) ∥ G ∥ 10 L 2 ( | t | + r 1 ) 3  1 / 2 d t ≲ 1  r 2 − r 1 R  1 2 ∥ G ∥ 5 L 2 ( R ) . Corollary 2.10. Assume that 0 < R 0 < R 1 < R 2 < · · · < R m +1 is a se quenc e of p ositive numb ers. L et ( u 0 , u 1 ) ∈ H ( R 0 ) b e r adial initial data with r adiation pr oﬁle G ( s ) . Then the c orr esp onding fr e e wave u satisﬁes ∥ χ R 0 ,R 1 u ∥ Y ( R ) ≲ 1 R − 1 / 2 0      Z R 0 − R 0 G ( s )d s      + ∥ G ∥ L 2 ( { s : R 0 < | s | R m +1 } ) . Pr o of. W e ma y split the linear free wa ve u in to several ones u ( r , t ) = 1 r Z R 0 − R 0 G ( s )d s + m +1 X j =0 v j ( r , t ) , r > R 0 + | t | . Here v j is the free w av e whose radiation proﬁle is exactly the restriction of G ( s ) on the set { s : R j < | s | < R j +1 } (or { s : | s | > R m +1 } for j = m + 1). A direct calculation then sho ws that   χ R 0 | x | − 1   Y ( R ) ≲ 1 R − 1 / 2 0 . The conclusion then follows from a combination of this upper b ound, the classic Stric hartz estimate and Lemma 2.9. Remark 2.11. L et 0 = R 0 < R 1 < R 2 < · · · < R m +1 b e a se quenc e and ( u 0 , u 1 ) ∈ ˙ H 1 × L 2 b e r adial initial data with r adiation pr oﬁle G ( s ) . Then the same ar gument as ab ove gives ∥ χ 0 ,R 1 u ∥ Y ( R ) ≲ 1 ∥ G ∥ L 2 ( { s :0 < | s | R m +1 } ) . 16 The following results give an upp er b ound of L 5 L 10 norm for solutions to the linear wa v e equation with lo calized data. Lemma 2.12. Assume that 0 ≤ r 1 < r 2 ≤ R . L et u 1 ∈ L 2 ( R 3 ) b e r adial function supp orte d in { x : | x | > R } . Then the fr e e wave v = S L (0 , u 1 ) satisﬁes ∥ χ r 1 ,r 2 v ∥ Y ( R ) ≲ 1  r 2 − r 1 R  1 10 ∥ u 1 ∥ L 2 ( R 3 ) . Pr o of. W e recall the explicit formula v ( r, t ) = 1 2 r Z r + t r − t su 1 ( s )d s, r > | t | . It follows that | v ( r, t ) | ≲ 1 | t | 1 / 2 r ∥ su 1 ( s ) ∥ L 2 ( R + ) ≲ 1 r − 1 / 2 ∥ u 1 ∥ L 2 ( R 3 ) . Again we alwa ys hav e v ( r, t ) = 0 if | t | < r < R/ 2. Thus w e also hav e | v ( r, t ) | ≲ 1 min n r − 1 / 2 , R − 1 / 2 o ∥ u 1 ∥ L 2 ( R 3 ) , r > | t | . A similar calculation to the pro of of Lemma 2.9 then completes the pro of. Corollary 2.13. Assume that 0 ≤ r 1 < r 2 ≤ R . L et F ∈ L 1 L 2 ( R × R 3 ) b e r adial and supp orte d in the r e gion Ω R . Then the solution v to the line ar fr e e wave  □ v = F, ( x, t ) ∈ R 3 × R ; ( v , v t ) | t =0 = (0 , 0) satisﬁes ∥ χ r 1 ,r 2 v ∥ Y ( R ) ≲ 1  r 2 − r 1 R  1 10 ∥ F ∥ L 1 L 2 ( R × R 3 ) . Pr o of. This is a direct consequence of Lemma 2.12 and Duhamel’s form ula v ( · , t ) = Z t 0 S L ( t − t ′ )(0 , F ( · , t ′ ))d t ′ . Finally we giv e a few estimates for the norm of a free w av e, as well as its interaction strength with ground states, in term of its radiation concentration. Lemma 2.14. L et v L b e a r adial line ar fr e e wave with r adiation pr oﬁle G ( s ) and λ ∈ R + . We deﬁne τ =  sup 0 0 1 r 1 / 2 Z r − r | G ( s ) | d s. Then • F or 0 ≤ r 1 < r 2 ≤ λ , we have ∥ χ r 1 ,r 2 v L ∥ Y ( R ) ≲ 1  r 2 − r 1 λ  1 / 10 τ . • If 1 ≤ r 1 < r 2 ≤ λ , then   χ r 1 ,r 2 W 4 v L   L 1 L 2 ≲ 1 τ λ − 1 / 2 r − 3 / 2 1 . 17 • If 0 ≤ r 1 < r 2 ≤ 2 < λ , then   χ r 1 ,r 2 W 4 v L   L 1 L 2 ≲ 1 τ λ − 1 / 2 ( r 2 − r 1 ) 1 / 2 . Pr o of. First of all, we giv e a point-wise estimate of v L ( r , t ) for r > | t | . If r + | t | < λ , then | v L ( r , t ) | = 1 r     Z t + r t − r G ( s )d s     ≲ 1 1 r 1 / 2 ∥ G ∥ L 2 ( t − r,t + r ) ≲ 1 1 r 1 / 2 ∥ G ∥ L 2 ( −| t |− r, | t | + r ) ≲ 1 1 r 1 / 2 ·  | t | + r λ  1 / 2 τ ≲ 1 τ λ 1 / 2 . On the other hand, we alwa ys ha ve | v L ( r , t ) | ≤ 1 r Z t + r t − r | G ( s ) | d s ≲ 1 1 r 1 / 2 ( | t | + r ) 1 / 2 Z | t | + r −| t |− r | G ( s ) | d s ≲ 1 τ r 1 / 2 . In summary we ha ve | v L ( r , t ) | ≲ 1 min n λ − 1 / 2 , r − 1 / 2 o τ , r > | t | . Th us we ma y conduct a direct calculation for 0 ≤ r 1 < r 2 ≤ λ ∥ χ r 1 ,r 2 v L ∥ 5 Y ( R ) ≲ 1 Z R Z r 2 + | t | r 1 + | t | min  λ − 5 , r − 5  τ 10 · r 2 d r ! 1 / 2 d t ≲ 1 Z λ − λ Z r 2 + | t | r 1 + | t | λ − 5 τ 10 · r 2 d r ! 1 / 2 d t + Z | t | >λ Z r 2 + | t | r 1 + | t | r − 3 τ 10 d r ! 1 / 2 d t ≲ 1  r 2 − r 1 λ  1 / 2 τ 5 . Next we assume that 1 ≤ r 1 < r 2 ≤ λ . A direct calculation shows that   χ r 1 ,r 2 W 4 v L   L 1 L 2 ≲ 1 Z R Z r 2 + | t | r 1 + | t | min  λ − 1 , r − 1  τ 2 r − 8 · r 2 d r ! 1 / 2 d t ≲ 1 Z R Z r 2 + | t | r 1 + | t | λ − 1 τ 2 r − 6 d r ! 1 / 2 d t ≲ 1 Z R λ − 1 / 2 τ ( r 1 + | t | ) − 5 / 2 d t ≲ 1 λ − 1 / 2 τ r − 3 / 2 1 . Finally if 0 ≤ r 1 < r 2 ≤ 2 < λ , then we hav e   χ r 1 ,r 2 W 4 v L   L 1 L 2 ≲ 1 Z R Z r 2 + | t | r 1 + | t | min  λ − 1 , r − 1  τ 2 W ( r ) 8 · r 2 d r ! 1 / 2 d t ≲ 1 Z 1 − 1 Z r 2 + | t | r 1 + | t | λ − 1 τ 2 d r ! 1 / 2 d t + Z | t | > 1 Z r 2 + | t | r 1 + | t | λ − 1 τ 2 r − 6 d r ! 1 / 2 d t ≲ 1 Z 1 − 1 λ − 1 / 2 τ ( r 2 − r 1 ) 1 / 2 d t + Z | t | > 1  λ − 1 τ 2 ( r 2 − r 1 ) ( r 1 + | t | ) 6  1 / 2 d t ≲ 1 τ λ − 1 / 2 ( r 2 − r 1 ) 1 / 2 . 18 Finally we give a Strchartz estimate with highly lo calized radiation proﬁle. Lemma 2.15 (see Lemma 5.1 of Shen [41]) . L et v L b e a r adial fr e e wave and I = [ a, b ] ⊂ R + b e an interval. Then ∥ χ 0 v L ∥ Y ( R ) ≲ 1 ∥ G + ∥ L 2 ( R \ I ) +  b − a a  1 / 2 ∥ G + ∥ L 2 ( I ) . Her e G + is the r adiation pr oﬁle of v L in the p ositive time dir e ction. 3 Soliton resolution estimates The following proposition gives an instantaneous soliton resolution of a radial solution to (CP1), as w ell as some quantitativ e prop erties of the soliton resolution, as long as the solution is asymp- totically equiv alen t to a free wa v e with a small Strichartz norm outside the main light cone. Prop osition 3.1. L et n b e a p ositive inte ger and c 2 ≫ 1 b e a suﬃciently lar ge c onstant. Then ther e exists a smal l c onstant δ 0 = δ 0 ( n, c 2 ) > 0 , such that if a r adial exterior solution u to (CP1) deﬁne d in Ω 0 is asymptotic al ly e quivalent to a ﬁnite-ener gy fr e e wave v L with δ . = ∥ χ 0 v L ∥ Y ( R ) < δ 0 , then one of the fol lowing holds: (a) Ther e exists a se quenc e ( ζ j , λ j ) ∈ { +1 , − 1 } × R + for j = 1 , 2 , · · · , J with 0 ≤ J ≤ n − 1 such that λ j +1 λ j ≲ j,c 2 δ 2 , j = 1 , 2 , · · · , J − 1;        u ( · , 0) − J X j =1 ζ j ( W λ j , 0) −  v L ( · , 0)       ˙ H 1 × L 2 +       χ 0   u − J X j =1 ζ j W λ j         Y ( R ) ≲ J,c 2 δ. (b) Ther e exists a se quenc e ( ζ j , λ j ) ∈ { +1 , − 1 } × R + for j = 1 , 2 , · · · , n satisfying λ j +1 λ j ≲ j,c 2 δ 2 , j = 1 , 2 , · · · , n − 1; such that u satisﬁes the fol lowing soliton r esolution estimate in the exterior r e gion        u ( · , 0) − n X j =1 ζ j ( W λ j , 0) −  v L ( · , 0)       H ( c 2 λ n ) +       χ c 2 λ n   u − n X j =1 ζ j W λ j         Y ( R ) ≲ n,c 2 δ. Remark 3.2. Ple ase r efer to [41] for the pr o of of this pr op osition. Ple ase note that the pr op o- sition her e is slightly diﬀer ent fr om the original one (Pr op osition 4.1 in [41]) in thr e e asp e cts • In Pr op osition 4.1 of [41] we ﬁx an absolute c onstant c 2 , thus the implicit c onstants in the ine qualities ther e only dep ends on the inte gers j, J or n . In the further ar gument of this work we pr ob ably have to cho ose a lar ger p ar ameter c 2 than the original one use d in [41], thus we al low to cho ose any suﬃciently lar ge p ar ameter c 2 her e. A brief r eview of the pr o of given in [41] r eve als that the pr o of applies to al l lar ge c onstants c 2 . • Pr op osition 4.1 in [41] applies to al l (we akly) asymptotic al ly e quivalent solutions of a fr e e wave with a smal l Strichartz norm, even if those solutions ar e not ne c essarily deﬁne d in the whole r e gion Ω 0 . F or simplicity we only c onsider exterior solutions deﬁne d in the whole r e gion Ω 0 in this work. Ple ase se e R emark 4.2 of [41] for mor e details. 19 • Inste ad of two p ar ameters ζ and λ , the original pr op osition utilize a single p ar ameter α to r epr esent a gr ound state W α = 1 α  1 3 + | x | 2 α 4  − 1 / 2 , α ∈ R \ { 0 } . These two ways of r epr esentation ar e c ompletely e quivalent. It is not diﬃcult to se e W α = ζ W λ ⇐ ⇒ α = ζ λ 1 / 2 . Remark 3.3. Ple ase note that the sc ale p ar ameters λ j not only dep end on the exterior solution u , but also the choic e of c 2 . In fact, a brief r eview of the pr o of shows that the p ar ameters λ j and ζ j ar e determine d inductively by the identities ( λ 0 = + ∞ ) λ j = c − 1 2 max ( r ∈ (0 , λ j − 1 ) : r 1 / 2      u ( r , 0) − v L ( r , 0) − j − 1 X k =1 ζ k W λ k ( r )      = c 1 / 2 2 W ( c 2 ) ) ; ζ j = sign u ( c 2 λ j , 0) − v L ( c 2 λ j , 0) − j − 1 X k =1 ζ k W λ k ( c 2 λ j ) ! ; which implies that u ( c 2 λ j , 0) − v L ( c 2 λ j , 0) − j X k =1 W λ k ( c 2 λ j ) = 0 . Nevertheless, the numb er of bubbles J , the signs ζ j do not dep end on the choic e of the lar ge p ar ameter c 2 , as long as δ < δ ( n, c 2 ) is suﬃciently smal l. In addition, we may also cho ose the implicit c onstant in the r atio ine quality λ j +1 λ j ≲ j δ 2 , j = 1 , 2 , · · · , J − 1 indep endent of c 2 , under the same assumption. In fact, we may ﬁx a lar ge c onstant c ∗ 2 and c onsider another c onstant c 2 > c ∗ 2 . A c c or ding to Pr op osition 3.1, if δ < δ ( n, c 2 , ε ) is suﬃciently smal l, wher e ε is a p ar ameter to b e determine d later, we may apply Pr op osition 3.1 with e ach p ar ameter c 2 , c ∗ 2 and obtain        u ( · , 0) − J X j =1 ζ j ( W λ j , 0) −  v L ( · , 0)       ˙ H 1 × L 2 ≤ ε, (Case a)        u ( · , 0) − n X j =1 ζ j ( W λ j , 0) −  v L ( · , 0)       H ( c 2 λ n ) ≤ ε ; (Case b) as wel l as        u ( · , 0) − J ∗ X j =1 ζ ∗ j ( W λ ∗ j , 0) −  v L ( · , 0)       ˙ H 1 × L 2 ≤ ε, (Case a)        u ( · , 0) − n X j =1 ζ ∗ j ( W λ ∗ j , 0) −  v L ( · , 0)       H ( c ∗ 2 λ ∗ n ) ≤ ε ; (Case b) with λ j +1 λ j ≤ ε 2 ; λ ∗ j +1 λ ∗ j ≤ ε 2 . 20 Her e ( ζ j , λ j ) and ( ζ ∗ j , λ ∗ j ) ar e the signs and sc ales with p ar ameters c 2 , c ∗ 2 , r esp e ctively. A c ombi- nation of the two soliton r esolution r epr esentations yields that       J X j =1 ζ j ( W λ j , 0) − J ∗ X j =1 ζ ∗ j ( W λ ∗ j , 0)       H ( R ) ≤ 2 ε. Her e we let J = n and/or J ∗ = n if the c orr esp onding soliton r esolution is in c ase b. The r adius R is chosen to b e R =        + ∞ , J, J ∗ < n ; c 2 λ n , J = n, J ∗ < n ; c ∗ 2 λ ∗ n , J < n, J ∗ = n ; max { c 2 λ n , c ∗ 2 λ ∗ n } , J = J ∗ = n. It is not diﬃcult to se e that if ε = ε ( n, c 2 ) is a suﬃciently smal l c onstant, then we must have J = J ∗ ; ζ j = ζ ∗ j ; λ j ≃ 1 λ ∗ j . It imme diately fol lows that λ j +1 λ j ≲ 1 λ ∗ j +1 λ ∗ j ≲ j δ 2 . J-bubble exterior solutions Fix a constan t c 2 as ab ov e. W e call a radial exterior solution u to (CP1) deﬁned in the region Ω 0 to b e a J -bubble exterior solution if it satisﬁes the following conditions. • u is asymptotically equiv alen t to a free wa ve v L with δ . = ∥ χ 0 v L ∥ Y ( R ) < δ 0 ( J + 1 , c 2 ). Here δ 0 ( J + 1 , c 2 ) is the constant giv en in Proposition 3.1. • The soliton resolution of u given in Prop osition 3.1 comes with exactly J bubbles. Please note that this deﬁnition only considers the instantaneous soliton resolution property given in Prop osition 3.1, th us is diﬀerent from the conception of J -bubble soliton resolution as time tends to a blow-up time or inﬁnity . Ho w ever, if u is a radial global solution whose soliton resolution comes with J bubbles as t → + ∞ , then the time-translated solution u ( · , · + t ) must b e a J -bubble exterior solution deﬁned here when t is suﬃciently large, as shown in the last section of this work. The situation of type I I blow-up solution is similar, if we apply a lo cal cut-oﬀ technique when necessary . More details are given in the last section. Remark 3.4. L et u b e a J -bubble exterior solution as deﬁne d ab ove. Then we may deﬁne w ∗ ( x, t ) = u ( x, t ) − J X j =1 ζ j W λ j ( x ) − v L ( x, t ) , which me ans □ w ∗ = F ( u ) − J X j =1 ζ j F ( W λ j ) , 21 and obtain the fol lowing estimate if δ < δ ( J, c 2 ) is suﬃciently smal l:       χ 0   F ( u ) − J X j =1 ζ j F ( W λ j )         L 1 L 2 ≤       χ 0   F ( u ) − F   J X j =1 ζ j W λ j           L 1 L 2 +       χ 0   F   J X j =1 ζ j W λ j   − J X j =1 ζ j F ( W λ j )         L 1 L 2 ≲ J  ∥ χ 0 w ∗ ∥ 4 Y ( R ) + ∥ χ 0 W ∥ 4 Y ( R ) + ∥ χ 0 v L ∥ 4 Y ( R )   ∥ χ 0 w ∗ ∥ Y ( R ) + ∥ χ 0 v L ∥ Y ( R )  + X 1 ≤ j c 5 , the el liptic e quation − ∆ φ = 5 W 4 φ + 5 W 4 admits a solution φ ∈ C 2 ( R 3 \ { 0 } ) satisfying the fol lowing c onditions 22 • φ is a r adial ly symmetric solution; • φ ( c ) = 0 ; • 1 / 2 ≤ r φ ( r ) /µ 0 ≤ 3 / 2 for al l r ∈ (0 , r 4 ) ; • | φ ( x ) | ≲ 1 | x | − 1 for al l x ∈ R 3 \ { 0 } ; her e the implicit c onstant is indep endent of c > c 5 ; • φ ∈ ˙ H 1 ( { x : | x | > r } ) for any r > 0 . Pr o of. It follo ws from a direct calculation that u satisﬁes the elliptic equation ab o ve if and only if w ( r ) = rφ ( r ) satisﬁes the one-dimensional elliptic equation − w rr = 5  1 3 + r 2  − 2 ( w + r ) . (6) W e ﬁrst construct a sp ecial solution w ∗ to this equation with the best decay near the inﬁnity . W e consider the map T : C ([1 , + ∞ )) → C ([1 , + ∞ )): ( T w )( r ) = − 5 Z ∞ r Z ∞ s  1 3 + τ 2  − 2 ( w ( τ ) + τ )d τ d s. A direct calculation shows that the norms in the Banac h space X = C ([1 , + ∞ )) satisfy ∥ T w ∥ X ≤ sup r ≥ 1  5 Z ∞ r Z ∞ s τ − 4 ( ∥ w ∥ X + τ )d τ d s  ≤ 5 2 + 5 6 ∥ w ∥ X and ∥ T w 1 − T w 2 ∥ X ≤ sup r ≥ 1  5 Z ∞ r Z ∞ s τ − 4 ∥ w 1 − w 2 ∥ X d τ d s  ≤ 5 6 ∥ w 1 − w 2 ∥ X . This v eriﬁes that T is a con traction map. The classic ﬁxed-p oin t argument immediately gives a solution w ∗ to (6) with the asymptotic b eha viour w ∗ ( r ) = − 5 2 r − 1 + O ( r − 2 ); w ∗ r ( r ) = 5 2 r − 2 + O ( r − 3 ) . Since (6) is a linear ordinary diﬀeren tial equation with bounded co eﬃcients, the solution w ∗ ma y extend to a solution deﬁned in [0 , + ∞ ). Now we claim that w ∗ (0)  = 0. Indeed, we observ e that • one of the solution to (6) is − r , whic h is zero at the origin; • v = r  r 2 − 1 3   1 3 + r 2  − 3 / 2 satisﬁes the homogeneous diﬀerential equation − v rr = 5  1 3 + r 2  − 2 v . As a result, all solutions to (6) with w (0) = 0 can be giv en by w ( r ) = − r + C v ( r ) = ⇒ lim r → + ∞ w ( r ) = −∞ . The solution w ∗ ( r ) is clearly not in this form, thus we m ust ha v e µ 0 . = w ∗ (0)  = 0. This v eriﬁes our claim. Finally w e may give the desired solution w in the following form and let φ ( x ) = | x | − 1 w ( | x | ). w ( r ) = w ∗ ( r ) + β v ( r ) , β = − w ∗ ( c ) v ( c ) . 23 This clearly solv es the equation (6) with w ( c ) = 0 and w ∈ C 2 ( R + ). By the asymptotic behaviour of w ∗ and v , we ma y choose a suﬃcien tly large absolute constant c 5 ≫ 1, suc h that | β | ≃ 1 c − 1 , c > c 5 . This immediately implies that | w ( r ) | ≤ | w ∗ ( r ) | + | β || v ( r ) | is uniformly bounded for all c > c 5 and r > 0; and that | w ( r ) − µ 0 | ≤ | w ∗ ( r ) − µ 0 | + | β || v ( r ) | ≤ | w ∗ ( r ) − µ 0 | + √ 3 | β | r < | µ 0 | / 2 , ∀ r ∈ (0 , r 4 ) , as long as the constant r 4 is suﬃciently small. The ﬁrst four prop erties of φ ( x ) = | x | − 1 w ( | x | ) immediately follows these properties of w . Finally the last prop erty of φ follo ws from the regularit y φ ∈ C 2 ( R \ { 0 } ) and the asymptotic b ehaviour | φ r ( r ) | ≤ 1 r | w ∗ r ( r ) | + 1 r 2 | w ∗ ( r ) | + | β |   ∂ r ( r − 1 v )   ≲ 1 r − 2 , r ≫ 1 . Remark 4.2. L et φ b e the solution given in the pr evious lemma. The e quation − ∆ φ = 5 W 4 φ + 5 W 4 and the uniform upp er b ound | φ ( x ) | ≲ 1 | x | − 1 implies that ∥ χ 0 ∆ φ ∥ L 1 L 2 ( R × R 3 ) ≲ 1 1 . The upp er b ound is indep endent of lar ge p ar ameter c > c 5 . Corollary 4.3. L et 0 ≤ r 1 < r 2 ≤ λ . Then the solution φ given in L emma 4.1 satisﬁes   χ r 1 ,r 2 W 4 λ φ   L 1 L 2 ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 . Ple ase note that the implicit c onstant is indep endent of c > c 3 . Pr o of. Let us recall that | φ ( x ) | ≲ 1 | x | − 1 . Th us a direct calculation shows that   χ r 1 ,r 2 W 4 λ φ   L 1 L 2 ≲ 1 Z ∞ −∞ Z r 2 + | t | r 1 + | t | 1 λ 4  1 3 + r 2 λ 2  − 4 r − 2 · r 2 d r ! 1 / 2 d t ≲ 1 Z λ − λ Z r 2 + | t | r 1 + | t | 1 λ 4 d r ! 1 / 2 d t + Z | t | >λ Z r 2 + | t | r 1 + | t | λ 4 r 8 d r ! 1 / 2 d t ≲ 1 Z λ − λ ( r 2 − r 1 ) 1 / 2 λ 2 d t + Z | t | >λ λ 2 ( r 2 − r 1 ) 1 / 2 ( r 1 + | t | ) 4 d t ≲ 1 λ − 1 ( r 2 − r 1 ) 1 / 2 . 5 Bubble In teraction In this section w e giv e a few appro ximation of a J -bubble exterior solution u with weak radiation concen tration, if suc h a solution existed. Lemma 5.1. Fix a p ositive inte ger J ≥ 2 . Ther e exists an absolute c onstant c 2 and two smal l c onstants c 1 = c 1 ( J ) and τ 0 = τ 0 ( J ) > 0 , such that if a r adial exterior solution u to (CP1) is a 24 J -bubble exterior solution deﬁne d in Se ction 3 (with the p ar ameter c 2 ) satisfying τ . = sup 0 0 1 r 1 / 2 Z r − r | G ( s ) | d s +       χ c 2 λ J   F ( u ) − J X j =1 ζ j F ( W λ j )         L 1 L 2 ( R × R 3 ) < τ 0 , wher e G is the r adiation pr oﬁle of the asymptotic e quivalent fr e e wave v L of u ; ζ j ’s and λ j ’s ar e the c orr esp onding signs and sc ales given by Pr op osition 3.1; then the err or term w = u − J X j =1 ζ j W λ j ( x ) − √ 3 ζ J − 1 λ − 1 / 2 J − 1 φ ( x/λ J ) − v L ( x, t ) and the r adiation pr oﬁle G ∗ asso ciate d to  w (0) satisfy the ine qualities ∥ G ∗ ∥ L 2 ( s :2 k c 2 λ J < | s | < 2 k +1 c 2 λ J ) + ∥ χ 2 k c 2 λ J , 2 k +1 c 2 λ J □ w ∥ L 1 L 2 ≤ 2 k − K 2 τ , k = 0 , 1 , 2 , · · · , K ; ∥  w (0) ∥ H ( c 1 λ J − 1 ) +   χ c 1 λ J − 1 □ w   L 1 L 2 ≤ 2 τ . Her e φ ( x ) is the solution to the el liptic e quation − ∆ φ = 5 W 4 φ + 5 W 4 given in L emma 4.1 with the p ar ameter c = c 2 ; K is the minimal p ositive inte ger such that 2 K +1 c 2 λ J ≥ c 1 λ J − 1 . Pr o of. By dilation w e ma y assume λ J = 1, without loss of generality . W e let c 2 ≫ 1 and c 1 ≪ 1 b e constants, which will b e determined later in the argument. According to Remark 3.3, the inequalities λ − 1 / 2 J − 1 ≲ J τ and 4 c 2 < c 1 λ J − 1 alw ays hold as long as τ < τ ( J, c 2 , c 1 ) is suﬃciently small. Please note that τ ( J, c 2 , c 1 ) here represents a constant determined by J , c 2 and c 1 only , whic h might be diﬀeren t at diﬀerent places. This kind of notations will b e frequently used in the subsequent. Now we compare u with S ∗ ( x, t ) = J X j =1 ζ j W λ j ( x ) + ζ J − 1 √ 3 λ − 1 / 2 J − 1 φ ( x ) + v L ( x, t ) . It is clear that w = u − S ∗ =   u − v L − J X j =1 ζ j W λ j ( x )   − √ 3 ζ J − 1 λ − 1 / 2 J − 1 φ ( x ) It immediately follows from our assumption and the inequality λ − 1 / 2 J − 1 ≲ J τ that ∥  w (0) ∥ H ( c 1 λ J − 1 ) + ∥ χ c 1 λ J − 1 ( □ w ) ∥ L 1 L 2 ≤ τ + √ 3 λ 1 / 2 J − 1  ∥ φ ∥ ˙ H 1 ( { x : | x | >c 1 λ J − 1 } ) + ∥ χ c 1 λ J − 1 ∆ φ ∥ L 1 L 2  ≤ τ + c ( J ) τ  ∥ φ ∥ ˙ H 1 ( { x : | x | >c 1 λ J − 1 } ) + ∥ χ c 1 λ J − 1 ∆ φ ∥ L 1 L 2  . Since we hav e lim r → + ∞  ∥ φ ∥ ˙ H 1 ( { x : | x | >r } ) + ∥ χ r ∆ φ ∥ L 1 L 2  = 0 , the following inequality holds as long as τ < τ ( J, c 2 , c 1 ) is suﬃciently small ∥  w (0) ∥ H ( c 1 λ J − 1 ) + ∥ χ c 1 λ J − 1 ( □ w ) ∥ L 1 L 2 ( R × R 3 ) ≤ 2 τ . (7) 25 A similar argumen t, as well as the uniform boundedness of ∥ χ 0 ∆ φ ∥ given in Remark 4.2, also giv es ∥ χ c 2 ( □ w ) ∥ L 1 L 2 ( R × R 3 ) ≤ τ + √ 3 λ 1 / 2 J − 1 ∥ χ c 2 ∆ φ ∥ L 1 L 2 ≲ J τ . (8) F or con venience w e utilize the notations Ψ k for the following c hannel-like regions Ψ k =  ( x, t ) : | t | + 2 k c 2 < | x | < | t | + 2 k +1 c 2  , k = 0 , 1 , 2 , · · · , K . In order to take the adv an tage of ﬁnite speed of wa v e propagation, we also deﬁne Ψ k,ℓ = { ( x, t ) ∈ Ψ k : | x | + | t | < 2 k + ℓ c 2 } , ℓ = 1 , 2 , · · · , K + 1 − k . Next we introduce a few notations for the norms: a k = ∥ χ Ψ k w ∥ Y ( R ) ; a k,ℓ = ∥ χ Ψ k,ℓ w ∥ Y ( R ) ; as well as b k = ∥ G ∗ ∥ L 2 ( { s :2 k c 2 < | s | < 2 k +1 c 2 } ) + ∥ χ Ψ k ( □ w ) ∥ L 1 L 2 . No w w e pro ve a few inequalities concerning a k , a k,ℓ and b k , which will ﬁnally lead to the con- clusion of Lemma 5.1. First of all, we observe that w is c 2 -w eakly non-radiative. Thus we ma y apply Lemma 2.2 on w , recall (8) and obtain b k ≲ 1 ∥ χ Ψ k ( □ w ) ∥ L 1 L 2 ≲ J τ . (9) A more delicate upp er b ound of b k can be given in terms of a k and a k,ℓ . W e calculate (we use the notation λ = λ J − 1 for conv enience b elo w) □ w = F ( u ) − J X j =1 ζ j F ( W λ j ) − ζ J − 1 √ 3 λ 1 / 2 (5 W 4 φ + 5 W 4 ) = F   w + v L + J X j =1 ζ j W λ j + √ 3 ζ J − 1 λ 1 / 2 φ   − J X j =1 ζ j F ( W λ j ) − ζ J − 1 5 √ 3 λ 1 / 2 ( W 4 φ + W 4 ) . It immediately follows that b k ≲ 1 I 1 + I 2 + · · · + I 7 . with I 1 = ∥ χ Ψ k W 4 w ∥ L 1 L 2 ; I 2 = c ( J )       χ Ψ k   | w | 5 + λ − 2 φ 4 | w | + | v L | 4 | w | + J − 1 X j =1 W 4 λ j | w |         L 1 L 2 ; I 3 = c ( J )       χ Ψ k   | v L | 5 + λ − 2 φ 4 | v L | + J X j =1 W 4 λ j | v L |         L 1 L 2 ; I 4 = c ( J )       χ Ψ k   λ − 5 / 2 | φ | 5 + λ − 1 W 3 | φ | 2 + λ − 1 / 2 J − 1 X j =1 W 4 λ j | φ |         L 1 L 2 ; I 5 =    χ Ψ k W 4 ( W λ − √ 3 λ − 1 / 2 )    L 1 L 2 ; I 6 = c ( J )       χ Ψ k   J − 2 X j =1 W 4 W λ j + J − 1 X j =1 W W 4 λ j + W 3 W 2 λ J − 1         L 1 L 2 ; I 7 = c ( J ) X 1 ≤ j 2 K +1 c 2 } ) + ∥ χ 2 K +1 c 2 □ w ∥ L 1 L 2  ≲ 1 k − 1 X m =0 2 m − k 2 b m + K X m = k 2 k − m 10 b m + 2 k − K 10 τ ; Here we utilize the exterior energy identit y (5) and and exterior upper b ound (7). Similarly we ma y apply ﬁnite sp eed of propagation and deduce a k,ℓ ≲ 1 k − 1 X m =0 2 m − k 2 b m + k + ℓ − 1 X m = k 2 k − m 10 b m , ℓ = 1 , 2 , · · · , K + 1 − k . No w we collect all inequalities ab ov e: b k ≤ c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ a k,ℓ + 2 − 12 5 ( K − k ) a k ! + c ∗ 2  a 5 k + c 2 / 5 1 2 2 5 ( k − K ) a k  + c ∗ 2 c 1 / 2 1 τ 2 k − K 2 ; b k ≤ c ∗ 3 τ ; a k ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 b m + K X m = k 2 k − m 10 b m + 2 k − K 10 τ ! ; a k,ℓ ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 b m + k + ℓ − 1 X m = k 2 k − m 10 b m ! . Here c ∗ 0 = c ∗ 0 (1), c ∗ 1 = c ∗ 1 (1) are absolute constants; and c ∗ 2 = c ∗ 2 ( J ), c ∗ 3 = c ∗ 3 ( J ) are t wo constants determined b y J only . The second inequality follows from (9). W e claim that w e may choose suitable constants c 2 = c 2 (1) and c 1 = c 1 ( J ), τ 0 = τ 0 ( J ), as w ell as another constant γ = γ (1), suc h that if τ < τ 0 is suﬃciently small, then b k ≤ 2 k − K 2 τ ; a k ≤ γ 2 k − K 10 τ a k,ℓ ≤ γ 2 k − K 2 2 2 5 ℓ τ . (11) Please note that the upper bound of b k here immediately giv es the ﬁrst inequality in the conclu- sion of Lemma 5.1. Indeed, w e may ﬁrst choose the constants γ , c 2 > 1 and c 1 < 1 one by one suc h that γ > 20 c ∗ 0 ; c ∗ 1 c − 2 2 γ < 1 10 ; γ c ∗ 2 c 2 / 5 1 < 1 10 ; then choose a suﬃciently small constant τ 0 = τ 0 ( J ) such that (and that τ 0 < τ ( J, c 2 , c 1 ), which guaran tees that all the argument abov e holds if τ < τ 0 ) 16 c ∗ 2 max  ( c ∗ 3 ) 4 , 1  γ 5 τ 4 0 < 1 10 ; Please note that the condition c 1 satisﬁes also implies that c ∗ 2 c 1 / 2 1 < 1 10 . 29 No w we pro ve the inequalities in (11). Let us deﬁne B k = max n 0 , b k − 2 k − K 2 τ o and A k = max n 0 , a k − γ 2 k − K 10 τ o ; A k,ℓ = max n 0 , a k,ℓ − γ 2 k − K 2 2 2 5 ℓ τ o ; and prov e that they satisfy the inequalities: B k ≤ c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ A k,ℓ + 2 − 12 5 ( K − k ) A k ! + 16 c ∗ 2 A 5 k + c ∗ 2 c 2 / 5 1 2 2 5 ( k − K ) A k ; (12) A k ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + K X m = k 2 k − m 10 B m ! ; (13) A k,ℓ ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + k + ℓ − 1 X m = k 2 k − m 10 B m ! . (14) These three inequalities can b e veriﬁed in the same manner. Since all B k , A k and A k,ℓ are nonnegativ e, the ﬁrst inequality is trivial if B k = 0. If B k > 0, then w e m ust hav e b k = B k + 2 k − K 2 τ Inserting this identit y , as well as the inequalities a k ≤ A k + γ 2 k − K 10 τ ; a k,ℓ ≤ A k,ℓ + γ 2 k − K 2 2 2 5 ℓ τ in to the inequalit y b k satisﬁes, we obtain B k + 2 k − K 2 τ ≤ c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ  A k,ℓ + γ 2 k − K 2 2 2 5 ℓ τ  + 2 − 12 5 ( K − k )  A k + γ 2 k − K 10 τ  ! + c ∗ 2   A k + γ 2 k − K 10 τ  5 + c 2 / 5 1 2 2 5 ( k − K )  A k + γ 2 k − K 10 τ   + c ∗ 2 c 1 / 2 1 τ 2 k − K 2 ≤ c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ A k,ℓ + 2 − 12 5 ( K − k ) A k ! + c ∗ 2  16 A 5 k + c 2 / 5 1 2 2 5 ( k − K ) A k  + c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 γ 2 k − K 2 2 − 2 ℓ τ + γ 2 − 5 2 ( K − k ) τ ! + 16 c ∗ 2 γ 5 2 k − K 2 τ 5 + c ∗ 2 c 2 / 5 1 γ 2 k − K 2 τ + c ∗ 2 c 1 / 2 1 2 k − K 2 τ ≤ c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ A k,ℓ + 2 − 12 5 ( K − k ) A k ! + c ∗ 2  16 A 5 k + c 2 / 5 1 2 2 5 ( k − K ) A k  + 2 k − K 2 τ  γ c ∗ 1 c − 2 2 2 − 2 k + γ c ∗ 1 c − 2 2 2 − 2 K + 16 c ∗ 2 γ 5 τ 4 0 + c ∗ 2 c 2 / 5 1 γ + c ∗ 2 c 1 / 2 1  ≤ c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ A k,ℓ + 2 − 12 5 ( K − k ) A k ! + c ∗ 2  16 A 5 k + c 2 / 5 1 2 2 5 ( k − K ) A k  + (1 / 2)2 k − K 2 τ . 30 This veriﬁes the inequalit y concerning B k . Similarly if A k > 0 (or A k,ℓ > 0), then w e ha ve A k + γ 2 k − K 10 τ ≤ c ∗ 0 k − 1 X m =0 2 m − k 2  B m + 2 m − K 2 τ  + K X m = k 2 k − m 10  B m + 2 m − K 2 τ  + 2 k − K 10 τ ! ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + K X m = k 2 k − m 10 B m + k − 1 X m =0 2 2 m − k − K 2 τ + K X m = k 2 4 m + k − 5 K 10 τ + 2 k − K 10 τ ! ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + K X m = k 2 k − m 10 B m ! + c ∗ 0 2 k − K 2 τ + c ∗ 0 2 k − K 10 τ 1 − 2 − 2 5 + c ∗ 0 2 k − K 10 τ ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + K X m = k 2 k − m 10 B m ! + 7 c ∗ 0 2 k − K 10 τ ; and A k,ℓ + γ 2 k − K 2 2 2 5 ℓ τ ≤ c ∗ 0 k − 1 X m =0 2 m − k 2  B m + 2 m − K 2 τ  + k + ℓ − 1 X m = k 2 k − m 10  B m + 2 m − K 2 τ  ! ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + k + ℓ − 1 X m = k 2 k − m 10 B m + k − 1 X m =0 2 2 m − k − K 2 τ + k + ℓ − 1 X m = k 2 4 m + k − 5 K 10 τ ! ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + k + ℓ − 1 X m = k 2 k − m 10 B m ! + c ∗ 0 2 k − K 2 τ + c ∗ 0 2 k − K 2 2 2 5 ( ℓ − 1) τ 1 − 2 − 2 5 ≤ c ∗ 0 k − 1 X m =0 2 m − k 2 B m + k + ℓ − 1 X m = k 2 k − m 10 B m ! + 6 c ∗ 0 2 k − K 2 2 2 5 ℓ τ . These verify the inequalities (12), (13) and (14). Finally we sho w that B k = A k = A k,ℓ = 0, whic h immediately v eriﬁes (11). Indeed, let us consider M = max k =0 , 1 , ··· ,K B k ≤ max k =0 , 1 , ··· ,K b k ≤ c ∗ 3 τ 0 . Inserting this upp er b ound into (13) and (14), we obtain max k,ℓ { A k , A k,ℓ } ≤ c ∗ 0  2 − 1 / 2 1 − 2 − 1 / 2 + 1 1 − 2 − 1 / 10  M ≤ 18 c ∗ 0 M ≤ γ M . W e then insert this into (12) and obtain M = max k B k ≤ max k " c ∗ 1 c − 2 2 2 − 2 k K − k +1 X ℓ =1 2 − 12 5 ℓ γ M + 2 − 12 5 ( K − k ) γ M ! + 16 c ∗ 2 γ 5 M 5 + c ∗ 2 c 2 / 5 1 2 2 5 ( k − K ) γ M # ≤ max k h 2 c ∗ 1 c − 2 2 2 − 2 k γ M + 16 c ∗ 2 γ 5 M 5 + c ∗ 2 c 2 / 5 1 γ M i ≤ 2 c ∗ 1 c − 2 2 γ M + 16 c ∗ 2 γ 5 ( c ∗ 3 τ 0 ) 4 M + c ∗ 2 c 2 / 5 1 γ M ≤ (2 / 5) M . As a result, w e must ha ve M = 0, whic h veriﬁes (11) and ﬁnishes the pro of of the ﬁrst inequality in the conclusion. Finally the second inequality in the conclusion immediately follows from (7). 31 Next we ma y further extend the domain of the approximation giv en in Lemma 5.1. Please note that from now on we alw ays apply the soliton resolution theory (Prop osition 3.1) with the parameter c 2 giv en in Lemma 5.1. In particular, a J -bubble exterior solution (see Section 3) is also deﬁned with this parameter c 2 . Corollary 5.2. Given any p ositive inte ger J ≥ 2 and a p ositive c onstant c 3 < c 2 , ther e exists a smal l c onstant τ 1 = τ 1 ( J, c 3 ) , such that if u is a J -bubble exterior solution to (CP1) deﬁne d in Se ction 3 with τ . = sup 0 0 1 r 1 / 2 Z r − r | G ( s ) | d s < τ 1 , then the err or function w = u − J X j =1 ζ j W λ j ( x ) − ζ J − 1 √ 3 λ − 1 / 2 J − 1 φ ( x/λ J ) − v L ( x, t ) and the r adiation pr oﬁle G ∗ asso ciate d to  w (0) satisfy ∥ G ∗ ∥ L 2 ( s : c 2 λ J < | s | 2 K +1 c 2 ) + ∥ χ 2 K +1 c 2 □ w ∥ L 1 L 2  ≲ 1 b + c ( J, n ) λ − 1 / 2 τ + K X k =0  α n − α n +1 2 k c 2  1 / 10 c ( J )2 k − K 2 τ +  α n − α n +1 2 K +1 c 2  1 / 10 c ( J ) τ ≲ 1 b + c ( n, J ) λ − 1 / 2 τ + c ( J )( α n − α n +1 ) 1 / 10 2 − K/ 10 τ . Similarly we may use the ﬁnite sp eed of propagation to deduce a ℓ ≲ 1 b + c ( n, J ) λ − 1 / 2 τ + ℓ − 1 X k =0  α n − α n +1 2 k c 2  1 / 10 c ( J )2 k − K 2 τ . Th us we ha ve a 0 ≲ 1 b + c ( n, J ) λ − 1 / 2 τ ; a ℓ ≲ 1 b + c ( n, J ) λ − 1 / 2 τ + c ( J )( α n − α n +1 ) 1 / 10 2 − ( ℓ − 1) / 10 2 ( ℓ − 1 − K ) / 2 τ , ℓ = 1 , 2 , · · · , K + 1 . Con versely we may also give an upp er b ound of b in terms of a and a ℓ . By Lemma 2.2, w e hav e b ≲ 1 ∥ χ α n +1 ,α n □ w ∥ L 1 L 2 ≲ 1 I 1 + I 2 + · · · + I 7 . 33 Here I 1 , I 2 , · · · , I 7 are deﬁned in the same manner as in the pro of of Lemma 5.1 I 1 = ∥ χ Ψ W 4 w ∥ L 1 L 2 ; I 2 = c ( J )       χ Ψ   | w | 5 + λ − 2 φ 4 | w | + | v L | 4 | w | + J − 1 X j =1 W 4 λ j | w |         L 1 L 2 ; I 3 = c ( J )       χ Ψ   | v L | 5 + λ − 2 φ 4 | v L | + J X j =1 W 4 λ j | v L |         L 1 L 2 ; I 4 = c ( J )       χ Ψ   λ − 5 / 2 | φ | 5 + λ − 1 W 3 | φ | 2 + λ − 1 / 2 J − 1 X j =1 W 4 λ j | φ |         L 1 L 2 ; I 5 =    χ Ψ W 4 ( W λ − √ 3 λ − 1 / 2 )    L 1 L 2 ; I 6 = c ( J )       χ Ψ   J − 2 X j =1 W 4 W λ j + J − 1 X j =1 W W 4 λ j + W 3 W 2 λ J − 1         L 1 L 2 ; I 7 = c ( J ) X 1 ≤ j 0 1 r 1 / 2 Z r − r | G ( s ) | d s < τ 2 , then the err or function w ∗ = u − J X j =1 ζ j W λ j ( x ) − v L ( x, t ) satisﬁes sup 0 c 3 ) ≲ c 3 λ − 1 / 2 ; (19) ∥ χ c 3 ( □ w ∗ − □ w ) ∥ L 1 L 2 ≲ 1 λ − 1 / 2 . W e deﬁne a = ∥ χ 0 ,c 3 w ∗ ∥ Y ( R ) ; b = ∥ G ∗ ∥ L 2 ( − c 3 ,c 3 ) + ∥ χ 0 ,c 3 □ w ∗ ∥ L 1 L 2 ; and ( ℓ = 0 , 1 , 2 , · · · , K + 1) Ψ ℓ =  ( x, t ) : | t | < | x | < c 3 + | t | , | x | + | t | < c 2 2 ℓ  ; a ℓ = ∥ χ Ψ ℓ w ∗ ∥ Y ( R ) . Let us ﬁrst give a rough upp er bound of b . W e may apply Prop osition 3.1 and Remark 3.4 to deduce b ≲ 1 ∥  w ∗ (0) ∥ ˙ H 1 × L 2 + ∥ χ 0 □ w ∗ ∥ L 1 L 2 ≲ J τ . (20) Next we give more dedicate upp er b ounds of b and a . In order to giv e an upper bound of a , we let w 1 , w 2 b e the radial free wa ve with radiation proﬁles G 1 , G 2 b elo w, resp ectively , G 1 ( s ) =  G ∗ ( s ) , | s | < c 3 ; G ∗ ( s ) , | s | > c 3 ; G 2 ( s ) =  0 , | s | < c 3 ; G ∗ ( s ) − G ∗ ( s ) , | s | > c 3 ; and w 3 , w 4 b e the solution to the follo wing wa ve equation with zero initial data, resp ectively , □ w 3 =    0 , | x | < | t | ; □ w ∗ , ( x, t ) ∈ Ω 0 ,c 3 ; □ w , ( x, t ) ∈ Ω c 3 ; □ w 4 =    0 , | x | < | t | ; 0 , ( x, t ) ∈ Ω 0 ,c 3 ; □ w ∗ − □ w , ( x, t ) ∈ Ω c 3 . It follows from the ﬁnite sp eed of propagation that w ∗ ( x, t ) = w 1 ( x, t ) + w 2 ( x, t ) + w 3 ( x, t ) + w 4 ( x, t ) , ( x, t ) ∈ Ψ . = Ω 0 ,c 3 . W e then apply Remark 2.11 on w 1 , Corollary 2.13 on w 3 , and the regular Strichartz estimates on w 2 , w 4 to deduce a ≤ ∥ χ Ψ w 1 ∥ Y ( R ) + ∥ χ Ψ w 2 ∥ Y ( R ) + ∥ χ Ψ w 3 ∥ Y ( R ) + ∥ χ Ψ w 4 ∥ Y ( R ) ≲ 1 ∥ G ∗ ∥ L 2 ( − c 3 ,c 3 ) + ∥ G ∗ ∥ L 2 ( { s : c 3 < | s | 2 K +1 c 2 } ) + ∥ G ∗ − G ∗ ∥ L 2 ( { s : | s | >c 3 } ) + ∥ χ Ψ □ w ∗ ∥ L 1 L 2 + ∥ χ c 3 ,c 2 □ w ∥ L 1 L 2 + K X k =0  c 3 2 k c 2  1 / 10 ∥ χ 2 k c 2 , 2 k +1 c 2 □ w ∥ L 1 L 2 +  c 3 2 K +1 c 2  1 / 10 ∥ χ 2 K +1 c 2 □ w ∥ L 1 L 2 + ∥ χ c 3 ( □ w ∗ − □ w ) ∥ L 1 L 2 ≲ 1 b + c ( J, c 3 ) λ − 1 / 2 τ + K X k =0  c 3 2 k c 2  1 / 10 c ( J )2 k − K 2 τ +  c 3 2 K +1 c 2  1 / 10 c ( J ) τ + c ( c 3 ) λ − 1 / 2 ≲ 1 b + c ( J, c 3 ) λ − 1 / 2 + c ( J ) c 1 / 10 3 λ − 1 / 10 τ . Here we use 2 K ≃ J λ . Similarly w e may incorp orate the ﬁnite propagation sp eed and obtain a ℓ ≲ 1 b + c ( J, c 3 ) λ − 1 / 2 τ + ℓ − 1 X k =0  c 3 2 k c 2  1 / 10 c ( J )2 k − K 2 τ + c ( c 3 ) λ − 1 / 2 37 Th us a 0 ≲ 1 b + c ( J, c 3 ) λ − 1 / 2 ; a ℓ ≲ 1 b + c ( J, c 3 ) λ − 1 / 2 + c ( J ) c 1 / 10 3 2 − ℓ − 1 10 2 ℓ − 1 − K 2 τ , ℓ = 1 , 2 , · · · , K + 1 No w we giv e an upp er b ound of b . W e apply Lemma 2.2 on w ∗ and obtain b ≲ 1 ∥ χ 0 ,c 3 □ w ∗ ∥ L 1 L 2 ≲ 1       χ 0 ,c 3   F   w ∗ + v L + J X j =1 ζ j W λ j   − J X j =1 ζ j F ( W λ j )         L 1 L 2 ≲ 1 I 1 + I 2 + I 3 + I 4 ; with I 1 = c ( J )   χ 0 ,c 3 W 4 w ∗   L 1 L 2 ; I 2 = c ( J )       χ 0 ,c 3   | w ∗ | 5 + | v L | 4 | w ∗ | + J − 1 X j =1 W 4 λ j | w ∗ |         L 1 L 2 ; I 3 = c ( J )       χ 0 ,c 3   | v L | 5 + J X j =1 W 4 λ j | v L |         L 1 L 2 ; I 4 = c ( J )       χ 0 ,c 3 X 1 ≤ j 0 , such that any r adial J -bubble exterior solution u to (CP1) must satisfy τ . = sup 0 0 1 r 1 / 2 Z r − r | G ( s ) | d s ≥ τ 3 . Her e v L is the r adiation p art of u ; G is the c orr esp onding r adiation pr oﬁle of v L ; λ J − 1 is the size of ( J − 1) -th bubble given by Pr op osition 3.1. Pr o of. Let τ 2 = τ 2 ( J ) and M = M ( J ) b e the constan ts given in Lemma 5.3. Let us consider the appro ximated solution w and w ∗ giv en in Corollary 5.2 and Lemma 5.3 resp ectively . W e recall the asymptotic b ehaviour | φ ( r ) | ≳ 1 r − 1 near the zero (see Lemma 4.1) and obtain r 1 / 2 | w ( r , 0) − w ∗ ( r , 0) | = r 1 / 2 √ 3 λ − 1 / 2 J − 1 | φ ( r /λ J ) | ≳ 1  r λ J  − 1 / 2  λ J λ J − 1  1 / 2 , as long as r /λ J ≪ 1 is suﬃciently small. Thus w e may c ho ose a small constan t c 4 = c 4 ( J ) < c 2 suc h that ( c 4 λ J ) 1 / 2 | w ( c 4 λ J , 0) − w ∗ ( c 4 λ J , 0) | > 2 M ( J )  λ J λ J − 1  1 / 2 . (22) No w let τ 1 = τ 1 ( J, c 4 ) is the constant given in Corollary 5.2, which is unique determined by J . If u is a J -bubble exterior solution with τ < min { τ 1 , τ 2 } , then it follows from Corollary 5.2 and Lemma 5.3 that ∥ G ∗ ∥ L 2 ( s : c 4 λ J < | s | 0 b e a large n umber suc h that ∥  u (0) ∥ H ( R ) ≪ 1. By the small data theory , a standard cut-oﬀ tec hnique and ﬁnite sp eed of propagation, there exists a ﬁnite-energy free wa v e u − L suc h that lim t →−∞ Z | x | >R + | t | |∇ t,x ( u − u − L )( x, t ) | 2 d x = 0 . Let G − b e the radiation proﬁle of u − L in the negativ e time direction. This immediately giv es the (nonlinear) radiation proﬁle in the negativ e time direction lim t →−∞ Z ∞ R − t  | G − ( r + t ) − r u t ( r , t ) | 2 + | G − ( r + t ) − r u r ( r , t ) | 2  d r = 0 . Next we consider the positive time direction. According to Lemma 3.7 of Duyck aerts-Kenig- Merle [9], there exists a ﬁnite-energy free wa v e u L , such that lim t → + ∞ Z | x | >t − A |∇ t,x ( u − u L )( x, t ) | 2 d x = 0 , ∀ A ∈ R . Th us the radiation proﬁle G + of u L in the p ositive time direction b ecomes the (nonlinear) radiation proﬁle of u , i.e. lim t → + ∞ Z ∞ t − A  | G + ( r − t ) − r u t ( r , t ) | 2 + | G + ( r − t ) + r u r ( r , t ) | 2  d r = 0 , ∀ A ∈ R . Let us consider the time-translated solution u ( · , · + t ) for t > R , which is deﬁned at least outside the main light cone, whose (nonlinear) radiation proﬁles can be giv en by G t, + ( s ) = G + ( s − t ) , s > 0; G t, − ( s ) = G − ( s + t ) , s > 0 . (24) W e let v t,L b e the free wa v e whose radiation proﬁles in tw o time directions are equal to G t, + and G t, − for s > 0, resp ectively . It is not diﬃcult to see that v t,L is exactly the asymptotically equiv alent free wa v e of u . W e claim that the following limit holds lim t → + ∞  ∥ χ 0 v t,L ∥ Y ( R ) + sup r> 0 1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s  = 0 . (25) 41 In fact, for any small constant ε > 0, we ma y ﬁnd an interv al [ a, b ] ⊂ R and a large num ber T 0 ≥ R , such that ∥ G + ∥ L 2 ( R \ [ a,b ]) < ε ; ∥ G − ∥ L 2 ([ T 0 , + ∞ )) < ε. No w let us consider very large time t ≫ max { T 0 , − a } . According to Lemma 2.15, we ha ve ∥ χ 0 v t,L ∥ Y ( R ) ≲ 1 ε +  b − a t + a  1 / 2 ∥ G + ∥ L 2 ([ a,b ]) . In addition, we also ha ve 1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s = 1 r 1 / 2  Z − t + r − t | G + ( s ) | d s + Z t + r t | G − ( s ) | d s  . If r < t + a , then the in terv al [ − t, − t + r ] do es not intersects with [ a, b ], thus the Cauc hy-Sc hw arz inequalit y gives 1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s ≲ 1 ε. On the other hand, if r > t + a , then we hav e 1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s ≲ 1 ε + 1 r 1 / 2 Z b a | G + ( s ) | d s In summary , we ha ve sup r> 0  1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s  ≲ 1 ε +  b − a t + a  1 / 2 ∥ G + ∥ L 2 ([ a,b ]) . As a result, the following inequality holds lim sup t → + ∞  ∥ χ 0 v t,L ∥ Y ( R ) + sup r> 0 1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s  ≲ 1 ε. Since ε > 0 is arbitrary , the limit (25) immediately follows. In addition, our assumption that the soliton resolution of u comes with J bubbles implies that the time translated solution u ( · , · + t ) is a J -bubble exterior solution as deﬁned at the end of Section 3 for suﬃcien tly large time. In fact, almost orthogonality of decoupled bubbles imply that the following energy estimates hold as long as t is suﬃcien tly large: • If the soliton resolution of u ( · , · + t ) given b y Prop osition 3.1 (with n = J + 1) is incom- plete(i.e. in case b), then ∥  u ( t ) −  v t,L (0) ∥ 2 ˙ H 1 × L 2 > J ∥ W ∥ 2 ˙ H 1 + 1 2 ∥ W ∥ 2 ˙ H 1 ( { x : | x | >c 2 } ) . • If the soliton resolution of u ( · , · + t ) given by Proposition 3.1 comes with exactly J 1 bubbles for some J 1 ∈ { 0 , 1 , · · · , J } , then    ∥  u ( t ) −  v t,L (0) ∥ 2 ˙ H 1 × L 2 − J 1 | W ∥ 2 ˙ H 1    < 1 2 ∥ W ∥ 2 ˙ H 1 ( { x : | x | >c 2 } ) . A comparison of radiation proﬁles also shows that ∥  v t,L (0) −  u L ( t ) ∥ ˙ H 1 × L 2 ≲ 1 ∥ G − ∥ L 2 ([ t, + ∞ )) + ∥ G + ∥ L 2 (( −∞ , − t ]) → 0 , t → + ∞ , 42 whic h, as w ell as the soliton resolution as t → + ∞ , implies that lim t → + ∞ ∥  u ( t ) −  v t,L (0) ∥ ˙ H 1 × L 2 = lim t → + ∞ ∥  u ( t ) −  u L ( t ) ∥ ˙ H 1 × L 2 = √ J ∥ W ∥ ˙ H 1 . As a result, u ( · , · + t ) must b e a J -bubble exterior solution for suﬃciently large time t . It immediately follows from Prop osition 6.1 that sup 0 0  1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s  ≥ τ 3 , ∀ t ≫ 1 . Here λ J − 1 ( t ) is the size of the ( J − 1)-th bubble, as given in Prop osition 3.1. Combining this with (24) and (25), we obtain sup 0 τ 3 2 , t ≫ 1 . No w we let g + and g − b e the (right hand side) maximal functions of | G + | 2 , | G − | 2 resp ectiv ely g + ( − t ) = sup r> 0 1 r Z − t + r − t | G + ( s ) | 2 d s ; g − ( t ) = sup r> 0 1 r Z t + r t | G − ( s ) | 2 d s. (26) The low er b ound ab o ve implies that the inequality g + ( − t ) + g − ( t ) ≥ τ 2 3 4 λ J − 1 ( t ) − 1 (27) holds for all time t > T . Here T ≥ R is a large time. W e recall | G + ( s ) | 2 ∈ L 1 ( R ), | G − ( s ) | 2 ∈ L 1 ([ R, + ∞ )) and the fact that the maximal function is of weak (1 , 1) type to deduce that there exists a constant C , such that |{ t ∈ [ T , + ∞ ) : g + ( − t ) > κ }| ≤ C κ , ∀ κ > 0; |{ t ∈ [ T , + ∞ ) : g − ( t ) > κ }| ≤ C κ , ∀ κ > 0 . Here the notation | · | represents the Leb esgue measure of a subset of R . A com bination of these inequalities with (27) yields      t ∈ [ T , + ∞ ) : τ 2 3 4 λ J − 1 ( t ) − 1 > 2 κ      ≤ 2 C κ , ∀ κ > 0 . W e ma y simplify it and write it in the form of |{ t ∈ [ T , + ∞ ) : λ J − 1 ( t ) < η }| ≤ C ∗ η , ∀ η > 0 . (28) Here C ∗ is a constant indep endent of η > 0. Since the soliton resolution implies that (see Remark 6.2 b elo w) lim t → + ∞ λ J − 1 ( t ) t = 0 , there exists a num b er T ∗ > T , suc h that λ J − 1 ( t ) < t 2 C ∗ , t ≥ T ∗ . As a result, we alwa ys hav e λ J − 1 ( t ) < η for all t ∈ [ T ∗ , 2 C ∗ η ], as long as η is suﬃciently large. Th us we ha ve |{ t ∈ [ T , + ∞ ) : λ J − 1 ( t ) < η }| ≥ 2 C ∗ η − T ∗ , ∀ η ≫ 1 . This gives a con tradiction with (28) and ﬁnishes the pro of in the case of global solution. 43 The type I I blow-up case W e ma y assume the radiation part u L in the soliton resolution comes with a very small energy norm ∥  u L ( T + ) ∥ ˙ H 1 × L 2 < ε. b y applying a cut-oﬀ technique if necessary . Here ε = ε ( J ) ≪ τ 2 ( J ) is a suﬃciently small constan t. W e might further reduce the upp er b ound of ε in the argumen t b elow but the upp er b ound alw ays dep ends on J only . W e may deﬁne the solution u in the exterior region { ( x, t ) : | x | > | T − T + |} b y solving (CP1) with initial data  u L ( T + ). Finite sp eed of propagation sho ws that the new exterior solution coincides with the original solution wherever b oth solutions are deﬁned. By small data theory , we also ha ve sup t ∈ R ∥  u ( t ) ∥ H ( | t − T + | ) ≤ 2 ε. Th us we ma y ﬁx a time t 0 sligh tly smaller than T + and ﬁnd a small num b er r 0 > 0, suc h that ∥  u ( t 0 ) ∥ H ( T + − t 0 − r 0 ) ≤ 3 ε. Again the small data theory implies that u can also b e deﬁned in the region { ( x, t ) : t < t 0 , | x | > T + − t − r 0 } with sup t ≤ t 0 ∥  u ( t ) ∥ H ( T + − t − r 0 ) ≤ 4 ε. In summary , we ma y deﬁne the (nonlinear) radiation proﬁle G ± of u with ∥ G + ∥ L 2 ( − T + , + ∞ ) ≲ 1 ε, ∥ G − ∥ L 2 ( T + − r 0 , + ∞ ) ≲ 1 ε. (29) As a result, the time-translated solution u ( · , · + t ) is asymptotically equiv alen t to a free wa v e v t,L , whose radiation proﬁles can b e giv en by G t, + ( s ) = G + ( s − t ) , s > 0; G t, − ( s ) = G − ( s + t ) , s > 0 . By the Strichartz estimates and (29), the following inequalit y ∥ χ 0 v t,L ∥ Y ( R ) + sup t> 0  1 r 1 / 2 Z r 0 ( | G t, + ( s ) | + | G t, − ( s ) | ) d s  ≲ 1 ∥ G + ∥ L 2 ( − t, + ∞ ) + ∥ G − ∥ L 2 ( t, + ∞ ) ≲ 1 ε. (30) holds for t ∈ [ T + − r 0 , T + ). F ollowing a similar argument to the global case and using the con tinuit y of ∥  u ( t ) −  v t,L (0) ∥ ˙ H 1 × L 2 , we may show that u ( · , · + t ) must b e a J -bubble exterior solution for these times t , as long as ε < ε ( J ) is suﬃcien tly sm all. It follows from Prop osition 6.1, our assumption ε ≪ τ 3 and (30) that sup 0 τ 3 2 , t ∈ [ T + − r 0 , T + ) . The same argument as in the case of global solutions shows that there exists a constant C ∗ > 0 suc h that |{ t ∈ [ T + − r 0 , T + ) : λ J − 1 ( t ) < η }| ≤ C ∗ η , ∀ η > 0 . (31) W e recall that the following holds in the soliton resolution lim t → T + λ J − 1 ( t ) T + − t = 0 , 44 th us there exists a small constant r 1 < r 0 , such that λ J − 1 ( t ) < 1 2 C ∗ ( T + − t ) , ∀ t ∈ [ T + − r 1 , T + ) . As a result, w e alwa ys ha ve λ J − 1 ( t ) < η for all t ∈ [ T + − 2 C ∗ η , T + ), as long as η is suﬃcien tly small, which immediately gives a contradiction with (31) and ﬁnishes the proof in the type I I blo w-up case. Remark 6.2. The sc ale λ J − 1 ( t ) in the ar gument ab ove is given by Pr op osition 3.1, which is not ne c essarily the same as the sc ale λ ∗ j ( t ) in the soliton r esolution  u ( t ) = J X j =1 ζ ∗ j ( W λ ∗ j ( t ) , 0) +  u L ( t ) + o (1) , t → T + . However, when t is suﬃciently close to T + , we may apply Pr op osition 3.1 and utilize the soliton r esolution to de duc e       J X j =1 ζ ∗ j ( W λ ∗ j ( t ) , 0) − J X j =1 ζ j ( W λ j ( t ) , 0)       ˙ H 1 × L 2 =        u ( t ) −  u L ( t ) − o (1) − J X j =1 ζ j ( W λ j ( t ) , 0)       ˙ H 1 × L 2 ≤        u ( t ) −  v t,L (0) − J X j =1 ζ j ( W λ j ( t ) , 0)       ˙ H 1 × L 2 + ∥  v t,L (0) −  u L ( t ) − o (1) ∥ ˙ H 1 × L 2 ≲ J ε ; and ( j = 1 , 2 , · · · , J − 1 ) λ j +1 ( t ) λ j ( t ) ≲ J ε 2 ; λ ∗ j +1 ( t ) λ ∗ j ( t ) < ε 2 . This imme diately gives λ j ( t ) ≃ 1 λ ∗ j ( t ) , as long as ε < ε ( J ) is suﬃciently smal l. As a r esult, we stil l have lim t → T + λ j ( t ) T + − t = 0 , j = 1 , 2 , · · · , J. The situation of glob al solutions is similar (and even b etter). In fact, in the glob al c ase we have lim t → + ∞       J X j =1 ζ ∗ j ( W λ ∗ j ( t ) , 0) − J X j =1 ζ j ( W λ j ( t ) , 0)       ˙ H 1 × L 2 = 0; lim t → + ∞ λ j +1 ( t ) λ j ( t ) + λ ∗ j +1 ( t ) λ ∗ j ( t ) ! = 0 , j = 1 , 2 , · · · , J − 1; which implies that lim t → + ∞ λ j ( t ) λ ∗ j ( t ) = 1; = ⇒ lim t → + ∞ λ j ( t ) t = 0 , j = 1 , 2 , · · · , J. Ac kno wledgemen t The author is ﬁnancially supported by National Natural Science F oundation of China Pro ject 12471230. 45 References [1] C. Collot, T. Duyc k aerts, C. Kenig and F. Merle. “Soliton resolution for the radial quadratic w av e e quation in space dimension 6.” Vietnam Journal of Mathematics 52(2024), no. 3: 735- 773. [2] R. Cˆ ote, and C. Laurent. “Concen tration close to the cone for linear wa ves.” R evista Matem´ atic a Ib er o americ ana 40(2024): 201-250. [3] R. Cˆ ote, C.E. Kenig and W. Schlag. “Energy partition for linear radial wa v e equation.” Mathematische A nnalen 358, 3-4(2014): 573-607. 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Nonexistence of multi-bubble radial solutions to the 3D energy critical wave equation

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