STRICHARTZ ESTIMATES AND STRAUSS CONJECTURE ON VARIOUS SETTINGS. Xin Yu. Philosophy. Baltimore, Maryland. April, All rights reserved

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1 STRICHARTZ ESTIMATES AND STRAUSS CONJECTURE ON VARIOUS SETTINGS by Xin Yu A dissertation submitted to The Johns Hokins University in conformity with the requirements for the degree of Doctor of Philosohy Baltimore, Maryland Aril, 2011 c Xin Yu 2011 All rights reserved

2 Abstract In this thesis, I include two works both focusing on existence and regularity of solutions of semilinear wave equations. In the first work I rovide general Strichartz estimates for certain erturbed wave equation, equied with these estimates I am able to verify the Strauss Conjecture for semilinear wave equations with 2 obstacles when n = 3. I also obtain the shar life san for the subcritical case n =3, 2 << c,byusinga real interolation method. In the second work (joint with Chengbo Wang), we verify the Strauss conjecture for semilinear wave equations on asymtotically Euclidean manifolds when n=3,4, we also give an almost shar life san for the subcritical case when n=3. The main ingredients include a KSS tye estimate with 0 <μ<1/2 and weighted Strichartz estimates of order two. Readers: Dr. Christoher Sogge (Advisor) and Dr. Chengbo Wang. ii

3 Acknowledgments First and for most, I would like to exress my dee gratitude to my advisor, Dr. Christoher Sogge. He has always been atient, warm-hearted and encouraging during this study. Secondly, I would like to thank my collaborator Chengbo Wang. His knowledge and great ersonality imressed me, and I have learned a lot from him. I would also like to thank Sinan Ariturk, I owe him eternal gratitude for roofreading two of my aers. Thirdly, I would like to thank the many staff and faculty at Johns Hokins who have assisted in large ways and small throughout this rocess: Jian Kong, Sabrina Raymond, Charlene Poole, Linda Buckner, Morris Hunt, Williams Minicozzi, Chikako Mese, Richard Brown, Hans Lindblad and many others. Additionally, I would like to extend heartfelt thanks to all my fellow graduate students including ost-graduates Jin-Cheng Jiang,Yifei Chen and Qi Zhong who have made the graduate life easier and more fun. I dedicate this dissertation to my arents as well as Jiming and Aria. iii

4 Contents Abstract ii Acknowledgments iii 1 Introduction Background Preliminaries and Main Results Part I Strauss Conjecture on Perturbed Wave Equations Local in time Strichartz Estimates in R n Strichartz Estimates on Ω Strauss Conjecture when n = 3, Part II Strauss Conjecture on Asymtotically Euclidean Manifolds A secial Case: 3-D, >1+ 2, the metric g is sherically symmetric Weighted Strichartz and Energy Estimates Local in time Strichartz Estimates Strauss Conjecture when < c,ρ>2,n= Bibliograhy 57 Vitae 61 iv

5 Chater 1 Introduction 1.1 Background The wave equation is a hyerbolic artial differential equation which occurs in many fields such as acoustics, electromagnetics, and fluid dynamics. For the Cauchy roblem of linear wave equations in Minkowski sace R 1+n, (1.1) u(t, x) =( t 2 Δ)u(t, x) =0, (t, x) R + R n, u(0,x)=f, t u(0,x)=g, x R n, it is known that we have a direct formula for the solution. Exlicitly, suose (f,g) C (n+3)/2 C (n+1)/2, when the dimension n is odd, by using the method of sherical means we can get a unique C 2 solution as follows u(t, x) = (n 2) [ t ( 1 t t ) n 3 2 t n 2 A t f(x)+( 1 t t ) n 3 2 t n 2 A t g(x)]. 1

6 When the dimension n is even, we can also get the solution by Hadamard s method of descent, (1.2) u(t, x) = 1 [ ( 1 ) n 2 2 t n (n 1)ω n t t t y <1 ( 1 t f(x + ty) 1 y 2 dy+ ) n 2 2 t n 1 t y <1 g(x + ty) 1 y 2 dy]. One can obtain much information such as Huygen s rincile and Comarison Theorem on the wave from the fundamental solutions above. However, in ractise, the initial data (f,g) usually do not have the required smoothness, in which case we need the following solution exression in distribution form u(t, x) =cos(td)f + sin(td) t D g + 0 sin(t s)d F (s, )ds, D where D = Δ, and we have added the forcing term F, i.e. we have u = F. This exression is obtained simly by Fourier transform on the equation. Note that when the initial data is smooth enough the two exressions of solutions coincide with each other. When the initial data has weaker regularity, we can still set u fine existence theory. Secifically, for the Cauchy roblem of the wave equation in Minkowski sace (1.3) u(t, x) =F (t, x), (t, x) R + R n u(0,x)=f, t u(0,x)=g, x R n, by utilizing the idea of weak solution and an energy estimate, we have a unique solution with Sobolev regularity H s as long as (f,g,f) has corresonding norm bound. A more comlicated case is when the forcing term F relies on the wave function itself. 2

7 In my graduate study I mainly study the system of semilinear wave equations, (1.4) ( t 2 Δ g )u(t, x) =F (u), (t, x) R + R n u(0,x)=f, t u(0,x)=g, x R n, where F (u) acts like u. There are two main kinds of existence roblems to consider, one is when the initial data is large, with techniques involving conservation laws and monotonicity formula methods. The other is when the initial data is small, with techniques involving erturbation methods and continuity arguments. I mainly work on the latter roblem with small initial data, which has the advantage that we can control the nonlinear term by emloying the finite seed of roagation roerty of wave, then combined with the result for linear systems, we can exect at least a local existence result of the nonlinear system. In Minkowski sace R 1+n, the work is initiated by John [11], who showed that for small C0 data global solutions of (1.4) always exist when >1+ 2 but not when <1+ 2 when the dimension is 3. Later on Strauss [22] conjectured that for a general n there should be a global solution for small data when > c, but blow u below c. Here the critical ower c is defined as the ositive root of the equation (n 1) 2 (n +1) 2=0. This conjecture is surrising but has been verified by subsequent works on Euclidean sace, and ended by Geogiev, Lindblad and Sogge [4] and Tataru [23]. It is known that > c is necessary for global existence, even with small data, see [18], [25], [29] and reference therein. Moreover, when n 3and c, the shar life san is known in Zhou [28] (see also [12] for lower bound of the life san c and n 3, and [30] for uer bound of the life san when < c and n 3). On more comlicated manifolds, 3

8 however, there has not been much work until very recently. My first work is to study the semilinear equation erturbed by obstacles, so we consider the initial-boundary roblem (1.5) ( t 2 Δ g)u(t, x) =F (u), (t, x) R + Ω Bu Ω =0 u(0,x)=f, t u(0,x)=g, where Ω is the comlement of several obstacles, and B is identity or inward normal derivative oerator. This work is insired by the work of Hidano, Metcalfe, Smith, Sogge and Zhou (HMSSZ)[6], who obtained global solutions for subcritical owers when there is one nontraing obstacle and n =3, 4. With an obstacle near the origin, we do not automatically get the same estimates as in Euclidean sace due to reflections and refractions, but with a star-shaed or nontraing assumtion, many works have shown that there will be a local energy decay. By utilizing the local decay one can get around the difficulty near the origin and verify the Strauss conjecture by combining the Strichartz estimates in classical theory. This is basically what HMSSZ [6] resents. My work generalizes the results by allowing some traed rays, the key oint is to use a weaker energy decay roved by Ikawa [9]. Moreover, I obtained the shar life san for the local solution when < c and n =3onR Ω, which corresonds the result in Euclidean sace by Lindblad, while uses a different method of real interolation. My second work is a collaboration with Chengbo Wang. We focus on another kind of erturbation of wave equations and consider variable coefficients wave equation. Secifically, we consider the roblem in asymtotically Euclidean sace with metric g, inwhich g ij δ ij acts like x ρ with ρ>0, hence the erturbation is bounded near the origin and tends to be flat away from the origin. There is not much activities on this toic yet, following revious works of [2] and [21], we verified the Strauss conjecture for solutions of (1.4) in low dimensions. The recise results will be stated in the next section. 4

9 1.2 Preliminaries and Main Results We use to denote the satial gradient u =( x1 u,, xn u). If α =(α 1,,α n )is multi-indexed and f is a function in R n,then α f = α f α. 1 x 1 x αn n The homogeneous Sobolev sace Ḣγ is defined as ( h Ḣγ (R n ) = D γ h L 2 x (R n ) = ξ γ ĥ(ξ) ) 1/2 2 dξ. R n We also define the mixed-norm sace ( ( ) q/ ) 1/q h L q r L = h(rω) dσ(ω) ω r n 1 dr 0 S n 1 for finite exonents and h L r L ω ( ) 1/. =su h(rω) dσ(ω) r>0 S n 1 Let Z be generators of Lorentz grou excluding the boost oerators, i.e. Z = { i, Ω jk, 1 i n, 1 j<k n}, where Ω ij = x i j x j i is the rotational vector field. Set Γ={ t,z}. 5

10 We write fg if f Cg,whereC is a constant. We also assume >1, and set s c = n 2 2 1, s d = In my first work we will use the following notations for the system (1.5) Ω=R n or R n \Ω is a subset of x <Rwith smooth boundary such that a secial local energy decay is satisfied. Δ g = ij 1 det g i det gg ij j is the Lalace-Beltrami oerator. The Riemannian metric g jk (x) =δ jk (x) for x >R. B = I or v, v is the inward normal vector. We also ose the assumtion on the domain below. Hyothesis B If (f,g,f) vanish for x >R,thenfor ɛ >0, u L 2 t H 1 ([0,S] { x <R}) + t u L 2 t,x ([0,S] { x <R}) f Ḣ1+ɛ + g Ḣɛ + F L 2 t Ḣ ɛ. Here S = T or +. Later on we will see that this Hyothesis is satisfied in the case I shall consider. Next we rovide a technical definition of a Sobolev-tye norm, h Hγ ɛ (R n ) = D γ (1 Δ) ɛ/2 h L 2 x (R n ) = ξ γ (1 + ξ 2 ) ɛ/2 ĥ(ξ) 2 dξ. R n Now the main existence result in my first work is as follows. Main Theorem 1. Let n =3, 4, assume Hyothesis B. Also assume 2 u i u i F (u) u. i=1 6

11 Then if > c,γ = s c, there exists a global solution with (Z α u, t Z α u) Ḣγ Ḣγ 1, whenever ( ) Z α f + Hγ Zα g 2ɛ H γ 1 <ɛ 2ɛ ɛ is sufficiently small. Moreover, when n =3,2<< c and γ = s d, we still have an almost global solution with life san T = Cɛ ( 1)/(2 2 1). As a remark, we state that Hyothesis B is satisfied when there are two obstacles far aart. Exlicitly, Ikawa [9] managed to show that solutions of (1.5) with n =3, Δ g =Δ, B = I, andf (u) = 0 have exonential decay estimates with a loss of 2 derivatives of data. Interolating between that estimate and the energy estimate we get an estimate of the form: u (t, x) L 2 x ( x <1) e ct u (0,x) Ḣɛ ( x <1), for any ositive number ɛ, which imlies the local energy decay (Hyothesis B), so we will have the global and local existence results in this case. We also note that the life san given here for < c is shar based on revious works of John, Lindblad and Zhou on Euclidean sace. The roof of the existence result relies on some generalized Strichartz estimates, which will be discussed in Chater 2. For the moment we turn to the second work and resent the main result. We study the semilinear wave equation system (1.4) on asymtotically Euclidean manifolds (R n,g) with small data. Secifically, for the metric n g = g ij (x) dx i dx j, i,j=1 7

12 we suose g ij (x) C (R n )and,forsomeρ>0, (H1) α N n α x (g ij δ ij )=O( x α ρ ), with δ ij = δ ij being the Kronecker delta function. We also assume that (H2) g is non-traing. Let g(x) =(det(g)) 1/4. The Lalace Beltrami oerator associated with g is given by Δ g = ij 1 g 2 ig ij g 2 j, where g ij (x) denotes the inverse matrix of g ij (x). We also define the corresonding vector fields i = i g 1, Ω ij =Ω ij g 1. Now we can state our main results. Main Theorem 2. Suose (H1) and (H2) hold with ρ>2, n =3, 4, and c << 1+4/(n 1). Then for any ɛ>0 such that (1.6) s = s c ɛ (s d, 1/2) there is a δ>0 deending on so that (1.4) has a global solution satisfying (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣs 1, α 2, t R +, whenever the initial data satisfies (1.7) ( Z α f Ḣs + Z α g Ḣs 1 ) <δ. On the other hand, if n =3and2 < c =1+ 2. Then there still exists a solution 8

13 in [0,T δ ] R 3 under the same data assumtion, with (1.8) s = s d,t δ = cδ ( 1) ɛ, for any small ɛ > 0. As a remark, we state that if we relax the condition on ρ to ρ>1, we still have the corresonding existence result when n = 3 with smallness of Cauchy data with first order derivatives. We will state this work in details in Chater 3. 9

14 Chater 2 Part I Strauss Conjecture on Perturbed Wave Equations As in the classical existence results, the key ingredient of the roof of Main theorem 1 is different variations of Strichartz estimates. Recall that the classical mix-norm Strichartz estimates for the solution of (1.3) in Euclidean sace is u L q f t Lr x Ḣ + g γ Ḣγ 1 + F L q t L r x where n 2, (q, r, γ) and( q, r, 1 γ) are admissible airs, i.e. 2/q +(n 1)/r =(n 1)/2 and 1/q + n/r = n/2 γ. The next two sections are devoted to local and global Strichartz estimates to be used. 2.1 Local in time Strichartz Estimates in R n The finite time Strichartz Estimates will be used to rove the local existence result of solutions, and is stated below. Theorem Let u be a solution of (1.3). Letγ = s d,if0 <a 1/2, 2 <<, and 10

15 let S T =[0,T] R n,then x ( n+2a)/+(n 1)/2 u L t L rl 2 ω (S T ) T 2a/ ( f Ḣγ + g Ḣγ 1 + x n/2+1 γ F L 1 t L 1 rl 2 ω(s T )). The roof of this Theorem involves an interolation between the following two estimates. KSS estimate: x 1/2+a e itd f L 2 t,x T a f L 2 x, 0 <a 1/2. Endoint Trace lemma: x (n 1)/2 e itd f L t,r L 2 f. ω Ḃ 1/2 2,1 The KSS estimate originated by Keel, Smith and Sogge is a direct result of local energy decay and a scaling argument for a artition of {x :0< x < 1}. See [5] for details. The endoint Trace lemma is due to Fang and Wang [3]. Also note that the the homogeneous Besov sace Ḃs,q is defined as f Ḃs,q = 2 js P j f l q j (j Z)L x, where f = j P jf is the Littlewood-Paley decomosition. Next we will cite some results and notations on real interolation theory, which can be found in [1] and [24]. The advantage of the real interolation over the comlex interolation is that we can get a Sobolev norm out of two Besov norms. Let A 0,A 1 be Banach saces, define the real interolation sace (A 0,A 1 ) θ,q for 0 <θ<1and1 q via the norm, ( 1/q, a (A0,A 1 ) θ,q = a (A0,A 1 ) θ,q;k = (t θ K(t, a)) dt/t) q 0 where K(t, a) = inf ( a 0 A0 + a 1 A1 ). a=a 0 +a 1 11

16 Then we have the fact ( (L 0 t,rl 2 w,w 0 (r)dtdrdω ), ( L 1 t,rl 2 ω,w 1 (r)dtdrdω )) θ, = ( L t,rl 2 ω,w(r)dtdrdω ), if 1/ =(1 θ)/ 0 + θ/ 1,w(r) =w (1 θ)/ 0 0 w θ/ 1 1.And (B s 0 q 0,B s 1 q 1 ) θ,r = B s r, if s 0 s 1,r,q 0,q 1 1,s =(1 θ)s 0 + θs 1. (B s 0 q 0,B s 1 q 1 ) θ,2 (B s 0 q 0,B s 1 q 1 ) θ,r, if r 2. Now we turn to the roof of Theorem Let θ =1 2/, notethatḣs = Ḃs 2,2,thenby the real interolation between the KSS estimate and Endoint Trace lemma given above with exonents (θ, ), we get for the homogeneous art, Homogeneous art of LHS = x ( n+2a)/+(n 1)/2 u L t L rl 2 ω (S T ) ) T ( f 2a/ ( + g Ḃ2,2 0,Ḃ1/2 2,1 ) θ, ( Ḃ 1 2,2,Ḃ 1/2 2,1 ) θ, ) T ( f 2a/ ( + g Ḃ2,2 0,Ḃ1/2 2,1 ) θ,2 ( Ḃ 1 2,2,Ḃ 1/2 2,1 ) θ,2 = T 2a/ ( f Ḣ1/2 1/ + g Ḣ 1/2 1/) =RHS. As for the inhomogeneous art F, we first aly Duhamel s rincile to get Inhomogeneous art of LHS F L 1 t Ḣ 1/2 1/. Then recall the Trace lemma r n/2 s v L r L 2 v ω Ḣs, 1/2 <s<n/2, 12

17 by duality we get F L 1 t Ḣ 1/2 1/ x 1+/2 F L 1 t L 1 r L2 ω (S T ), which finishes the roof of the local in time Strichartz estimates. 2.2 Strichartz Estimates on Ω In Minkowski sace, it is seen from [3] and [6] that Strauss conjecture when n =3, 4can be verified by utilizing the following weighted Strichartz estimates x n 2 n+1 γ u L t L rl (f,g) 2 ω (Ḣγ,Ḣγ 1 ) + n x 2 +1 γ F L 1 t L 1 r L2 ω if 1/2 1/ < γ < n/2 1/. When there is an obstacle, however, the classical Strichartz estimate does not naturally hold in the long run due to reflection rays. Borrowing the idea of [6], we observe that if we localize the time, by finite roogation roerty of wave, the solution away from the origin will behave as in the Minkowski sace. As for the solution near the origin, there is an energy estimate available. Combining the two kinds of estimates, we will have control of the solution at least local in time. Next based on the argument in Smith and Sogge [19], by using the local energy decay, we exect to get a Strichartz-tye estimate for erturbed wave equation from the corresonding one local in timeandglobalintimeinr 1+n. Now I will resent the idea discussed above in details, starting with the definition of the generalized admissible air. Definition We say that (X, γ, η, ) is almost admissible if it satisfies i), Minkowski almost Strichartz estimates ( ) (2.1) u L t )A(S) X([0,S] Rn f Ḣγ (R n ) + g Ḣ γ 1 (R n ), where A(S) is a function of S and equals a constant when S =. 13

18 ii), Local almost Strichartz estimates for Ω (2.2) u L t X([0,1] Ω) f Hγ η (Ω) + g H γ 1 η (Ω). The generalized Strichartz estimates to be obtained is as follows. Theorem Assume Hyothesis B, n>2, >2, γ [ n 3 admissible, then ( (2.3) u L t X([0,S] Ω) A(S) f Hγ ɛ+η + g Hγ 1 ɛ+η, n ). ), and (X, γ, η, ) When the forcing term F is added, by a TT argument and the fact ( H γ ɛ ) = H γ ɛ, we can easily get the inhomogeneous version of the Strichartz estimates. Corollary Under the same condition, and (Y,1 γ,η,r) admissible, then ( ) u L t X([0,S] Ω) A(S) f + g Hγ ɛ+η H γ 1 ɛ+η + A 2 (S) (1 Δ g ) ɛ+η F L r t Y ([0,S] Ω). We will use the idea stated at the begining of the section to rove Theorem 2.2.2, so the difficulty lies in the control of so-called commutator terms, which comes out when we try to commute and function multilier β. For examle, if u =0,then(1 β)u solves the wave equation (1 β)u =Δβ u +2 β u, which roduces extra commutator term G =Δβ u +2 β u to be controlled. We mainly use the following two Proositions to fulfil this job. 14

19 Proosition Let w be a solution of (1.3) with f = g = 0, and assume that (2.1) is valid whenever v is a solution of the homogeneous wave equation. Assume further that >2,γ (n 3)/2. Then, if F (t, x) =0 if x > 2R, we have (2.4) w L t X([0,S] Rn )A(S) F L 2 t Ḣ γ 1 ([0,S] R n ). To rove the roosition, we use the distributional form of solution u(t, x) =cos(td)f +sin td D g + t 0 sin(t s)d F (s, )ds. D By Christ-Kiselev Lemma we can aly the Minkowski Strichartz estimates, then it suffices to show S e is D D 1+γ β( )(1 Δ) (1 γ)/2 H(s, )ds L 2 (R n ) H L 2 (R + R n ). 0 But this is just the duality of the local energy decay βe itd f L 2 t H γ f Ḣ n 1 γ, if γ, 2 whichisrovidedin[19]. The second roosition to bound the commutator terms rely on Hyothesis B. Proosition (Energy Decay) IfF is suorted in x <R, γ < n 1 2. Assume 15

20 Hyothesis B. Let β C 0 (R n ) equals 1 on a neighborhood of R n \Ω. (2.5) u L t Ḣ γ B ([0,S] Ω) + tu L t Ḣ γ 1 B ([0,S] Ω) + βu L 2 t H γ B ([0,S] Ω) + β t u L 2 t H γ 1 B ([0,S] Ω) f Hγ ɛ (Ω) + g H ɛ γ 1 (Ω) + F L 2 t Ḣγ+ɛ 1 B ([0,S] Ω). FirstnotethatthesaceH γ B (Ω) is the usual Dirichlet sace with comatibility conditions on boundary of Ω satisfied, and we are assuming that the required comatibility conditions on data are meet throughout the work(see for examle in [19]), therefore write H γ (Ω) for short elsewhere. To rove this estimate, note that f,g, and F are suorted in a ball, the L 2 t estimate in the case γ = 1 is just Hyothesis B, and then by ellitic regularity of the oerator Δ g, βu L 2 t H 3 x + β tu L 2 t H 2 x Δ g (βu) L 2 t H 1 x + βu L 2 t H1 x + Δ g (β t u) L 2 t L 2 x + β tu L 2 t L 2 x βδ g u L 2 t H 1 x + [Δ g,β]u L 2 t H 1 x + βu L 2 t H1 x (2.6) + β t Δ g u L 2 t L 2 x + [Δ g,β] t u L 2 t L 2 x + β tu L 2 t L 2 x Since Δ g u solves the equation with data (Δ g f,δ g g)andforcingtermδ g F,weget βδ g u L 2 t H 1 x + β tδ g u L 2 t L 2 x Δ g f Ḣ1+ɛ x + Δ g g Ḣɛ x + Δ g F L 2 t Ḣ ɛ x (2.7) f Ḣ3+ɛ x + gḣ2+ɛ x + F L 2 t Ḣ 2+ɛ x. Also, notice that [Δ g,β]u = β 1 x u + β 2 u,whereβ i C 0,i=1, 2 have suort belonging 16

21 to su(β). Thus [Δ g,β]u L 2 t H 1 x + [Δ g,β] t u L 2 t L 2 x β 3 u L 2 t H 2 x + β 3 t u L 2 t H 1 x (2.8) (2.9) β 3 u θ L 2 t H1 β 3 u 1 θ L 2 t H3 + β 3 t u θ L 2 t,x β 3 t u 1 θ L 2 t H2 x, where β 3 C 0 has suort in su(β 1 ) su(β 2 ), and θ is any real number in (0, 1). Based on (3.51), (3.52) and (2.8), we get that the L 2 t estimate is true for γ =3,and similarly holds for γ =5, 7, 9... andmoreoverforγ R by duality and interolation, i.e. (2.10) βu L 2 t H γ B ([0,S] Ω) + β t u L 2 t H γ 1 B ([0,S] Ω) f Ḣγ+ɛ (Ω) + g Ḣ γ+ɛ 1 (Ω) + F L 2 t Ḣγ+ɛ 1 B ([0,S] Ω). By Duhamel s rincile, the inhomogeneous solution v satisfies βv L 2 t H γ B ([0,S] Ω) + β t v L 2 t H γ 1 ([0,S] Ω) F B L 1 t Ḣγ+ɛ 1 B ([0,S] Ω), by duality of the above estimate, energy estimates and ellitic regularity, we get (2.11) u L t Ḣ γ B ([0,S] Ω) + tu L t Ḣ γ 1 B ([0,S] Ω) f Ḣγ (Ω) + g Ḣ γ 1 (Ω) + F L 2 t Ḣγ+ɛ 1 B ([0,S] Ω). Now (2.5) is a result of (2.10) and (2.11). We have finished the roof of Proosition Now we turn to the roof of Theorem Fix β C 0 (R n ) satisfying β(x) =1, x 3R and write u = v + w, where v = βu, w =(1 β)u. 17

22 Then w solves the free wave equation ( t 2 Δ)w =[β,δ]u w t=0 =(1 β)f, t w t=0 =(1 β)g. Notice that [β,δ]u is comactly suorted, so an alication of Proosition and the Minkowski Strichartz estimates shows that w L t X is dominated by A(S) ρu L 2 t Ḣ γ lus B good terms on the Cauchy data, if ρ C0 equals one on the suort of β. Therefore, by (2.5), w L t X is dominated by the right hand side of (2.3) with η =0. For v = βu, wedecomoseitintimetand write v = j= ϕ(t j)v, where ϕ C0 (( 1, 1)). Let v j = ϕ(t j)v for j 1andv 0 = v j=1 v j.thenv j solves ( t 2 Δ g)v j = G j Bv j (t, x) =0, x Ω v j (0, ) = t v j (0, ) =0, where G j = ϕ(t j)[δ g,β]u +[ 2 t,ϕ(t j)]βu + ϕ(t j)f.alsov 0 solves the equation with G 0 = ϕ[δ g,β]u +[ t 2, ϕ]βu + ϕf and initial date (βf,βg). Since G j with j 0vanishesift is not in [j 1,j +1] orif x > 3R, bythelocal Strichartz estimates (2.2) and Duhamel s Princile, we get for j = 1, 2,..., S v j L t X([0,S] Ω) G j (s, ) Hγ 1 η 0 ds G j L 2 t H γ+η 1. B Similarly, v 0 L t X(R + Ω) f Hγ η + g H γ 1 η + G 0 L 2 t H γ+η 1 B. 18

23 Since >2, by (2.5) and the disjoint suort of G j,wehave v 2 L t X([0,S] Ω) v j 2 L t X([0,S] Ω) j=0 j=1 G j 2 + v L 2 t Hγ+η 1 B ([0,S] Ω) 0 L t X(R + Ω) f 2 Ḣγ+ɛ+η + g 2 Ḣγ+ɛ+η 1 + F 2 L 2 t Ḣ γ+ɛ+η 1, A 2 (S)( f 2 Ḣγ+ɛ+η + g 2 Ḣγ+ɛ+η 1 + F 2 L 2 t Ḣ γ+ɛ+η 1), which finishes the roof of Theorem Strauss Conjecture when n =3, 4 In this section we rovide the roof of Main Theorem 1. We use the classical iteration methods with suitable sace X chosen, then aly the Strichartz estimates roved in the revious two sections to get the results we want. Since the argument is somewhat the same for local and global cases, we only resent the subcritical case > c and n =3, 4. Define X = X γ, (R n ) to be the sace with the norm defined by (2.12) h Xγ, = h L sγ ( x <2R) +(A(S)) 1 x n/2+1 γ/ h L r L 2 ω ({ x >2R}), where s γ =2n/(n 2γ). When n =3,< c,γ = 1 1,wehaveS = T and A(T )isas 2 defined in the last section; When n =3, 4, > c and γ = n/2 2/ 1wehaveS = and A(S) isaconstant. Using the sace X defined just now, we can rove the following estimate: (2.13) u L t X(R + R 3 ) f Ḣγ + g Ḣγ 1, when u solves u = 0 with initial data (f,g). 19

24 Indeed, the contribution of the second art of the norm in (2.12) is roved by interolation between Morawetz estimates and Trace Lemma, and the contribution of the first term is due to Sobolev estimates and an interolation between L 2 t and L t bound in the following energy estimate (2.14) u L t Ḣ γ B (R + Ω) + tu L t Ḣ γ 1 B (R + Ω) + βu L 2 t Hγ B (R + Ω) + β t u L 2 t H γ 1 B (R + Ω) f Hγ ɛ (Ω) + g Hγ 1 ɛ (Ω) + F L 2 t Ḣγ+ɛ 1 B (R + Ω), where F is suorted in x <R, γ< n 1. The estimate is a variation of the energy 2 estimate (2.5) and has similar roof, which we will neglect here. Furthermore, by finite roagation seed of the wave equation, Sobolev estimates and interolation between (2.14), we have the local estimate for solutions of (1.5) with F = 0: (2.15) u L t X([0,1] Ω) ( f Ḣγ + g Ḣγ 1), where 2. From (2.13) and (2.15), we see that (X, γ, 0,) is admissible. By Theorem 2.2.2, we therefore obtain the following Proosition: Proosition Under the conditions of Main Theorem 1, (2.16) u L t X(R + Ω) f Hγ ɛ + g H ɛ γ 1. Based on the above roosition, it is easy to get the following corollary with forcing term added. Corollary For n =3, 4, let u be a solution of (1.5), and assume Hyothesis B, 20

25 γ = s c, > c.then (2.17) u L t Lsγ x (R + { x <2R}) + x n/2+1 γ/ u L t L rl 2 ω (R + { x >2R}) f Hγ ɛ + g H γ 1 ɛ + F + x L n/2+1 γ F 1 t Ls 1 γ ɛ L 1 x (R + { x <2R}) t L 1 rl 2 ω(r + { x >2R}). Proof. By (2.3.1), we can assume f = g = 0. By Duhamel s rincile, we have LHS F L 1 t H ɛ γ 1 (R + Ω) F L 1 t Ḣ γ 1 (R + Ω) + F L 1 t Ḣγ+ɛ 1 (R + Ω). Recall that the dual version of the trace lemma and Sobolev embedding gives (see (3.16) of [6]): (2.18) g Ḣγ 1 x n/2+1 γ g L 1 r L 2 ({ x >2R})+ g, if 1/2 < 1 γ <n/2. ω L s 1 γ ({ x <2R}) Here the condition 1/2 < 1 γ<n/2 is satisfied owing to γ = s c and > c. If we use (2.18), then we get F L 1 t Ḣ γ 1 (R + Ω) + F L 1 t Ḣγ+ɛ 1 (R Ω) x n/2+1 γ F L 1 t L 1 r L2 ω (R + { x >2R}) + F L 1 t Ls 1 γ x (R + { x <2R}) + x n/2+1 γ ɛ F + F L 1 t L 1 rl 2 ω(r + { x >2R}) L 1 t Ls 1 γ ɛ x (R + { x <2R}) x n/2+1 γ F + F L 1 t L 1 rl 2 ω(r +, { x >2R}) L 1 t Ls 1 γ ɛ x (R + { x <2R}) when ɛ>0 is small enough, which comletes the roof. Now we are only left with adding the 2nd order derivatives on the solution for the 21

26 Strichartz estimates and energy estimates above. Secifically, we want to rove that (2.19) ( ) Γ α u L t Ḣ γ + t Γ α u B L t Ḣ γ 1 ( ) Z α f + Hγ Zα g B 2ɛ H γ 1 2ɛ + ( ) x n 2 +1 γ Γ α F L 1 t L 1 rl 2 ω(r + { x >2R}) + Γ α F, L 1 t Ls 1 γ 2ɛ x (R + {x Ω: x <2R}) and (2.20) ( x n 2 n+1 γ Γ α u L t L rl 2 ω (R + { x >2R}) + Γ α u L ( ) Z α f + Hγ Zα g 2ɛ H γ 1 2ɛ t Lsγ x (R + {x Ω: x <2R}) + ( ) x n 2 +1 γ Γ α F L 1 t L 1 rl 2 ω(r + { x >2R}) + Γ α F. L 1 t Ls 1 γ 2ɛ x (R + {x Ω: x <2R}) ) The idea is based on the fact that Γ commute with g away from the origin, ellitic roerty of Δ g and local energy decay near the origin. We will first deal with the Cauchy data for Γ α u.thisisclearifγ α is relaced by Z α. On the other hand, the Cauchy data is (g, Δ g f + F (0, )) for t u and (Δ g f + F (0, ), Δ g g + t F (0, )) for 2 t u, sowehave g Hγ ɛ + Δ g f H ɛ γ 1 H ɛ γ + F L t H ɛ γ 1 L t ( Z α f Hγ ɛ H ɛ γ + tf L Hγ 1 t ɛ ) + Z α g Hγ 1 ɛ + Δ g g H ɛ γ 1 + Γ α F L 1 Hγ 1 t ɛ, where we use Sobolev embedding in the time variable t for (F, t F ). If we use (2.18) to control the last term Γα F L 1 t H ɛ γ 1, then we get (2.19) and (2.20) for the Cauchy data art of Γu. Let us now give the argument for (2.20). Fix β 0 C 0 satisfying β 0 =1for x R 22

27 and vanishing for x > 2R. Let Γ α u =(1 β 0 )Γ α u + β 0 Γ α u = v + w. Since Γ commutes with g when x R, wehave g v =(1 β 0 )Γ α F [β 0, Δ g ]Γ α u, v(0, ) = ((1 β 0 )Γ α u(0, ), t v(0, ) = t (1 β 0 )Γ α u(0, ). The initial data has been taken care of from the discussion above, and the first nonlinear term is dominated by the right hand side of (2.20) by (2.17). For the second nonlinear term, we use Proosition and control it by (2.21) [β 0, Δ g ]Γ α u L 2 t H γ 1 B j2 β 1 j t u L 2 t H γ+2 j, B assuming that β 1 equals one on the suort of β 0 andissuortedinr< x < 2R. Note that [ g, t 2] = 0, if we use (2.14) for 2 t u and Duhamel s rincile for the forcing term t 2F, we can control β 1 t 2u L 2 by the right hand side of (2.20). On the other hand, t Hγ B by Cauchy-Schwarz and Parseval s Formula, β 1 t u 2 L 2 t Hγ+1 B β 1 t 2 u L 2 t H γ β 1u B L 2 t H γ+2. B So it suffices to dominate β 1 u L 2 t H γ+2. By ellitic regularity of the oerator Δ g,wehave B β 1 u L 2 t H γ+2 β 2 Δ g u B L 2 t H γ + β 2u B L 2 t H γ B β 2 2 t u L 2 t Hγ B + β 2u L 2 t H γ B + β 2F L 2 t H γ B, where β 2 C 0 equals one on suort of β 1 and is suorted in the set where x < 2R. 23

28 The first two terms are dominated as above using (2.14) and Duhamel s rincile. For the last term, Sobolev embedding and duality yields (2.22) β 2 F L 2 t H γ α B x F L 2 t Ls 1 γ (R + {x Ω: x 2R}) α 1 t,xf α. L 1 t L s 1 γ ɛ (R + {x Ω: x 2R}) Thus we are done with the roof of (2.20) when Γ α u is relaced by v. For w = β 0 Γ α u, the coefficients of Γ are bounded on suort of β 0, so by Sobolev embedding β 0 Γ α u L t Lsγ x (R + Ω) j 2 β 1 Γ α u L t Ḣγ B ( ) β 1 j t u L 2 t H γ+2 j + β 1 Γ j u B L t Ḣ γ. B The first term is dominated as above, and the bound for the second term comes from (2.19), so we are done with roof of (2.20). Now we turn to the roof of (2.19). As before we first consider the inequality where Γ α u is relaced by v =(1 β 0 )Γ α u in (2.19). The inequality involving initial data has been taken care of in the first aragrah of the roof, and the first nonlinear term is from energy estimates in R n, Duhamel s rincile and (2.18). For the remaining term by (2.14) we see that it is controlled by (2.23) [β 0, Δ g ]Γ α u L 2 t H γ+ɛ 1 B j2 β 1 j t u L 2 t H γ+ɛ+2 j. B By almost the same argument as above we get the desired bound in (2.19). Now we are only left with w = β 0 Γ α u. First notice that the left hand side of (2.19) 24

29 with w is dominated by j3 β 1 j t u L t H 2+γ j B g (β 1 u)=β 1 F +[Δ g,β 2 ]u (β 1 u, t β 1 u) t=0 =(β 1 f,β 1 g),. For the case j =0, 1, since we use (2.5) with the Duhamel formula to bound β 1 u L t H γ+2 B + β 1 t u L t H γ+1 B β 1 f H γ+2 B + β 1 g H γ+1 B + β 2 u L 2 t H γ+ɛ+2 B + β 1 F L 1 t H γ+ɛ+1. B The term on the right involving u was controlled reviously; on the other hand, by Sobolev embedding, β 1 F L 1 t H γ+ɛ+1 x α F. B L 1 t Ls 1 γ ɛ x To handle the terms for j =2, 3 we use the equation to bound β 1 j t u L t j=2,3 H 2+γ j B j=0,1 ( β 1 j t Δ g u L t H γ j B + β 1 j t F L t H γ j B ). The terms involving Δ g u are dominated by β 2 j t u L t H γ+2 j with j =0, 1. The terms B involving F are controlled for j = 1 by Sobolev Embedding Theorem, and for j =0by observing that (2.22) holds with L 2 t relaced by L t. This comletes the roof of (2.19). Now we finally get to the roof of Strauss conjecture stated in Main Theorem 1. As stated we only resent the case when n =3and> c. The argument is adated from [6]. First, let f solve the Cauchy roblem (1.5) with F = 0. We iteratively define u k,for 25

30 k 1, by solving ( t 2 Δ g )u k (t, x) =F (u k 1 (t, x)), (t, x) R + Ω u k (0, ) =f, t u k (0, ) =g (Bu k )(t, x) =0, on R + Ω. Our aim is to show that if the constant ε > 0 in Cahchy data bound is small enough, then so is M k = ( Γ α u L k + t Ḣ γ t Γ α u L B (R + Ω) k t Ḣ γ 1 B (R + Ω) + n 2 +1 γ x Γ α u L k + t L rl 2 ω (R + { x >2R}) Γα u k L t Lsγ x (R + {x Ω: x <2R}) ) for every k =0, 1, 2,... For k = 0, it follows by (2.19) and (2.20) that M 0 C 0 ε,withc 0 a fixed constant. More generally, (2.19) and (2.20) yield that (2.24) M k C 0 ε + C 0 ( x n 2 +1 γ Γ α F (u k 1 ) L 1 t L1 r L2 ω (R + { x >2R}) ) + Γ α F (u k 1. ) L 1 t Ls 1 γ 2ɛ x (R + {x Ω: x <2R}) Note that our assumtion on the nonlinear term F imlies that for small v Γ α F (v) v 1 Γ α v + v 2 Γ α v 2. α 1 Furthermore, since u k will be locally of regularity H γ+2 B L and F vanishes at 0, it follows that F (u k ) satisfies the B boundary conditions if u k does. Since the collection Γ contains vectors sanning the tangent sace to S n 1, by Sobolev 26

31 embedding for n =3, 4wehave v(r ) L ω + Γ α v(r ) L 4 ω Γ α v(r ) L 2 ω. α 1 Consequently, for fixed t, r > 0, Γ α F (u k 1 (t, r )) L 2 ω Γ α u k 1 (t, r ) L. 2 ω Thus the first summand in the right hand side of (2.24) is dominated by C 1 M k 1. We next observe that, since s γ > 2andn 4, it follows by Sobolev embedding on {Ω x < 2R} that v L (x Ω: x <2R) + Γ α v L 4 (x Ω: x <2R) Γ α v L sγ (x Ω: x <2R). α 1 Since s 1 γ 2ɛ < 2, it holds for each fixed t that (2.25) Γ α F (u k 1 (t, )) s Γ α F (u k 1 (t, )) L L 1 γ 2ɛ (x Ω: x <2R) 2 (x Ω: x <2R) Γ α u k 1 (t, ) L sγ (x Ω: x <2R). The second summand in the right side of (2.24) is thus dominated by C 1 M k 1, and we conclude that M k C 0 ɛ +2C 0 C 1 M k 1. For ɛ sufficiently small, by the definition of A(S), we obtain (2.26) M k 2 C 0 ε, k =1, 2, 3,... To finish the roof of Strauss Conjecture we need to show that u k converges to a solution 27

32 of the equation (1.5). For this it suffices to show that A k =(A(S)) 1 n 2 +1 γ x (u k u k 1 ) L t L rl 2 ω(r + { x >2R}) + u k u k 1 L t Lsγ x (R + {x Ω: x <2R}) tends geometrically to zero as k. Since F (v) F (w) v w ( v 1 + w 1 ) when v and w are small, the roof of (2.26) can be adated to show that, for small ε > 0, there is a uniform constant C so that A k CA k 1 (M k 1 + M k 2 ) 1, which, by (2.26), imlies that A k 1 2 A k 1 for small ε. Since A 1 is finite, the claim follows, which finishes the roof of Main Theorem 1. 28

33 Chater 3 Part II Strauss Conjecture on Asymtotically Euclidean Manifolds This chater is devoted to the joint work with Chengbo Wang, which is subsequent to [21]. In the work of Sogge and Wang [21], a global existence result is obtained for the system of (1.4) with symmetrical metric g ij. In our work we remove the radial assumtion and also show the suercritical case < c when n = 3 by roving a local in time Strichartz estimate. We will first go over the argument resented in [21] in Section 3.1, then get the required estimates without radial assumtion in Section 3.2 and 3.3. Finally we briefly give the argument to rove the existence results resented in Main Theorem A secial Case: 3-D, >1+ 2,themetricg is sherically symmetric Theorem Suose (H1) and (H2) hold with ρ>1, n =3,and>1+ 2. Assume g ij (x) =g ij ( x ), x. Then for any ɛ>0 such that (3.1) s = s c ɛ (s d, 1/2) 29

34 there is a δ>0 deending on so that (1.4) has a global solution satisfying (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣs 1, α 1, t R +, whenever the initial data satisfies (3.2) ( Z α f Ḣs + Z α g Ḣs 1 ) <δ. α 1 We exect to aly the idea in [6] to get the existence results in Theorem Thus, it suffices to obtain the generalized Strichartz estimates as follows. (3.3) Z α u L 2 t Y s,ɛ + x n/2 (n+1)/ s ɛ Z α u L α 1 t L x L2+η ω ({ x >1}) ( Z α f Ḣs + Z α g Ḣs 1), α 1 and for s [0, 1], (3.4) α 1 ( ) Z α u L t Ḣ + s Zα u L t Ḣ + s 1 Zα u L t Lqs x ( x 1) ( Z α f Ḣs + Z α g Ḣs 1), where q s =2n/(n 2s). Note that for convenience we have defined the norm Y s,ɛ as f(x) Ys,ɛ = x (1/2) s ɛ f(x) L 2 x. On Asymtotically Euclidean manifolds we do not automatically have the above estimates which are roved to be true in the flat metric case, due to the lack of Fourier transform techniques. However, in the work of [2] and [21], an imortant Keel-Smith- Sogge (KSS) tye estimate is roven for (M,g) by sectral methods. Equied with the KSS estimates and energy estimates, we are able to roceed and eventually obtain the estimates (3.3) and (3.4). 30

35 We first resent the KSS estimates on this setting. KSS estimates was originated by Keel, Smith and Sogge [15] and state that (3.5) (log(2 + T )) 1/2 x 1/2 u L 2 ([0,T ] R 3 ) u (0, ) L 2 (R 3 ) + T 0 F (s, ) L 2 (R 3 ) ds, where u solves the equation u = F and u =( t u, x u). This estimate has been generalized for general weight of form x a with a 0 (see [10] and references therein). Recently, Bony and Häfner [2] obtained a weaker version of the KSS estimates for asymtotically Euclidean sace when the metric is non-traing. With this estimate, they were able to show the global and long time existence for quadratic semilinear wave equations with dimension n 4andn = 3. Then Sogge and Wang [21] roved the almost global existence for 3-D quadratic semilinear equations by obtaining the shar KSS estimates for a =1/2. Now we resent the KSS estimates as a lemma here. Lemma (KSS estimates). Assume that (H1) and (H2) hold with ρ>1. LetN 0, μ 1/2 and (log(2 + T )) 1/2 μ =1/2, A μ (T )= 1 μ>1/2. Then the solution of (1.1) satisfies (3.6) su 0tT 1k+jN+1 k t P j/2 gu(t, ) L 2 + x α N α N A μ (T ) ( ) x μ (Γ α u) + Γα u L2 x T L2 x (Z α u) (0, ) L 2 + x α N Γ α F (s, ) L 1 T L 2 x, where L q T Lr x = Lq ([0,T]; L r (R n )). We also make the following variation of solutions. Remark 1. Set P = gδ g g 1 = g 1 g ij g 2 g 1,whereg =(detg) 1/4. We will rove the estimates if u is the solution of ( 2 + P )u = F, which has the benefit that the solution 31

36 can be reresented by the following formula u(t) =cos(tp 1/2 )f + P 1/2 sin(tp 1/2 )g + t 0 P 1/2 sin((t s)p 1/2 )F (s)ds. All of the oerators occurring in this formula commutates with the wave oerator 2 + P. In general, an estimate for Δ g will corresonds another estimate for P. For examle, if we have the estimate (3.3) for P, consider the equation (3.7) ( t 2 Δ g)v(t, x) =G(t, x), (t, x) R + R n u(0,x)=v 0 (x), t v(0,x)=v 1 (x), x R n. Notice that if we let u = gv and F = gg, then (3.8) ( 2 Δ g )v = G ( 2 + P )u = F. Thus we have also the estimate (3.3) for Δ g. Now we turn to the roof of (3.3) and (3.4). When α = 0, i.e. the order is zero, the L 2 t Y s,ɛ bound in (3.3) is from interolation between x 1/2 ɛ e itp 1/2 f L 2 t L 2 x f L 2 x, and x 3/2 ɛ e itp 1/2 f L 2 t L 2 x f Ḣ 1 x, which hold due to the KSS estimates. The L t L x L2+η ω bound is from interolation between the L 2 t Y s,ɛ and the following Sobolev inequalities with angular regularity (see Corollary 32

37 1.2 in [3]), (3.9) x d 2 α e itp 1/2 f(x) L t, x L 2+η e itp 1/2 f(x) ω L t Ḣ α x f Ḣ α x for α (1/2, 1] and some η>0. As for (3.4), the first two terms in the left hand side are just the energy estimates. For the last term we first use a Sobolev embedding, then aly an interolation between =2and =+. The case when = is the energy estimates. The case when = 2 is from the local energy decay below. Lemma (Local Energy Decay). For the linear equation (1.4), if F (t, x) = 0for x >Rwith R fixed, then for fixed β C0 (Rd ), we have (3.10) βu L 2 t H 1 f Ḣ1 + g L 2 x + F L 2 t L 2 x. Moreover, if F 0ands [0, 1], then (3.11) βu L 2 t H s f Ḣs + g Ḣs 1 x. The local energy decay also lays an imortant role in the roof of the KSS estimates when μ =1/2 and can be roven by the KSS estimates when μ>1/2. So we are finished with the case α =0. When α = 1, since we assume a radial metric, it is trivial for Z =Ω ij due to the fact that [P, Ω ij ] = 0, so we are left with Z =. The roof relies largely on the equivalence of P 1/2 and in secifical norms, which we state here as a lemma. Lemma [Relation between P 1/2 and ] i, If s [ 1, 1], then u Ḣs P s/2 u L 2 x. ii, If s [0, 1], then j u Ḣ s P 1/2 u Ḣ s, and P 1/2 u Ḣs j j u Ḣs. iii, If s (0, 2] and 1 <q<d/s,then P s/2 u L q x u Ḣs,q. 33

38 iv, If 3/2 μ <μ 3/2, then (3.12) x μ l u L 2 (R d ) x μ P 1/2 u L 2 (R d ), (3.13) x μ P 1/2 u L 2 (R d ) d x μ l u. L 2 (R d ) l=1 iv, If u H 1 (R d ), (3.14) P 1/2 u L 2 (R d ) g 1 u L 2 (R d ) P 1/2 u L 2 (R d ). With the above lemma and the estimates with order 0, we can estimate L 2 t Y s,ɛ art as follows. x α u L 2 α t Ys,ɛ x u L 2 t Ys,ɛ α 1 α 1 j1 P j/2 u L 2 t Y s,ɛ/2 j1 j1 α 1 ( ) P j/2 f Ḣs + P j/2 g Ḣs 1 x ( P (j+s)/2 f L 2 x + P (j+s 1)/2 g L 2 x ) ( α x f Ḣs + x α g ) Ḣx s 1. The roof for the first order energy norm L t Ḣ s is similar so we neglect it here. The L t L r L2+η ω estimate is direct consequence of the energy estimates and (3.9). So we are left with the local decay for φ x u L 2 t Ḣ s,whereφ C 0, but this is again from the KSS estimates for μ>1/2. Now we are finished with the roof of (3.3) and (3.4), with these two estimates at hand, we are able to use similar argument as in [6] to obtain the existence result with n = 3 under symmetrical assumtion on the metric. 34

39 3.2 Weighted Strichartz and Energy Estimates In what follows, remainder terms, r j, j N, will denote any smooth functions such that (3.15) α x r j (x) =O ( x ρ j α ), α, thus P = gδ g g 1 = Δ+r r 1 + r 2. In order to extend the existence result in Section 3.1 to n = 4 and nonsymmetric metric, there are several difficulties to overcome. Firstly, we no longer have [Ω,P] = 0, thus have to figure out a way to handle with the extra commutator terms to get (3.3) and (3.4). Secondly, in order to extend the result to dimension n = 4, we hoe to obtain the estimates with second order derivatives, hence need exlore the relation between P and Z. Secifically, we are aimed at showing the following estimates. Theorem Let u be the solution of (1.4) with F =0. Assume that (H1) and (H2) hold with ρ>2, n 3, 2 < and s (s d, 1). For all ɛ>0 and η>0small enough, we have (3.16) Z α u L 2 t Y s,ɛ + x n/2 (n+1)/ s ɛ Z α u L ( Z α f Ḣs + Z α g Ḣs 1), t L x L2+η ω ({ x >1}) and for s [0, 1], (3.17) ) ( Z α u L t Ḣ + s Zα u L t Ḣ s 1 + Z α u L t Lqs x ( x 1) ( Z α f Ḣs + Z α g Ḣs 1), where q s =2n/(n 2s). 35

40 Note that Theorem 3.2.1, with order 0 ( α = 0) and ρ>0, has been roved in Section3.1. In order to deal with the commutators coming from the commutator of P and the Z, we will need the following three lemmas to gain control on forcing terms. Lemma Let u solve the wave equation (1.4). Then for any s [0, 1] and ɛ>0, we have: (3.18) u L 2 t Y s,ɛ f Ḣs + g Ḣs 1 + x (1/2)+ɛ F L 2 t Ḣ s 1 The homogeneous art is roved in Section 3.1. For the inhomogeneous art, when s = 1, it is just Remark 2.1 in [21] which states that (3.19) x 3/2 ɛ u L 2 (R R n ) x (1/2)+ɛ F L 2 (R R n ). When s =0,itisequivalentto x 1/2 ɛ u L 2 t Ḣ 1 (R R n ) x (1/2)+ɛ F L 2 (R R n ) which is true due to (3.19) and the KSS estimates. Now (3.18) is from interolation between s =0ands =1. Lemma Let w solve the wave equation (1.4) with f = g =0. Thenfors [0, 1] and ɛ>0, (3.20) w L t Ḣ s x x 1/2+ɛ F L 2 t Ḣ s 1 x. Again we use interolation. When s =1,notethat w Ḣ1 P 1/2 w L 2, we can get the corresonding estimate by the KSS estimate with μ>1/2 andtt argument. When s = 0, we use duality as in the roof of (3.18), then it is just the KSS estimates. On the basis of the above two lemmas, we can control the commutator terms by a 36

41 kind of weighted L 2 t Ḣs 1 x norm. Then with the following lemma we will be able to bound this norm by the good terms, thus we can use the argument as in [21] to get over the difficulty on error terms. Lemma Let n 3, N 1andu be the solution to (1.4) with F = 0. Then for any s [0, 1], ɛ>0and α = N, wehave (3.21) α =N x (1/2) ɛ α x u L 2 t Ḣs 1 f ḢN+s 1 Ḣ s + g ḢN+s 2 Ḣ s 1. The estimate for s = 1 follows directly from the KSS estimates (3.6). Moreover, we have the following estimate (3.22) x (1/2) ɛ u L 2 t L 2 x = x (1/2) ɛ P 1/2 (P 1/2 u) L 2 t L 2 x P 1/2 f Ḣ1 + P 1/2 g L 2 x f L 2 x + g Ḣ 1. For s = 0, first notice that since n 3, we have Hardy s inequality x 2 xh L 2 x h Ḣ1, and the duality gives x 2 xf Ḣ 1 f L 2 x. Using the above estimate together with the KSS estimates and (3.22), we get x (1/2) ɛ α x u L 2 t Ḣ 1 x (5/2) ɛ x α 1 x u L 2 t Ḣ + 1 x (1/2) ɛ x α 1 u L 2 t L 2 x x (1/2) ɛ x α 1 u L 2 t L 2 x f ḢN 1 + g ḢN 2 Ḣ 1. Now (3.21) follows from an interolation between s =0ands =1. 37

42 The next two lemmas are to develo relation between P and 2. Lemma For 0 <μ 3/2 andk 2, we have (3.23) x μ j1 jk u L 2 x [ k 1 2 ] x μ P j u L 2 + x μ P j u x L 2, x j=0 [ k 2 ] j=1 where [a] denotes the integer art of a (max{k Z,k a}). Lemma For f Ḣs (R n ) Ḣs+2 (R n )withn 3ands [0, 1], we have (3.24) 2 xf Ḣs Pf Ḣs + f Ḣs. On the other hand, (3.25) Pf Ḣs x α f Ḣs. The first lemma is just Lemma 4.8 in [2]. The second lemma is technical and will need the following fractional Lebniz rule. Fractional Leibniz rule. Let 0 s<n/2, 2 i < and 1/2 =1/ i +1/q i (i =1, 2). Then fg Ḣs f L q 1 g Ḣs,1 + f Ḣs,2 g L q 2. Moreover, for any s ( n/2, 0) (0,n/2), fg Ḣs f L Ḣ s,n/ s g Ḣ s. The first inequality above is well known, see, e.g., [16]. The second inequality with s 0 is an easy consequence of the first inequality together with Sobolev embedding. Then the result for negative s follows by duality. Now we resent the roof of Lemma

43 First, we give the roof for the estimate (3.25). When s = 0, notice that Pf = g ij i j f + r 1 x f + r 2 f,wehave Pf L 2 x 2 x f L 2 x + xf L 2 x + f L 2 x f Ḣ2 L 2 x. When s = 1, recalling that j r i = O( x ρ i j ), by Hardy s inequality, x (r 2 f) L 2 x x (r 2 )f L 2 x + r 2 x f L 2 x x f L 2 x. Thus Pf Ḣ1 x = x Pf L 2 x x (g ij i j f) L 2 x + x (r 1 x f) L 2 x + x (r 2 f) L 2 x 3 x f L 2 x + xf L 2 x f Ḣ3 Ḣ 1. Our estimate (3.25) is obtained by an interolation between the above two estimates on Pf. Now we turn to the roof of the estimate (3.24). First, when s = 0, by ellitic roerty of P,wehave (3.26) 2 x f L 2 x Pf L 2 x + f L 2 x. 39

44 Second, for s =1,using(3.26), 3 xf L 2 x P x f L 2 x + x f L 2 x [P, x ]f L 2 x + x Pf L 2 x + x f L 2 x r 3 α x α f L 2 x + Pf Ḣ1 + f Ḣ1 Pf Ḣ1 + f Ḣ1 + f Ḣ2 Pf Ḣ1 + f Ḣ1 + ɛ f Ḣ3 +(1/ɛ) f Ḣ1, ɛ >0. Here we have used Hardy s inequality and the fact that Ḣ3 Ḣ1 Ḣ2.Nowifwechoose ɛ>0 small enough and use (3.25) with s =0,wehave (3.27) 2 xf Ḣ1 Pf Ḣ1 + f Ḣ1 PP 1/2 f L 2 x + f Ḣ1 P 1/2 f Ḣ2 + P 1/2 f L 2 x On the basis of (3.26) and (3.27), by an interolation for the oerator 2 P 1/2 and making use of Lemma 3.1.4, we have, (3.28) 2 x f Ḣ s P 1/2 f Ḣ1+s + P 1/2 f Ḣs 1 P 1/2 f Ḣ1+s + P 1/2+(s 1)/2 f L 2 x P 1/2 f Ḣ1+s + f Ḣs. We need only to deal with the term P 1/2 f Ḣ1+s. Note that for s [0, 1], we have P 1/2 v Ḣ1+s v Ḣs + v Ḣ s, which is true for s = 0 (see (3.1.4)) and s = 1 (see (3.26)). Recalling that P g ij i j = 40

45 r 1 x + r 2, and by Sobolev embedding, we have, P 1/2 f Ḣ1+s Pf Ḣs + Pf Ḣ s Pf Ḣs + f Ḣ2 s + r 1 x f Ḣ s + r 2 f Ḣ s Pf Ḣs + f Ḣ2 s + f Ḣ1 Pf Ḣs + f θ 1Ḣs f 1 θ 1 Ḣ 2+s + f θ 2Ḣs f 1 θ 2 Ḣ 2+s, where θ i (0, 1]. (3.29) Pf Ḣs + f θ 1Ḣs 2 xf 1 θ 1 Ḣ s + f θ 2Ḣs 2 xf 1 θ 2 Ḣ s, where in the third inequality we used duality of Sobolev embedding and hölder s inequality, and in fourth inequality we used the fact that s 1 < 2+s and s 2 s<2+s(so that θ i > 0) for s (0, 1]. Now our estimate (3.24) (for s>0) follows from (3.28) and (3.29). Now we have obtained all the ingredients needed to show Theorem The roof consists of the following four estimates. Proosition (Generalized Morawetz estimates). Let n 3, s [0, 1) and ρ>2. Then for the solution u of the equation (1.4) with F =0,wehave (3.30) Z α u L 2 t Y s,ɛ ( Z α f Ḣs + Z α g Ḣs 1) Moreover, if we assume only ρ>1ands [0, 1], the estimate still holds with α 1. Proof. Since Z α = x is already roven in Section 3.1, we first check with Z α = Ω. Recall that by the interolation of (3.19) and the duality of (3.19), we have (3.31) u L 2 t Y s,ɛ F L 2 t Y 1 s,ɛ, if u is a solution of (1.4) with vanishing initial data. Since [P, Ω]u = r 2 α α x u, by 41

46 using a combination of the estimate with order 0 and Lemma for Ωu, wehave (3.32) Ωu L 2 t Y s,ɛ Ωf Ḣs + Ωg Ḣs 1 + x 3/2 s+ɛ r 2 α x α u L 2 + r 0 x 1/2+ɛ 2 t,x x u L 2 t Ḣs 1 α 1 Now since ρ>1, by (3.12) and Lemma 3.2.2, x 3/2 s+ɛ r 2 α x α u L 2 t,x α 1 x 1/2 s ɛ x α u L 2 t,x α 1 x 1/2 s ɛ α x u L 2 t,x α 1 i1 x 1/2 s ɛ /2 P i/2 u L 2 t,x (3.33) i1 ( P i/2 f Ḣs + P i/2 g Ḣs 1) x α f Ḣs + x α g Ḣs 1 α 1 where in the last inequality we have used fractional Leibniz rule and Lemma Let r(x) =r 0 x 1/2+ɛ = O( x ρ+1/2+ɛ ). Then r (x) =O( x ρ 1/2+ɛ ). Since n 3, by Hardy s inequality with duality, the KSS estimates (3.6) and interolation, r 2 xu L 2 t Ḣ s 1 x(r x u) L 2 t Ḣ s 1 + r x u L 2 t Ḣ s 1 (3.34) r x u L 2 t Ḣ + x s r x u L 2 t Ḣ s x α f Ḣs + x α g Ḣs 1. α 1 α 1 On the basis of (3.32), (3.33) and (3.34), we are done with Z α = Ω. This comletes the roof of the first order estimates under the condition ρ>1. For the second order art, we first consider the case Z α = x 2.Sinces [0, 1), we can always find ɛ>0 such that 1/2+s + ɛ 3/2. By Lemma 3.2.5, the roof for Z α = x, 42

47 Lemma and Lemma 3.2.6, we have xu 2 L 2 t Y s,ɛ x α u L 2 t Y s,ɛ α u L 2 t Y s,ɛ + Pu L 2 t Y s,ɛ α 1 ( x α f Ḣs + x α g Ḣs 1)+ Pf Ḣs + Pg Ḣs 1 α 1 ( x α f Ḣ + α s x g Ḣ s 1 )+ Pf Ḣs + P 1/2 g Ḣs α 1 ( x α f Ḣ + s α x g Ḣ s 1)+ x α f Ḣ + α s g Ḣs α 1 ( x α f Ḣs + x α g Ḣs 1), α 1 where the fractional Leibniz rule is used in the last inequality. Next, we consider the case Z α =Ω 2. Since [P, Ω 2 ]u = α 3 ( r2 α α x u),andω 2 u solves the wave equation with initial data (Ω 2 f,ω 2 g)andforcingterm[p, Ω 2 ]u, by (3.31), Lemma 3.2.2, Lemma

48 and the higher order estimates we have roved, Ω 2 u L 2 t Y s,ɛ Ω 2 f Ḣs + Ω 2 g Ḣs 1 + x 3/2 s+ɛ r 2 α x α u L 2 + x 1/2+ɛ r t,x 2 α x α u L 2 t Ḣs 1 α =3 Ω 2 f Ḣs + Ω 2 g Ḣs 1 + x α u L 2 t Y s,ɛ + x 1/2+ɛ r 1 x α u L 2 t Ḣ s 1 α =3 ( Z α f Ḣs + Z α g Ḣs 1)+ x 1/2+ɛ r 1 x α u L 2 t Ḣs 1 ( Z α f Ḣs + Z α g Ḣs 1) α =3 + x 1+2ɛ r 1 L Ẇ 1,n x 1/2 ɛ xu 3 L 2 t Ḣ s 1 ( Z α f Ḣs + Z α g Ḣs 1)+ ( x α f Ḣ + s α x g Ḣ s 1) ( Z α f Ḣs + Z α g Ḣs 1) wherewehaveusedthefactthatρ>2. Since the commutator term [P, Ω]u =[P, Ω ]u = ( α 3 r3 α x αu) corresonds to an even better case than what for Ω 2, the roof roceeds in the same way. This comletes the roof of the higher order estimates under the conditions ρ>2ands [0, 1). Proosition (Higher order energy estimates). Let n 3, s [0, 1] and ρ>2. Then for the solution u of the equation (1.4) with F =0,wehave (3.35) Z α u(t, x) L t Ḣ s ( Z α f Ḣs + Z α g Ḣs 1). Moreover, if we assume only ρ>1, the estimate still holds with α 1. 44

49 Proof. By Lemma and ellitic regularity for P,weknow x u Ḣ1 2 xu L 2 Pu L 2 + u L 2 P 1/2 u Ḣ1 + P 1/2 u Ḣ 1. Interolating this estimate with (3.1.4) with s =1, x u L 2 P 1/2 u L 2,wegetthatfor s [0, 1], (3.36) x u Ḣs P 1/2 u Ḣs + P 1/2 u Ḣ s P 1/2 u Ḣs + u Ḣ1 s. Thus by Lemma we have for s [0, 1/2] (such that s 1 s and Ḣs Ḣ1+s Ḣ1 s ), (3.37) x α u L t α 1 Ḣ P j/2 u s L t Ḣ + u s L t Ḣ 1 s j1 ( x α f Ḣ + α s x g Ḣ s 1 )+ f Ḣ1 s + g Ḣ s α 1 ( x α f Ḣ + s α x g Ḣ s 1). α 1 Now we can deal with Ωu. Noticing that Ω ij f = g 1 Ω ij f +(x i j g 1 x j i g 1 )f, by the fractional Leibniz rule, we have Ωf Ḣs Ω α f Ḣs, s <n/2. α 1 We have similar relationshi between x u and x u. By the Sobolev embedding, for any 45