Energy-critical semi-linear shifted wave equation on the hyperbolic spaces

Energy-critical semi-linear shifted wave equation on the hyperbolic spaces Ruipeng Shen Center for Applied Mathematics Tianjin University Tianjin, P.R.China January 1, 16 Abstract In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space H n with 3 n 5: t u H n + ρ )u = ζ u 4/n ) u, x, t) H n R. Here ζ = ±1 and ρ = n 1)/ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space H,1 L H n ) and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality T+ T Hn ρcosh x ) ux, t) n/n ) dµx)dt ne, sinh x in the defocusing case ζ = 1, where E is the energy. Moreover, if the initial data are also radial, we can prove the scattering of the corresponding solutions by combining the Morawetz-type inequality, the local theory and a pointwise estimate on radial H,1 H n ) functions. 1 Introduction In this work we continue our discussion on a semi-linear shifted wave equation on H n : t u H n + ρ )u = ζ u p 1 u, x, t) H n R; u t= = u ; t u t= = u 1. 1) Here the constants satisfy ρ = n 1)/, ζ = ±1 and p > 1. We call this equation defocusing if ζ = 1, otherwise if ζ = 1 we call it focusing. The energy-subcritical case p < p c := 1+4/n ), n 6) has been considered by the author s recent joint work with Staffilani [5]. As a continuation, this work is concerned with the energy critical case p = p c, 3 n 5. An Analogue of the wave equation in R n The equation 1) discussed in this work is the H n analogue of the semi-linear wave equation defined in Euclidean space R n : t u u = ζ u p 1 u, x, t) R n R. This similarity can be understood in two different ways, as we have already mentioned in [5]. MSC classes: 35L71, 35L5 1

I) The operator H n ρ in the hyperbolic space and the Laplace operator in R n share the same Fourier symbol λ, as mentioned in Definition.1 below. II) There is a transformation between solutions of the linear wave equation defined in a forward light cone in R n R and solutions of the linear shifted wave equation defined in the whole space-time H n R. Please see Tataru [7] for more details. The author would also like to mention one major difference between these two equations. The symmetric group of the solutions to R n wave equation includes the natural dilations T λ u)x, t). = λ /p 1) ux/λ, t/λ), where λ is an arbitrary positive constant. The shifted wave equation 1) on the hyperbolic spaces, however, does not possess a similar property of dilation-invariance. The energy Suitable solutions to 1) satisfy the energy conservation law: [ 1 Eu, u t ) = u ρ u ) + 1 tu ζ ] p + 1 u p+1 dµ = const, H n where dµ is the volume element on H n. Since the spectrum of H n is [ρ, ), it follows that the integral of u ρ u above is always nonnegative. We can also rewrite the energy in terms of certain norms. Please see Definition.1 for the definition of Ḣ,1 norm) E = 1 u H,1 H n ) + 1 tu L H n ) ζ p + 1 u p+1 L p+1 H n ). Please note that a solution in the focusing case may come with a negative energy. Previous results on R n Let us first recall a few results regarding the energy-critical wave equation on Euclidean spaces. In 199 s Grillakis [1, 11] and Shatah-Struwe [, 3] proved in the defocusing case that the solutions with any Ḣ1 L initial data exist globally in time and scatter. The focusing case is more subtle and has been the subject of many more recent works such as Kenig-Merle [16] dimension 3 n 5, energy below the ground state), Duyckaerts- Kenig-Merle [4, 5] radial case in dimension 3), Krieger-Nakanishi-Schlag [18, 19] energy slightly above the ground state). Previous results on H n Much less has been known in the case of hyperbolic spaces. Strichartztype estimates for shifted wave equations have been discussed by Tataru [7] and Ionesco [14]. More recently Anker, Pierfelice and Vallarino gave a wider range of Strichartz estimates and a brief description on the local well-posedness theory for the energy-subcritical case p < p c ) in their work [1]. The author s joint work with Staffilani [5] improved their local theory and proved the global existence and scattering of solutions in the defocusing case with any H 1/,1/ H 1/, 1/ H n ) initial data using a Morawetz type inequality, if n 6. Finally some global existence and scattering results have also been proved by A. French [7] in the energy-supercritical case, but only for small initial data. Goal and main idea of this paper This paper is divided into two parts. The first part is concerned with the local theory of the energy-critical shifted wave equation in hyperbolic spaces H n with 3 n 5: t u H n + ρ )u = ζ u 4/n ) u, x, t) H n R; u t= = u ; CP 1) t u t= = u 1. We will first introduce a family of new Strichartz estimates via a T T argument and then establish a local well-posedness theory for any initial data in the energy space H,1 L H n ).

The second part is about the global behaviour of solutions in the defocusing case. We will prove a Morawetz-type inequality T+ T Hn ρcosh x ) ux, t) n/n ) dµx)dt ne. ) sinh x As in the Euclidean spaces, global space-time integral estimates of this kind are a powerful tool to discuss global behaviour of solutions. Although we are still not able to show the scattering of solutions with arbitrary initial data in H,1 L H n ), which we expect to be true, the Strichartz estimate above is sufficient to prove the scattering in the radial case, thanks to a point-wise estimate on radial H,1 H n ) functions as given in Lemma.6. Main Results For the convenience of readers, we briefly describe our main results as follows. We always assume that 3 n 5 in this paper. I) For any initial data u, u 1 ) H,1 L H n ), there exists a unique solution to the equation CP1) in a maximal time interval T, T + ). II) In addition, if u is a solution to CP1) in the defocusing case with initial data u, u 1 ) H,1 L H n ), then it satisfies the Morawetz-type estimate ). III) Moreover, if the initial data u, u 1 ) H,1 L H n ) are radial, then the solution to the equation CP1) in the defocusing case exists globally in time and scatters. It is equivalent to saying that the maximal lifespan of the solution u is R and there exist two pairs u ±, u± 1 ) H,1 L H n ), such that lim u, t), t u, t)) S L t)u ±, u± 1 ) =. H,1 L H n ) t ± Here S L t) is the linear propagation operator for the shifted wave equation on H n as defined in Section.1. Notations and Preliminary Results.1 Notations The notation We use the notation A B if there exists a constant c such that A cb. Linear propagation operator Given a pair of initial data u, u 1 ), we use the notation S L, t)u, u 1 ) to represent the solution u of the free linear shifted wave equation t u H n + ρ )u = with initial data u, t u) t= = u, u 1 ). If we are also interested in the velocity t u, we can use the notations S L t)u, u 1 ) =. ) ) u.= u, t) u, t), t u, t)), S L t). u 1 t u, t). Fourier Analysis In order to make this paper self-contained, we make a brief review on the basic knowledge of the hyperbolic spaces and the related Fourier analysis in this subsection. Model of hyperbolic space We use the hyperboloid model for hyperbolic space H n in this paper. We start by considering Minkowswi space R n+1 equipped with the standard Minkowswi metric dx ) +dx 1 ) + +dx n ) and the bilinear form [x, y] = x y x 1 y 1 x n y n. The hyperbolic space H n can be defined as the upper sheet of the hyperboloid x x 1 x n = 1. The Minkowswi metric then induces the metric, covariant derivatives D and measure dµ on the hyperbolic space H n. 3

Fourier transform Please see [1, 13] for more details) The Fourier transform takes suitable functions defined on H n to functions defined on λ, ω) R S n 1. We can write down the Fourier transform of a function f C H n ) and the inverse Fourier transform by fλ, ω) = fx)[x, bω)] iλ ρ dµx); H n fx) = const. fλ, ω)[x, bω)] iλ ρ cλ) dωdλ; S n 1 where bω) and the Harish-Chandra c-function cλ) are defined by C n is a constant determined solely by the dimension n) bω) = 1, ω) R n+1 ; cλ) = C n Γiλ) Γiλ + ρ). It is well known that cλ) λ 1 + λ ) n 3. The Fourier transform f f defined above extends to an isometry from L H n ) onto L R + S n 1, cλ) dλdω) with the Plancheral identity: f 1 x)f x)dx = f1 λ, ω) f λ, ω) cλ) dωdλ. H n S n 1 We also have an identity H nf = λ + ρ ) f for the Laplace operator H n. Radial Functions Let us use the polar coordinates r, Θ) [, ) S n 1 to represent the point cosh r, Θ sinh r) H n R n+1 in the hyperboloid model above. In particular, the r coordinate of a point in H n represents the metric distance from that point to the origin H n, which corresponds to the point 1,,, ) in the Minkiwski space. As in Euclidean spaces, for any x H n we also use the notation x for the same distance from x to. Namely r = x = dx, ), x H n. A function f defined on H n is radial if and only if it is independent of Θ. By convention we may use the notation fr) for a radial function f. If the function fx) in question is radial, we can rewrite the Fourier transform and its inverse in a simpler form fλ) = fλ, ω) = fx)φ λ x)dµx); H n fx) = const. fλ)φ λ x) cλ) dλ. Here the function Φ λ x) is the elementary spherical radial) function of x H n defined by Φ λ x) = [x, bω)] iλ ρ dω. S n 1 One can use spherical coordinates on S n 1 to evaluate the integral and rewrite Φ λ x) into Φ λ x) = Φ λ r) = Γ n ) πγ n 1 ) π cosh r sinh r cos θ) iλ ρ sin n θ dθ. 3) The change of variables u = lncosh r sinh r cos θ) then gives another formula of Φ λ r) if r > : Φ λ r) = n 3 Γ n ) πγ n 1 r) n )sinh These integral representations imply that r r cosh r cosh u) n 3 e iλu du. 4

The function Φ λ r) is a real-valued function for all r and λ R. The function Φ λ r) has an upper bound independent of λ Φ λ r) Φ r) Ce ρr r + 1). 4) In the 3-dimensional case, the function Φ λ r) is particularly easy and can be given by an explicit formula Φ λ r) = sin λr)/λ sinh r). Convolution integral If f, K C H n ) and K is radial, we can define the convolution f K by an f K)x) = G fg )Kg 1 x)dg. Here G = SO1, n) is the connected Lie Group of n+1) n+1) matrices that leave the bilinear form [x, y] = x y x 1 y 1 x n y n invariant. The notations g and g 1 x represent the natural action of G on H n defined by the usual left-multiplication of matrices on vectors. The measure dg is the Haar measure on G normalized in such a way that the identity G fg )dg = H n fx)dµ holds for any f C H n ). The Fourier transform of f K satisfies the identity f K) λ, ω) = fλ, ω) Kλ). The Fourier transform K does not depend on ω since we have assumed that K is radial. The author would like to emphasize that there is no simple identity of this type without radial assumption on K. Please see [15] for more details..3 Sobolev Spaces Definition.1. Let D γ = H n ρ ) γ/ and D σ = H n + 1) σ/. These operators can also be defined by Fourier multipliers m 1 λ) = λ γ and m λ) = λ + ρ + 1) σ/, respectively. We define the following Sobolev spaces and norms for γ < 3/. H σ q H n ) = D σ L q H n ), u H σ q H n ) = D σ u Lq H n ); H σ,γ H n ) = D σ D γ L H n ), u H σ,γ H n ) = D γ Dσ u L H n ). Remark.. If σ is a positive integer, one can also define the Sobolev spaces by the Riemannian structure. For example, we can first define the W 1,q norm as u W 1,p = u q dµ H n for suitable functions u and then take the closure. Here u = D α ud α u) 1/ is defined by the covariant derivatives. It turns out that these two definitions are equivalent to each other if 1 < q <, see [7]. In other words, we have u H σ q u W σ,q. In particular, we can rewrite the definition of Ḣ,1 norm into ) 1/q u Ḣ,1 R n ) = H n u ρ u ) dµ. Definition.3. Let I be a time interval. The space-time norm is defined by ux, t) Lq L r I H n ) = I ) ) q/r 1/q ux, t) r dµ dt. H n 5

Proposition.4 Sobolev embedding). Assume 1 < q 1 q < and σ 1, σ R. If σ 1 n q 1 σ n q, then we have the Sobolev embedding Hq σ1 1 H n ) Hq σ H n ). For the proof see [1, 6] and the references cited therein. Proposition.5. See Proposition.5 in [5]) If q >, < τ < 3 and σ + τ n n q, then we have the Sobolev embedding H σ,τ H n ) L q H n )..4 Technical Lemma In this subsection we introduce a point-wise estimate for radial H,1 H n ) functions. Lemma.6. Let n 3. We have a point-wise estimate fr) n r 1/ sinh r) ρ f H,1 H n ) for any radial function f Ḣ,1 H n ). Proof. Without loss of generality we assume that f is smooth and has compact support, since functions of this kind are dense in the space of radial Ḣ,1 H n ) functions. Let us first pick up a large radius R so that Suppf) B, R) and then calculate = = = = R R R R R [ ] d dr f sinhρ r) dr [ fr sinh ρ r + ρf sinh r) ρ 1 cosh r ] dr [ ] f r ) sinh ρ r + ρ f sinh r) ρ 1) cosh r dr + [ ] f r ) sinh ρ r + ρ f sinh r) ρ 1) cosh r dr R R [ fr ) ρ f ] R sinh ρ r dr + ρ ρ ) f sinh r) ρ dr n H n [ f ρ f ]dµ = f Ḣ,1 H n ). ρsinh r) ρ 1 cosh r df ) ρf d [ sinh r) ρ 1 cosh r ] dr dr As a result, we have fr ) sinh ρ r = and finish the proof. r [ ] d dr f sinhρ r) n r ) 1/ f Ḣ,1 H n ) { r dr r ) 1/ [ ] 1/ d dr f sinhρ r) dr} Remark.7. The upper bound given in Lemma.6 is optimal. Given a smooth cut-off function ϕ : R [, 1] satisfying { 1, if 1/ r 3/; ϕr) =, if r < 1/4 or r > 7/4; we consider a family of radial functions defined in H n : { r f R r) = 1/ ρ ϕr/r), if R 1; e ρr r 1/ ϕr/r), if R > 1. One can check that f R r) H,1 H n ) 1 and f R R) R 1/ sinh R) ρ. 6

3 Strichartz Estimates In this section we introduce a family of Strichartz estimates compatible with initial data in the energy space H,1 H n ) L H n ) for all dimensions n 3. This immediately leads to a local theory when 3 n 5, which will be introduced in the next section. 3.1 Preliminary Results Definition 3.1. Let n 3. A couple p 1, q 1 ) is called admissible if 1/p 1, 1/q 1 ) belongs to the set { 1 T n =, 1 ), 1 ], 1 ) + n 1 n 1 }. p 1 q 1 p 1 q 1 Let us recall the Strichartz estimates with inhomogeneous Sobolev norms. Theorem 3.. See Theorem 6.3 in [1]) Let p 1, q 1 ) and p, q ) be two admissible pairs. The real numbers σ 1 and σ satisfy σ 1 βq 1 ) = n + 1 1 1 ) ; σ βq ) = n + 1 1 q 1 1 ). q Assume ux, t) is the solution to the linear shifted wave equation t u H n + ρ )u = F x, t), x, t) H n I; u t= = u ; t u t= = u 1. 5) Here I is an arbitrary time interval containing. Then we have u L p 1 L q 1 I H n ) + u, t u) CI;H σ 1 1, 1 H σ 1 1, 1 H n )) C u, u 1 ) H σ 1 1, 1 H σ 1 1, 1 + F H n ) L p I;H σ 1 +σ 1 q H n )) The constant C above does not depend on the time interval I. The following lemma see lemma 5.1 in [1]) plays an important role in the proof of the Strichartz estimates above. It is obtained by a complex interpolation and the Kunze-Stein phenomenon. Lemma 3.3. There exists a constant C > such that, for every radial measurable function κ on H n, every q, q < and f L q H n ), we have Here γ = min{q, q} q+ q 1/Q f κ L q C f L q sinh r) n 1 Φ r)) γ κr) dr) Q. and Q = q q q+ q. The following lemma comes from a basic Fourier analysis Lemma 3.4. If F L 1 L R H n ), then we have e ±isd F, s) ds F L 1 L R H n ). L H n ) ). 7

3. Strichartz Estimates for H,1 L data Definition 3.5. We fix χ : R [, 1] to be an even, smooth cut-off function so that { 1, r < 1; χr) =, r > 3/. Lemma 3.6. Given any < p, q < and σ R we have χd)d 1 D1 σ e ±itd f f L H n ). L p L q R H n ) Proof. Consider the operator χ D)D D σ e itd defined by the Fourier multiplier and its kernel κ σ t r) = const. λ χ λ)λ λ + ρ + 1) 1 σ e itλ If t, we recall cλ) λ 1 + λ ) n 3 and obtain κ σ t r) χ λ)λ λ + ρ + 1) 1 σ e itλ Φ λ r) cλ) dλ. χ λ)λ λ + ρ + 1) 1 σ e itλ Φ r) cλ) dλ Φ r). 6) Now let us consider the other case t >. By the definition of cλ) we have Thus we can rewrite the kernel κ σ t Here the function cλ) Γiλ + ρ) = C n Γiλ) = C n Γiλ + ρ) λ Γiλ + 1). into κ σ t r) = aλ)e itλ Φ λ r)dλ. aλ) = const.χ λ)λ + ρ 1 σ Γiλ + ρ) + 1) Γiλ + 1) is smooth in R. In addition, the function Φ λ r) satisfies π Φ λ r) =c n cosh r sinh r cos θ) iλ ρ sin θ) n dθ; π Φ λ r) Φ r) = c n cosh r sinh r cos θ) ρ sin θ) n dθ e ρr r + 1); λ Φ λ r) = c n i π [ cosh r sinh r cos θ) iλ ρ lncosh r sinh r cos θ) ] sin θ) n dθ; π λ Φ λ r) c n cosh r sinh r cos θ) ρ lncosh r sinh r cos θ) sin θ) n dθ ) sup lncosh r sinh r cos θ) θ [,π] We apply integration by parts on κ σ t and obtain κ σ t r) = 1 aλ)φ λ r)de itλ ) it = 1 [ a)φ r)e it a)φ r) ] 1 it it = i t a)φ r) + i t Φ r) e ρr rr + 1). λ [aλ)φ λ r)] e itλ dλ [ λ aλ))φ λ r) + aλ) λ Φ λ r))] e itλ dλ. 8

As a result we have κ σ t r) a) Φ r) + 1 t t [ λ aλ) Φ λ r) + aλ) λ Φ λ r) ] dλ t 1 e ρr r + 1) + t 1 Φ r) + λ Φ λ r) ) dλ t 1 e ρr r + 1). Now let us apply Lemma 3.3 with kernel κ σ t and q = q >. In this case γ = 1 and Q = q/ > 1. The integral in Lemma 3.3 can be estimated by: If t, we have + If t >, we have + sinh r) n 1 Φ r)) γ κ σ t r) Q dr sinh r) n 1 Φ r)) γ κ σ t r) Q dr + + + sinh r) ρ Φ r) Φ r) q/ dr sinh r) ρ e ρr r + 1) ) 1+q/ dr 1. sinh r) ρ e ρr r + 1)[ t 1 e ρr r + 1) ] q/ dr + t q/ sinh r) ρ e 1+q/)ρr r + 1) q+1 dr t q/. According to Lemma 3.3, we immediately have q > ) χ D)D D σ e ±itd L q L q { 1, t ; t 1, t >. 7) Let us consider the operators Tf = χd)d 1 D1 σ e ±itd f; T F = TT F = χd)d 1 D1 σ e isd F, s) ds; χ D)D D σ e ±it s)d F, s) ds; This is clear that T is an operator from L 1 R, H 1 σ, 1 H n )) L p L q to L H n ), and that T is an operator from L H n ) to L R, H σ 1,1 H n )). Furthermore, the estimate 7) guarantees the inequality TT F L p L q R H n ) F L p L q R H n ) holds as long as p >. By the TT argument see [9], for instance), we obtain χd)d 1 D1 σ e ±itd f = Tf L p L q R H n ) f L H n ) L p L q R H n ) thus finish the proof. 9

Theorem 3.7 Strichartz estimates for H,1 L initial data). Let n 3. If < p, q < and σ satisfy p + n 1 n 1 ; σ n + 1 1 q 1 ) ; q then there exists a constant C, so that the solution u to linear shifted wave equation t u H nu = F, x, t) H n I with initial data u, u 1 ) satisfies D 1 σ u Lp L q I H n ) + u, t u) CI;H,1 L H n )) C u, u 1 ) H,1 L H n ) + F L1 L I H n )). Proof. Without loss of generality, let us assume I = R. We start with the free linear propagation u L with a pair of arbitrary initial data u, u 1 ). In fact we have u L, t) = costd)u + sintd) D u 1. 8) Since D 1 σ u L solves the free linear shifted wave equation with initial data D 1 σ u, D 1 σ u 1 ), Theorem 3. immediately gives D 1 σ u L L p L q R H n ) D 1 σ u, u 1 ) H σ 1, 1 H σ 1, 1 H n ) = u, u 1 ) H 1, 1 H 1, 1 H n ). We rewrite this in the form of operators by the identity 8) and obtain D 1 D1 σ e ±itd f L p L q R H n ) f H 1, 1 H n ). 9) Given an arbitrary f L H n ), the combination of 9) and Lemma 3.6 gives D 1 D1 σ e ±itd f Lp L q R H n ) χd)d 1 D1 σ e ±itd f L p L q + D 1 D1 σ e ±itd 1 χd))f L p L q f L H n ) + 1 χd))f 1 H, 1 H n ) f L H n ). 1) We combine this with Lemma 3.4 and obtain D 1 D1 σ e ±it s)d F, s) ds F L 1 L R H n ); Lp L q R H n ) D 1 D1 σ e ±it s)d F, s) ds F L1 L R H n ). Lp L q R H n ) According to Theorem 1.1 in [3], we also have a truncated version of the first inequality above t D 1 D1 σ e ±it s)d F, s) ds F L 1 L R H n ). Lp L q R H n ) Therefore we have t D 1 D1 σ e ±it s)d F, s) ds F L 1 L R H n ). 11) L p L q R H n ) 1

By the identities u, t) = costd)u + sintd) D u 1 + t t u, t) = D sintd)u + costd)u 1 + sint s)d F, s) ds; D t [cost s)d] F, s) ds; we can combine the estimates 1), 11) and Lemma 3.4 to finish the proof. If we choose σ = n+1 1 1 q ) in the Theorem 3.7 and apply the Sobolev embedding, we obtain another version of Strichartz estimates. Theorem 3.8. Assume n 3. If p, q) satisfies 1 p, 1 q, 1 ) 1 ; p + n q n 1; then there exists a constant C, such that the solution u to the linear shifted wave equation I) t u H n + ρ )u = F x, t), x, t) H n I; u t= = u ; t u t= = u 1 satisfies u L p L q I H n ) + u, t u) CI;H,1 L H n )) C u, u 1 ) H,1 L H n ) + F L 1 L I H n )). Here we attach two figures, in which the grey regions illustrate all possible pairs p, q) that satisfy the conditions in Theorem 3.8, for two different cases: dimension 3 Figure 1) and higher dimensions Figure ).The lighter grey regions represent the pairs allowed in Theorem 3.7, while the darker grey regions show new admissible pairs, which are obtained via the Sobolev embedding. n Remark 3.9. Theorem 3.8 also holds for the pair p, q) =, n ) by the Sobolev embedding H,1 H n ) L n/n ) n given in Proposition.5. Thus the pair, n ) is also marked as admissible in the figures. 4 Local Theory Definition 4.1. Assume 3 n 5. We define the following space-time norm if I is a time interval u Y I) = u L n+ n+) = u L pc L pc ). n L n I H n ) I Hn Theorem 3.8 claims that if u is a solution to the linear equation t u H n + ρ )u = F with initial data u, u 1 ), then we have u Y I) + u, t u) CI;H,1 L H n )) C u, u 1 ) H,1 L H n ) + F L1 L I H n )). Furthermore, a basic computation shows F u) L 1 L I H n ) u pc Y I) ; ) F u 1 ) F u ) L1 L I H n ) c n u 1 u Y I) u 1 pc 1 Y I) + u pc 1 Y I). Combining these estimates with a fixed-point argument, we obtain the following local theory. Our argument is standard, see for instance, [, 8, 16, 17,, 1, 3] for more details.) 11

1/q 1/ 1/6 3/q+1/p=1/ 1/ 1/p Figure 1: Admissible pairs p, q) in dimension 3 Definition 4. Local solution). Assume 3 n 5. We say ut) is a solution of the equation CP1) in a time interval I, if u, t), t u, t)) CI; H,1 L H n )), with a finite norm u Y J) for any bounded closed interval J I so that the integral equation holds for all time t I. t u, t) = S L, t)u, u 1 ) + sin[t τ)d] F u, τ)) dτ D Theorem 4.3 Unique existence). For any initial data u, u 1 ) H,1 L H n ), there is a maximal interval T u, u 1 ), T + u, u 1 )) in which the equation CP1) has a unique solution. Proposition 4.4 Scattering with small data). There exists a constant δ 1 > such that if u, u 1 ) H,1 L H n ) < δ 1, then the Cauchy problem CP1) has a solution u defined for all t R with u Y R) u, u 1 ) H,1 L H n ). Proposition 4.5 Standard finite time blow-up criterion). If T + <, then u Y [,T+)) =. Similarly if T <, then u Y T,]) =. Proposition 4.6 Finite Y norm implies scattering). Let u be a solution to CP1). If u Y [,T+)) <, then T + = and there exists a pair u +, u+ 1 ) H,1 L H n ), such that lim u, t), t u, t)) S L t)u +, u+ 1 ) H,1 L H n ) =. t + A similar result holds in the negative time direction as well. Theorem 4.7 Long-time perturbation theory). See also [17, 4]) Let M be a positive constant. There exists a constant ε = ε M) >, such that if ε < ε, then for any approximation solution 1

1/q 1/ 1 1 n 1/p + n/q = n/-1 n-3 n-1) n-3 n 1/ 1/p Figure : Admissible pairs p, q) in dimension 4 or higher ũ defined on H n I I) and any initial data u, u 1 ) H,1 L H n ) satisfying t ũ H n + ρ )ũ = F ũ) + ex, t), x, t) H n I; ũ Y I) < M; ũ, ), t ũ, )) H,1 L H n ) < ; ex, t) L 1 L I H n ) + S L, t)u ũ, ), u 1 t ũ, )) Y I) ε; there exists a solution ux, t) of CP1) defined in the interval I with the given initial data u, u 1 ) and satisfying ux, t) ũx, t) Y I) CM)ε; ) ) u, t) ũ, t) u ũ, ) S t u, t) t ũ, t) L t) CM)ε. u 1 t ũ, )) sup t I 5 A Second Morawetz Inequality H,1 L H n ) In my recent joint work with Staffilani [5], we proved a Morawetz-type inequality T+ u p+1 4p + 1) dµdt < H p 1 E, n T if u is a solution to the energy sub-critical, defocusing, semi-linear shifted wave equation t u H n + ρ )u = u p 1 u on H n. The main idea is to choose a suitable function a and then apply the informal computation d t u D α ad α u + u a dt H n = Dβ ud β D α ad α u ) dµ 1 H 4 n ) dµ H n u a ) dµ + p 1 p + 1) H n u p+1 a ) dµ 13

on a solution u. In this section we prove a second and stronger Morawetz inequality by choosing a different function ar) = r and applying the same informal computation. The calculation turns out to be a little more complicated since the singularity of r at the origin make it necessary to apply a smooth cut-off technique at this point. Theorem 5.1. Let 3 n 5 and u, u 1 ) H,1 L H n ) be initial data. Assume u is the solution of CP1) in the defocusing case with initial data u, u 1 ). Then the energy E = 1 u, t) H,1 H n ) + 1 tu, t) L H n ) + 1 u, t) pc+1 p c + 1 L pc+1 H n ) is a constant in the maximal lifespan T, T + ). In addition, we have a Morawetz-type inequality T+ T Hn ρcosh x ) ux, t) n/n ) dµx)dt ne. sinh x Remark 5.. Throughout this section we will only consider real-valued solutions for convenience. Complex-valued solutions can be handled in the same manner. Remark 5.3. It suffices to prove Theorem 5.1 with an additional assumption u H 1 H n ). This is a consequence of the standard approximation techniques. Given any initial data u, u 1 ) Ḣ,1 L H n ), we can find a sequence u,n, u 1,n ) H 1 L H n ), such that u,n, u 1,n ) u, u 1 ) Ḣ,1 L Eu,n, u 1,n ) Eu, u 1 ). Let u and {u n } n Z + be the corresponding solutions to CP1) with these initial data. According to the perturbation theory we have u n u Y J) ; u, t), t u, t)) u n, t), t u n, t)) CJ; Ḣ,1 L ) for any closed bounded interval J = [ T 1, T ] contained in the maximal lifespan of u. A limiting process n shows that the energy conservation law and the Morawetz inequality hold for u as long as they hold for the solutions {u n } n Z +. In the rest of the section we always assume that u is a solution to CP1) with initial data u, u 1 ) H 1 L H n ). 5.1 Preliminary Results Lemma 5.4. We have u, t u) C T, T + ); H 1 L H n )). Proof. We have already known u, t u) C T, T + ); H,1 L H n )) by Definition 4.. Therefore it is sufficient to show u C T, T + ); L ). This is clearly true since t u C T, T + ); L ) and u L. Lemma 5.5. Assume n 3. Let r, Θ) be the polar coordinates on H n. Then the function ar) = r is smooth in H n except for r = and satisfies ar) = 1; D a ; ρ cosh r H na = sinh r ; r H na = ρ sinh r ; r 4ρ cosh r H na = sinh 3 r ; H n 4ρ1 ρ) cosh r Hna = sinh 3. r 14

The proof follows an explicit calculation, thus we omit the details. In fact, the inequality D a is a well-known consequence of the fact that the hyperbolic space H n has a negative constant sectional curvature. The following formula also helps in the calculation regarding the Laplace operator H n H n = ρ cosh r + r sinh r r + 1 sinh r S n 1. Definition 5.6. Let ψ : [, ) [, 1] be a non-increasing smooth cut-off function satisfying { 1, r < 1; ψr) =, r >. If δ, 1/1], we define a radial cut-off function ψ δ r) = ψδr)1 ψr/δ)) on H n. It is clear that δ, 1/δ < r < /δ; ψ δ r, Θ) 1/δ, δ < r < δ;, otherwise. Lemma 5.7. See Lemma 4.11 in [5], also [6] for more general case) Let P ε be the smoothing operator defined by the Fourier multiplier λ e ε λ. Given any q <, we have P ε L q H n ) L q H n ) C q < for any ε, 1) 1. Furthermore, if v L q H n ), then v P ε v L q as ε. Space-time smoothing operator Let φt) be a smooth, nonnegative, even function compactly supported in [ 1, 1] with 1 1 φt)dt = 1. Given a closed interval [ T 1, T ] T, T + ), we define u ε and F ε as the smooth version of u and F if ε < ε = 1/) min{1/1, T + T, T T 1 }. u ε t) = +1 1 φs) P ε u, t + sε)ds; F ε t) = +1 The function u ε is a smooth solution to the shifted wave equation 1 φs) P ε F u, t + sε))ds. 1) t u ε H n + ρ )u ε = F ε 13) in the time interval [ T 1, T ]. Combining the fact u Y [ T1 ε,t +ε ]) <, the inequality F u ε ) F ε L1 L I) F u ε ) F u) L1 L I) + F u) F ε L1 L I) ) C u ε u Y I) u ε pc 1 Y I) + u pc 1 Y I) + F u) F ε L1 L I), with Lemma 5.4 and Lemma 5.7, we immediately have Lemma 5.8. Let t be an arbitrary time in [ T 1, T ]. The functions u ε and F ε satisfy lim F u ε) F ε L1 L ε [ T 1,T ] H n ) = ; lim u ε, t ), t u ε, t )) u, t ), t u, t )) H ε 1 L H n ) = ; lim u ε, t ) u, t ) L ε pc+1 H n ) = ; M 1 := sup u ε, t u ε ) C[ T1,T ];H 1 L H n )) <. ε<ε The third line is a combination of the Sobolev embedding H 1 L pc+1 and the second line. 15

5. Energy Conservation Law Proposition 5.9 Energy Conservation Law). The energy Et) = 1 u, t) H,1 H n ) + 1 tu, t) L H n ) + 1 u, t) pc+1 p c + 1 L pc+1 H n ) is a finite) constant independent of t T, T + ). Proof. Without loss of generality, let us assume t, T + ). We can choose a time interval [ T 1, T ] T, T + ) so that t < T, smooth out the solution u as in 1) and define 1 E ε,δ t) = u εx, t) ρ u εx, t) + 1 tu ε x, t) + 1 ) p c + 1 u εx, t) pc+1 ψ δ dµx). H n We differentiate in t and obtain E ε,δt) = [F ε F u ε )] t u ε )ψ δ dµ D α ψ δ D α u ε ) t u ε )dµ H n H n Here we need to use the fact that u ε solves 13) and follow the same calculation we carried on in Section 4. of [5]. A basic integration shows t t E ε,δ t ) E ε,δ ) F ε F u ε ) t u ε ψ δ dµdt + D α ψ δ D α u ε ) t u ε dµdt H n H n F ε F u ε ) L 1 L tu ε L L + δt u ε L [,t ];H 1 H n )) t u ε L L + δ n 1 t u ε L B,1) [,t ]) t u ε L B,1) [,t ]). If we send first δ, then ε and apply Lemma 5.8, we obtain the energy conservation law. Remark 5.1. The energy of a solution to CP1) in the focusing case is also a constant under the same assumptions, because the defocusing assumption has not been used in the argument above. 5.3 Proof for the Morawetz inequality Since the argument is similar to the one we carried on in Section 4 of [5], we will give an outline of the proof only. Given an arbitrary time interval [ T 1, T ] T, T + ), we can smooth out u, the non-linear term F u) as in 1) and define M ε,δ t) = t u ε x, t) H n D α ad α u ε x, t) + u ε x, t) a ) ψ δ dµx) for t [ T 1, T ]. Here we choose the function a = r whose properties have been given in Lemma 5.5 and the smooth cut-off function ψ δ in Definition 5.6. We first combine a basic differentiation in t, integration by parts, the facts D β ψ δ D β a in B, 1) and D a to obtain M ε,δt) 1 n u ε pc+1 aψ δ dµ + 1 H 4 n [F ε F u ε )] H n H n a) u ε ψ δ dµ eε, δ) D α ad α u ε + u ε a ) ψ δ dµ. 14) Throughout the proof we use the notation eε, δ) to represent possibly different) error terms satisfying eε, δ) δm 1 + M pc+1 1 ) + δ n u ε, u ε ) C B,1) [ T 1,T ]) + δ n 1 u ε pc+1 C B,1) [ T 1,T ]) + δn 1 t u ε C B,1) [ T 1,T ]). 16

Next we apply integration by parts again and make an estimate for an arbitrary t [ T 1, T ]. Dα ad α u ε, t) + u ε, t) a L H n ;ψ δ dµ) ) ) u ε ρ + ρ ρ H n sinh u ε ψ δ dµ + eε, δ) x uε ρ u ε ) ψ δ dµ + eε, δ) M1 + eε, δ). 15) H n As a result we have T [F ε F u ε )] D α ad α u ε + u ε a ) ψ δ dµdt T 1 H n F ε F u ε ) L 1 L [ T 1,T ] H n ) Dα ad α u ε + u ε a M 1 + eε, δ)) 1/ F ε F u ε ) L 1 L [ T 1,T ] H n ). L [ T 1,T ];L H n ;ψ δ dµ)) We substitute the last integral in 14) by the lower bound above, integrate both sides and obtain an inequality M ε,δ T ) M ε,δ T 1 ) 1 T u ε pc+1 aψ δ dµdt + 1 T a) u ε ψ δ dµdt n H 4 n H n T 1 M 1 + eε, δ)) 1/ F ε F u ε ) L1 L [ T 1,T ] H n ) [T + T 1 ]eε, δ). 16) On the other hand, we can find an upper bound of Mt ) for each t [ T 1, T ] by 15). Mt ) = t u ε x, t ) D α ad α u ε x, t ) + 1 ) H u εx, t ) a ψ δ dµx) n 1 t u ε + D α ad α u ε + 1 ) ) H u ε a ψ δ dµ n 1 t u ε + u ε ρ u ε ) ψ δ dµ + eε, δ). H n We combine this with the inequality 16), substitute a, a by their specific expressions by Lemma 5.5 and then let δ to obtain 1 n T T 1 Hn ρcosh x ) u ε pc+1 dµdt + sinh x T T 1 T 1 Hn ρρ 1)cosh x ) 3 u ε dµdt sinh x E u ε, T 1 ), t u ε, T 1 )) + E u ε, T ), t u ε, T )) + M 1 F ε F u ε ) L 1 L [ T 1,T ] H n ). Here E is the energy of the linear shifted wave equation on H n defined as E v, v 1 ) = 1 v1 + v ρ v ) dµ = 1 H v H,1 H n ) + 1 v 1 L H n ). n Sending ε to zero gives 1 n T Hn ρcosh x ) u pc+1 T Hn ρρ 1)cosh x ) 3 u dµdt + dµdt sinh x T 1 sinh x E u T 1 ), t u T 1 )) + E ut ), t ut )) Eu, u 1 ). T 1 Since the argument above is valid for any time interval [ T 1, T ] satisfying T < T 1 < < T < T +, we can finally finish the proof by letting T 1 T and T T +. 17

6 Scattering Results with Radial Initial Data Theorem 6.1. Assume 3 n 5. Let u, u 1 ) be a pair of radial initial data in H,1 L H n ). Then the solution u to CP1) in the defocusing case with initial data u, u 1 ) exists globally in time and scatters. More precisely, the solution u satisfies Its maximal lifespan T u, u 1 ), T + u, u 1 )) = R. There exist two pairs u ±, u± 1 ) H,1 L H n ) such that lim u, t), t u, t)) S L t)u ±, u± 1 ) H,1 L H n ) =. t ± Proof. First of all, the Morawetz inequality see Theorem 5.1) reads T+ T Hn ρcosh x ) ux, t) n/n ) dµx)dt neu, u 1 ) <. 17) sinh x According to Lemma.6 and the energy conservation law we obtain a point-wise estimate ) ρ 1/ ux, t) x 1/ ρ cosh x sinh x ) ρ u, t) H,1 H n ) u, t) H,1H sinh x n) ) ρ 1/ ρ cosh x [Eu, u 1 )] 1/. sinh x As a result, the inequality 17) implies T+ T u n n + 1 ρ 1/ dµdt [Eu, u 1 )] H n 1/ 1+ ρ 1/. A basic calculation shows n/n ) + 1/ρ 1/) = n + 1)/n ). Thus we have u L n+1) n+1) n L n T, T + ) H n ). 18) For a small positive number κ, we define a pair p, q) by 1 p c κ) p, 1 ) ) n + κ q n + 1), n n + 1) = 1, 1 ). 19) It is clear that p, q) p c, p c ) as κ +. As a result, if we fix κ to be a sufficiently small positive number, we have 1/p, 1/q, 1/). The definition 19) also guarantees that 1/p + n/q = n/ 1. We apply Strichartz estimates Theorem 3.8), the energy conservation law, as well as an interpolation between L p L q and L n+1) n+1) n L n norms to obtain u Y [a,b]) + u Lp L q [a,b] H n ) C u, a), t u, a)) H,1 L H n ) + C F u) L 1 L [a,b] H n ) CEu, u 1 )) 1/ + C u κ n+1) n+1) u pc κ L n L n [a,b] H n L ) p L q [a,b] H n ). ) Here [a, b] may be any sub-interval of T, T + ). Let us define M = Eu, u 1 )) 1/ and choose a small constant η > so that CM > CM + Cη κ CM) pc κ. According to the fact 18) we can fix a time a, T + ) sufficiently close to T +, so that the inequality u n+1) n+1) < η L n L n [a,t +) H n ) 18

holds. Given any time b [a, T + ), the inequality ) implies u Y [a,b]) + u Lp L q [a,b] H n ) CM + Cη κ u Y [a,b]) + u Lp L q [a,b] H n )) pc κ. By a continuity argument in b we obtain the following upper bound independent of b [a, T + ). u Y [a,b]) + u L p L q [a,b] H n ) < CM. We make b T + and finally conclude that u Y [a,t+)) CM <. The global existence and scattering in the positive time direction immediately follows Proposition 4.5 and Proposition 4.6. The other time direction can be handled in the same way since the shifted wave equation is time-reversible. Acknowledgements I would like to express my sincere gratitude to Professor Gigliola Staffilani for her helpful discussions and comments. References [1] J-P. Anker, V. Pierfelice, and M. Vallarino. The wave equation on hyperbolic spaces Journal of Differential Equations 51): 5613-5661. [] H. Bahouri, and P. Gérard. High frequency approximation of solutions to critical nonlinear equations. American Journal of Mathematics 111999): 131-175. [3] M. Christ and A. Kiselev. Maximal functions associated to filtrations Journal of Functional Analysis 1791): 49-45. [4] T. Duyckaerts, C.E. Kenig, and F. Merle. Classification of radial solutions of the focusing, energy-critical wave equation. Cambridge Journal of Mathematics 113): 75-144. [5] T. Duyckaerts, C.E. Kenig, and F. Merle. Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation. The Journal of the European Mathematical Society 13, Issue 311): 533-599. [6] M. Cowling, S. Giulini, S. Meda L p L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces I Duke Mathematical Journal 71993): 19-15. [7] A. French. Existence and Scattering for Solutions to Semilinear Wave Equations on High Dimensional Hyperbolic Space 14), Preprint Arxiv: 147.695v1. [8] J. Ginibre, A. Soffer and G. Velo. The global Cauchy problem for the critical nonlinear wave equation Journal of Functional Analysis 11199): 96-13. [9] J. Ginibre, and G. Velo. Generalized Strichartz inequality for the wave equation. Journal of Functional Analysis 1331995): 5-68. [1] M. Grillakis. Regularity and asymptotic behaviour of the wave equation with critical nonlinearity. Annals of Mathematics 13199): 485-59. [11] M. Grillakis. Regularity for the wave equation with a critical nonlinearity. Communications on Pure and Applied Mathematics 45199): 749-774. [1] S. Helgason. Radon-Fourier transform on symmetric spaces and related group representations, Bulletin of the American Mathematical Society 711965): 757-763. 19

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