Blow up for critical wave equations on curved backgrounds
aa r X i v : . [ m a t h . A P ] M a r BLOW UP FOR CRITICAL WAVE EQUATIONS ON CURVEDBACKGROUNDS
J. NAHAS AND S. SHAHSHAHANI
Abstract.
We extend the slow blow up solutions of Krieger, Schlag, andTataru to semilinear wave equations on a curved background. In particular,for a class of manifolds (
M, g ) we show the existence of a family of blow-upsolutions with finite energy norm to the equation ∂ t u − ∆ g u = | u | u, with a continuous rate of blow up. In contrast to the case where g is theMinkowski metric, the argument used to produce these solutions can onlyobtain blow up rates that are bounded above. Introduction
We study the nonlinear focusing wave equation on a curved, three dimensionalbackground manifold (
M, g ),(1.1) ∂ t u − ∆ g u = | u | u, where u : M × R → R . Much is known about well posedness and blow up of solutionsto this equation when the metric g is flat. Some exciting work–for example [8], [16],[1], and references therein, has been done for hyperbolic backgrounds, but to ourknowledge little is known about it in the case of a more general metric. We constructa continuum of blow up solutions for this equation such that the energy norm, k u k H ≡ Z M ( | ∂ t u | + h∇ u, ∇ u i g ) dvol, is finite, like the work of Krieger, Schlag, and Tataru in [14]. The scenario ofsolutions that blow up with bounded energy is referred to as type II blow up. It iswidely believed that type II blow up solutions are intimately connected to the timeindependent, finite energy solutions of an equation, the soliton solutions. Since blowup is a local phenomenon, and M is approximately flat near the point r = 0, wealso base our construction on perturbations of the rescaled soliton of the equationfrom the Minkowski space, rather than from M . This soliton can be thought of asa function on the tangent space to M at r = 0, identified with a neighborhood ofthe point via the chosen coordinates. We will denote it by W ( | x | ) = 1 q | x | . That the relevant time independent solution has the flat space as background issimilar to the bubbling off of harmonic maps from R , in the study of the harmonicmap heat flow on compact Riemann surfaces by Struwe in [20]. Using finite speed ofpropagation for the wave equation, we are able to modify this ’local’ flat-backgroundsolution, to a solution on M × R .Our main result is the following theorem: Theorem 1.1.
Let ( M, g ) be a smooth three dimensional manifold, and ( r ( p ) , θ ( p ) , φ ( p )) be a coordinate chart in some open set U ⊆ M such that the metric g satisfies g ( r, θ, φ ) = dr + g ΩΩ ( r )( dθ + sin θ dφ ) , where g ΩΩ ( r ) is analytic in U , and obeys the estimate (cid:12)(cid:12)(cid:12)(cid:12) ˙ g ΩΩ ( r ) g ΩΩ ( r ) − r (cid:12)(cid:12)(cid:12)(cid:12) . r. (1.2) Then for ν ∈ (1 / , , and arbitrary δ > , there exists a solution u ∈ H ν ( M ) to (1.1) such that for small enough t , u ∈ C ( U ) , and u ( r ) = t − − ν W ( t − − ν r ) + ε ( r, t ) , (1.3) with Z r ≤ t ( 12 |∇ ε | + 12 | ∂ t ε | + 16 | ε | ) p det g dr dθ dφ → , and Z r>t ( 12 |∇ u | + 12 | ∂ t u | + 16 | u | ) p det g dr dθ dφ ≤ δ. These solutions are of considerable interest, because it is conjectured that forthe set of all initial data that lead to blow up for the nonlinear wave equation, typeII solutions are the boundary of this set (see Remark 1.2 of [5]). This remains achallenging problem to investigate.
Remark 1.1.
Nontrivial examples of a pair ( M, g ) that satisfies (1.2) include thethree sphere and three hyperbolic space. Remark 1.2.
Let R rrrr (0) denote the diagonal radial component of the curvaturetensor as r → . It is a straightforward exercise to show that for small enough r ,the difference between g and the flat metric, r − g ΩΩ ( r ) , satisfies r − g ΩΩ ( r ) = 13 R rrrr (0) r + o ( r ) . Remark 1.3.
In contrast to [14], where there was no upper bound on the rate ofthe blow up, our methods only produce blow up for ν ≤ . Whether these rates canbe extended above this threshold is an interesting open question. We expect the corresponding local well posedness, global well posedness, andblow up results developed for the flat case to hold for this equation, which webriefly recall. Global well posedness for initial data in the energy space up tothe energy of the first soliton, W ( | x | ), was proven in [10] using the concentrationcompactness techniques developed in [2]. This result is optimal, in the sense thatfor any energy above this threshold, there exist blow up solutions, see [14] (andalso [15]).Duyckaerts and Merle obtained a type II blow up classification in [7] for thesemilinear wave equation. They showed that up to the symmetries of the equation,for solutions with free energy equal to that of W ( | x | ), there is only one solution otherthan W ( | x | ) that does not scatter. This solution, W − ( x, t ), scatters backwards intime, and converges exponentially to W ( | x | ) forwards in time. This was followed bya result of Duyckaerts, Kenig, and Merle in [5] and [6] that for the nonlinear waveequation, type II blow up solutions must have a profile that is a rescaled solitonplus radiation, assuming the free energy of the initial data is close enough to thatof W ( | x | ). This fundamental theorem for type II blow up solutions allows one tofocus on determining the rescaling parameter λ ( t ) and the radiation term.This construction of slow blow up has been adapted to a number of differentsituations. It was originally produced for the charge one equivariant wave mapsfrom R to a sphere by Krieger, Schlag, and Tataru in [12] and wave maps to a LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 3 surface of rotation by Cˆarstea in [3], then for the focusing nonlinear wave equationin [14], and for the Yang-Mills equation in [13]. The construction has also beenmodified to produce blow up at infinity for the nonlinear wave equation in threedimensions by Donninger and Krieger in [4], and the range the blow up exponent ν was extended by Krieger and Schlag in [11]. We also note that in an upcomingpaper [19], the second author has further extended this machinery to the case of awave map between two dimensional spheres, with a similar upper bound on ν .While the solutions produced by this machinery are not smooth, Rodnianski andSterbenz in [18] prove smooth type II blow up for k -equivariant wave maps to asphere, where k ≥
4. Moreover, these solutions are stable within the equivariantclass of data. This was extended to all k ≥
1, as well as equivariant solutions to theYang-Mills equation by Rodnianski and Rapha¨el in [17]. The result correspondingto the nonlinear wave equation was proven by Hillairet and Rapha¨el in [9].From our ansatz (1.3), we treat the laplacian term in (1.1) as a flat laplacianplus a perturbation. Just as in [14], we cannot solve (1.1) directly for ε , but musttake several renormalization steps before performing a perturbation step. We mustmodify the procedure from [14] to account for an extra term arising from the curvedlaplacian ∆ g . It is this extra term, see Lemma 3.5, which bounds from above ourblow up rates in the perturbation step.In what follows we first introduce some notation, describe the renormalizationsteps in Section 2, and prove the main result in Section 3, which mostly consists ofthe perturbation step.1.1. Notation.
To solve (1.1), we will use the assumptions in Theorem 1.1 to picka coordinate chart U where the metric has the desired form (1.2). Choose r > U ′ = { ( r, θ, φ ) | r < r } is a compactly contained open subset of U , and let ϕ ( r ) ∈ C ∞ ( B r (0)) be suchthat ϕ ( r ) = 1 for r ≤ r . The conserved ’energy’ of (1.1) will be denoted by E ( u ) = 14 π Z M (cid:16) | ∂ t u | + 12 |∇ u | − | u | (cid:17) dvol but for functions compactly supported in U ′ , this will become E ( u ) = Z R + (cid:16) | ∂ t u | + 12 | ∂ r u | − | u | (cid:17) g ΩΩ ( r ) dr. Instead of solving (1.1) directly, we will first solve(1.5) ∂ t u − ∂ r u − r ∂ r u − ϕ ( r ) (cid:18) ˙ g ΩΩ ( r ) g ΩΩ ( r ) − r (cid:19) ∂ r u = | u | u for r ≥
0, then modify these solutions to satisfy (1.1). For convenience, we define κ ( r ) = ϕ ( r ) (cid:18) ˙ g ΩΩ ( r ) g ΩΩ ( r ) − r (cid:19) r . Renormalization step
Our aim in this section is to prove the following theorem.
Theorem 2.1.
Given N ≥ there is an approximate solution u k − to the equation − ∂ t u + △ g u + u = 0 , which has the form u k − ( t, r ) = λ / ( t ) (cid:20) W ( R ) + 1( tλ ) O ( R ) (cid:21) , J. NAHAS AND S. SHAHSHAHANI such that the corresponding error e k − := − ∂ t u k − + △ g u k − + u k − satisfies Z r ≤ t | e k − ( r, t ) | r dr = O ( t N ) t → . Here the O ( · ) are uniform in r ∈ [0 , t ] and t ∈ (0 , t ) for a small fixed t , and λ = t − − ν , R = λr. Remark 2.1.
Recall that △ g is given by ∂ r + ˙ g ΩΩ g ΩΩ ∂ r . Note that this construction islocal, and in particular restricted to the light cone at t . By choosing t sufficientlysmall, we may therefore assume that we are in the region where ϕ ( r ) ≡ , andignore ϕ in (1.5). We start by giving an outline of the strategy. The idea is to construct approxi-mate solutions u k by iteratively adding correction terms v k so that u k = u k − + v k . The starting point will be the rescaled soliton u = λ / W ( λr ) , where λ = λ ( t ) = t − − ν . The error at step k will be e k = (cid:3) g u k + u k . If we let ε = u − u k − and linearize the corresponding equation around ε = 0 andreplace u k − by u (or in other words look at the approximate solution for thelinearized operator around u ) we get (cid:0) − ∂ t + ∂ r + ˙ g ΩΩ g ΩΩ ∂ r + 5 u (cid:1) ε + e k − ≈ . We split this up into two cases. If r ≪ t we ignore the time derivative (which weexpect to be smaller) and replace the linearized equation by (cid:0) ∂ r + ˙ g ΩΩ g ΩΩ ∂ r + 5 u (cid:1) ε + e k − ≈ . If r ≈ t we expect u to be small and replace it by zero in the linearized equationto get (cid:0) − ∂ t + ∂ r + ˙ g ΩΩ g ΩΩ ∂ r (cid:1) ε + e k − ≈ . LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 5
We will use these two simplified equations to construct our approximate solutions.This will be made more precise shortly, but first we need some notation and defi-nitions. The following notation will be used throughout this section. N k ( v ) = X j =0 (cid:18) j (cid:19) u j k − v − j ,N k +1 ( v ) = 5( u k − u ) v + X j =0 (cid:18) (cid:19) u j k v − j ,K ( v ) = κ ( r ) r∂ r v,λ = t − − ν ,R = λr,a = rt ,b = ( tλ ) − = t ν ,b = r ,b = ( t ν λ ) − = t ,B = ( b , b , b ) ,β = ν − > − , C = { ( r, t ) (cid:12)(cid:12) ≤ r < t, < t < t } .b = b a and is not technically necessary, but we keep it as an extra variablebecause it arises naturally. As such b and b can be thought of as a gain of t and b as a gain of t ν . Note that since our calculations will be done in the light cone C , there are constants B , B , and B such that b j ∈ [0 , B j ] , and a ∈ [0 , . LetΩ := [0 , × [0 , ∞ ) × [0 , B ] × [0 , B ] × [0 , B ] , and denote the projection of Ω onto the last four factors by Ω a , and the projectiononto the last three factors by Ω a,R . Also, keep in mind that5 u ( t, R ) = 45 λ (3 + R ) . Defintion 2.1. Q is the algebra of continuous functions q : [0 , → R with thefollowing properties: : (i) q is analytic in [0 , with an even expansion at . : (ii) Near a = 1 we have an absolutely convergent expansion of the form q = q ( a ) + i = ∞ X i =1 (1 − a ) β ( i )+1 ∞ X j =0 q i,j ( a )(log(1 − a )) j with analytic coefficients q , q i,j , such that for each i only finitely many ofthe q i,j are not identically equal to zero. The exponents β ( i ) are of the from X k ∈ K ((2 k − / ν − /
2) + X k ∈ K ′ ((2 k − / ν − / , where K and K ′ are finite sets of positive integers. Defintion 2.2. Q ′ is the space of continuous functions q : [0 , → R with thefollowing properties: : (i) q is analytic in [0 , with an even expansion at . J. NAHAS AND S. SHAHSHAHANI : (ii) Near a = 1 we have an absolutely convergent expansion of the form q = q ( a ) + i = ∞ X i =1 (1 − a ) β ( i ) ∞ X j =0 q i,j ( a )(log(1 − a )) j with analytic coefficients q , q i,j , such that for each i only finitely many ofthe q i,j are not identically equal to zero. β ( i ) are as above. Defintion 2.3. S m ( R k (log R ) l ) is the class of analytic functions v : [0 , ∞ ) → R with the following properties: : (i) v vanishes of order m at R = 0 and v ( R ) = R m P j = ∞ j =0 c j R j for small R. : (ii) v has a convergent expansion near R = ∞ ,v = ∞ X i =0 l + i X j =0 c ij R k − i (log R ) j . Finally,
Defintion 2.4. a) S m ( R k (log R ) l , Q ) is the class of analytic functions v : Ω → R so that : (i) v is analytic as a function of R, b i v : Ω a → Q : (ii) v vanishes to order m at R = 0 and is of the form v ≈ R m j = ∞ X j =0 c j ( a, b , b , b ) R j around R = 0 . : (iii) v has a convergent expansion at R = ∞ ,v ( · , R, b , b , b ) = ∞ X i =0 l + i X j =0 c ij ( · , b , b , b ) R k − i (log R ) j where the coefficients c i : Ω a,R → Q are analytic with respect to b , b , b . b) IS m ( R k (log R ) l , Q ) is the class of analytic functions w on the cone C whichcan be represented as w ( t, α ) = v ( a, R, b , b , b ) , v ∈ S m ( R k (log R ) l , Q ) . Note that this representation is in general not unique. We are now ready to startthe proof of the theorem.
LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 7
Proof.
We prove that the corrections v k can be chosen so that they and the corre-sponding errors e k satisfy v k − ∈ k − X j =0 λ / ( t ν λ ) j ( tλ ) k − j ) IS ( R (log R ) m k , Q )(2.1) t e k − ∈ k − X j =0 λ / ( t ν λ ) j ( tλ ) k − j ) IS ( R (log R ) p k , Q ′ )(2.2) v k ∈ k − X j =0 λ / ( t ν λ ) j ( tλ ) k +1 − j ) IS ( R (log R ) p k , Q )(2.3) t e k ∈ k − X j =0 λ / ( t ν λ ) j ( tλ ) k − j ) (cid:2) IS ( R − (log R ) q k , Q )+ b IS ( R (log R ) q k , Q ′ )+ X i =2 b i IS ( R (log R ) q k , Q ′ ) (cid:3) . (2.4)Here m k , p k , q k are integers whose exact value is not important for us, and whichsatisfy m = p = 0 , q = 1 . The two step construction of the correction termshere corresponds to the different approximations of the linearized equation in theregimes r ≪ t and r ≈ t which we hinted at earlier. It is convenient for future useto introduce the operator D := + r∂ r = + R∂ R . Step 0 : e = u − (cid:3) u = − ∂ t (cid:20) λ (cid:18) λ ′ λ (cid:19) D W (cid:21) + κ ( r ) r∂ r h λ W i = λ " − (cid:18) λ ′ λ (cid:19) ′ D W − (cid:18) λ ′ λ (cid:19) D W + κ ( r ) R∂ R W . Therefore, there are constants c , c , and c , depending on ν and whose exact valueis irrelevant for us, such that t e = λ / (cid:20) c (cid:18) − R / R / (cid:19) + c (cid:18) − R + R (1 + R / (cid:19) + c t κ ( r ) R (1 + R / (cid:21) ∈ λ (cid:0) IS ( R − ) + t κ ( r ) IS ( R − ) (cid:1) ⊆ λ IS ( R − ) . Step 1 :Write e k − = P j e k − ,j where t e k − ,j ∈ λ / ( t ν λ ) j ( tλ ) k − − j ) (cid:2) IS ( R − (log R ) q k − , Q )+ X i =1 b i IS ( R (log R ) q k − , Q ′ ) (cid:3) . If e k − ,j = e k − ,j ( t, a, B, R ) , we let e k − ,j ( t, a, R ) := e k − ,j ( t, a, , R ) , and e k − ,j = e k − ,j − e k − ,j (unless k = 1 in which case we let e = e ). Wealso let e k − = P j e k − ,j and similarly for e k − . Define v k − ,j by taking t and J. NAHAS AND S. SHAHSHAHANI a as parameters and requiring that v k − ,j solve the following ODE in R, subjectto vanishing boundary conditions at R = 0 :( tλ ) (cid:16) − ∂ R − r ∂ R − R ) (cid:17) v k − = t e k − ,j . It follows from Lemma 2.1 in [11] that v k − ,j ∈ λ / ( t ν λ ) j ( tλ ) k − j ) IS ( R (log R ) m k , Q ) . Letting v k − = P j v k − ,j we see that (2.1) is satisfied. Step 2 :The error from the previous step is e k − = e k − + N k − ( v k − ) + K ( v k − ) + E t v k − + E a v k − , where N k − and K are defined above, E t v k − contains the terms in ∂ t v k − whereno derivatives fall on a and E a v k − the terms in ( ∂ t − ∂ r − r ∂ r ) v k − where atleast one derivative falls on a. We study t e k − term by term, and for this we workon the level of v j − ,j and sum over j at the end. e k − belongs to the right hand side of (2.2) because the b j s contribute the neces-sary gain of time decay. Indeed, for b and b this follows from the definition ofthese variables, and for b from the observation that b = t a . For the other terms in the error, write v k − ,j = t ˜ β w k − ,j ( a, R ) where ˜ β = ˜ β ( k, j )is defined by t ˜ β = λ / ( t ν λ ) j ( tλ ) k − j ) , and w k − ,j ∈ IS ( R (log R ) m k , Q ) . For t E t wejust need to note that t ∂ t ( t ˜ β IS ( R (log R ) m k )) ⊆ t ˜ β IS ( R (log R ) m k ) . To simplify the notation, for t E a we drop the indices 2 k − j and write w = w k − ,j . Note that t E a is then a linear combination the following terms t ( ∂ t t ˜ β ) aw a , t ˜ β aw a , t ˜ β aRw aR , t ˜ β a − Rw aR , t ˜ β (1 − a ) w aa . That t E a has the right form follows from the fact that a∂ a , a − ∂ a , (1 − a ) ∂ a map Q to Q ′ .With the same notation as before we have t κr∂ r v = t κt ˜ β aw a + t κt ˜ β Rw R , which implies that K ( v k +1 ) is of the right form.For N k − ( v k − ) we work on the level of v k − (rather than v k − ,j ) and begin bynoting that u k − − u ∈ λ / ( tλ ) IS ( R (log R ) n , Q ) , for some integer n depending on k. Linear term (in v k +1 ): note that u k − − u =( u k − − u ) + 4( u k − − u ) u + 6( u k − − u ) u + 4( u k − − u ) u . LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 9
We compute (suppressing Q for simplicity of notation) t ( u k − − u ) v k − ∈ k − X j =0 t ˜ β ( k,j ) ( tλ ) ( tλ ) IS ( R (log R ) m k ) IS ( R (log R ) n ) ⊆ k − X j =0 t ˜ β ( tλ ) IS ( R (log R ) p k ) ⊆ k − X j =0 t ˜ β a IS ( R − (log R ) p k ) ⊆ k − X j =0 t ˜ β IS ( R − (log R ) p k ) . Similarly, t ( u k − − u ) u v k − ∈ k − X j =0 t ˜ β ( tλ ) ( tλ ) S ( R − ) S ( R (log R ) m k ) S ( R (log R ) n ) ⊆ k − X j =0 t ˜ β ( tλ ) S ( R (log R ) p k ) ⊆ k − X j =0 t ˜ β S ( R − (log R ) p k ) , and t ( u k − − u ) u v k − ∈ k − X j =0 ( tλ ) t ˜ β ( tλ ) S ( R − ) S ( R (log R ) n ) S ( R (log R ) m k ) ⊆ k − X j =0 t ˜ β ( tλ ) S ( R (log R ) p k ) ⊆ k − X j =0 t ˜ β S ( R − (log R ) p k ) . Finally, t ( u k − − u ) u v k − ∈ k − X j =0 t ˜ β ( tλ ) ( tλ ) S ( R − ) S ( R (log R ) m k ) S ( R (log R ) n ) ⊆ k − X j =0 t ˜ β S ( R − (log R ) p k ) . Quintic term: t v k − ∈ k − X j ...j =0 t λ / IS ( R , Q )( t ν λ ) j + ··· + j ) ( tλ ) k − j + ··· + j ) ⊆ k − X j,j ,...j =0 a t ˜ β ( k,j ) IS ( R − , Q )( t ν λ ) j + ··· + j ) ( tλ ) k − − j + ··· + j ) ⊆ k − X j,j ,...j =0 a b k − − j + ··· + j )1 b ( j + ··· + j )3 t ˜ β ( k,j ) IS ( R − , Q ) ⊆ k − X j =0 t ˜ β ( k,j ) IS ( R − , Q ) . Quadratic term: note that u k − ∈ λ / IS ( R − ) (we are suppressing the algebra Q again). t u k − v k − ∈ k − X j,l =0 ( tλ ) λ IS ( R − ) IS ( R (log R ) m k )( t ν λ ) j + l ) ( tλ ) k − l − j ) ⊆ k − X j,l =0 t ˜ β ( k,j ) b l b k − − l )1 IS ( R − (log R ) p k , Q ) ⊆ k − X j =0 t ˜ β ( k,j ) IS ( R − (log R ) p k , Q ) . Cubic term: t u k − v k − ∈ k − X j ,j ,j =0 t t ˜ β ( k,j ) t ˜ β ( k,j ) t ˜ β ( k,j ) λIS ( R − ) IS ( R (log R ) m k ) ⊆ k − X j,j ,j =0 b ( j + j )3 b k − − j − j )1 t ˜ β ( k,j ) IS ( R (log R ) p k ) , which has the right form. The quartic term is similar. Step 3 :Near R = ∞ we isolate the principal part ˜ e k − of e k − by ignoring terms thatinvolve a factor of b , b or b or decay at least as fast as R − (log R ) p k +2 . Let usbe more precise and write t ˜ e k − = t P k − j =0 ˜ e k − ,j , where t ˜ e k − ,j = t ˜ β ( k,j ) h p k X i =0 q i,j ( a ) R (log R ) i + p k +1 X i =0 ˜ q i,j ( a )(log R ) i b p k X i =0 ˜ q i,j ( a ) R (log R ) i + b p k +1 X i =0 ˜˜ q i,j ( a )(log R ) i i = ( tλ ) t ˜ β ( k,j ) h p k X i =0 aq i,j ( a )(log R ) i + b p k +1 X i =0 (˜ q i,j ( a ) + a ˜ q i,j )(log R ) i + b p k +1 X i =0 ˜˜ q i,j ( a )(log R ) i i , (2.5)for some q i,j , ˜ q , i,j , ˜˜ q i,j ∈ Q ′ , with ˜ q p k +1 ,j = 0 (to be precise q i,j , etc, depend on k aswell, but we gloss over this to simplify the notation). We want to define ˜ v k,j by t ( − ∂ t + ∂ r + 2 r ∂ r )˜ v k,j = − t ˜ e k − ,j . Comparing with (2.5) we seek a solution of the form ˜ v k,j ( t, a ) = t ˜ β − ν w k,j ( a )where w k,j has the form w k,j = p k X i =0 W i k,j ( a )(log R ) i + b X l =1 , p k +1 X i =0 ˜ W i,l k,j (log R ) i + b p k +1 X i =0 ˜˜ W i k,j ( a )(log R ) i . (2.6) LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 11
We match the powers of the logarithms in (2.5) and (2.6) to obtain the followingequations for W i k,j t (cid:18) − ∂ t + ∂ r + 2 r ∂ r (cid:19) (cid:16) t ˜ β − ν W i k,j ( a ) (cid:17) = t ˜ β − ν ( aq i,j ( a ) − F i,j ( a )) ,t (cid:18) − ∂ t + ∂ r + 2 r ∂ r (cid:19) (cid:16) t ˜ β ˜ W i,l k,j ( a ) (cid:17) = t ˜ β ( a l − ˜ q i,j ( a ) − ˜ F li,j ( a )) , l = 1 , ,t (cid:18) − ∂ t + ∂ r + 2 r ∂ r (cid:19) (cid:16) t ˜ β + ν ˜˜ W i k,j ( a ) (cid:17) = t ˜ β + ν (˜˜ q i,j ( a ) − ˜˜ F i,j ( a )) . Here F i,j (and similarly ˜ F li,j, and ˜˜ F i,j ) can be determined form equations (2.5)and (2.6), and depends only on a, W i +12 k,j , ∂ a W i +12 k,j and W i +22 k,j , where we are usingthe convention W i k,j = 0 for i > p k (and ˜ W i,l k,j, = ˜˜ W i k,j = 0 for i > p k + 1).Note that we are again suppressing the k dependency in the notation. After somerearrangement, this can be written as a system of equations in the variable a as L β W i k,j = aq i,j ( a ) − F i,j ( a ) , β = ˜ β − ν,L β ˜ W i,l k,j = a l − ˜ q li,j ( a ) − ˜ F li,j ( a ) , β = ˜ β, (2.7) L β ˜˜ W i k,j = ˜˜ q i,j ( a ) − ˜˜ F i,j ( a ) , β = ˜ β + ν, where L β is defined by L β = (1 − a ) ∂ a + 2( a − + βa − a ) ∂ a − β + β. It is proved in equation (2,70) in [11] that (2.7) can be solved with zero Cauchydata at a = 0 yielding solutions which satisfy W i k,j ∈ a Q , i = 0 , . . . , p k , ˜ W i,l k,j ∈ a l +1 Q , i = 0 , . . . , p k + 1 , l = 1 , , (2.8) ˜˜ W i k,j ∈ a Q , i = 0 , . . . , p k + 1 . We are now almost ready to define the next correction v k . However, we need tomodify the expression for ˜ v k,j to ensure an even expansion in R and eliminate thesingularity of log R at R = 0 . With the notation h R i = √ R , we let v k,j = t β ( k,j ) R h R i − p k X i =0 a − W i k,j (cid:18)
12 log(1 + R ) (cid:19) i + p k +1 X i =0 ˜ W i,l k,j (cid:18)
12 log(1 + R ) (cid:19) i + bR h R i − p k +1 X i =0 a − ˜ W i, k,j (cid:18)
12 log(1 + R ) (cid:19) i + b p k +1 X i =0 ˜˜ W i k,j (cid:18)
12 log(1 + R ) (cid:19) i ! , (2.9)and define v k = P k − j =0 v k,j . That (2.3) is satisfied is a consequence of (2.8).
Step 4 :The error from the previous step is e k − + K ( v k ) + N k ( v k ) + ( e k − − ( − ∂ t + ∂ r + 2 r ∂ r ) v k ) , where K and N k are defined above, and e k − = e k − − e k − with e k − = P k − j =0 e k − ,j and t e k − ,j = t ˜ β ( k,j ) " p k X i =0 q i,j R h R i (cid:18)
12 log(1 + R ) (cid:19) i + p k +1 X i =0 ˜ q i,j (cid:18)
12 log(1 + R ) (cid:19) i b p k +1 X i =0 ˜ q i,j R h R i (cid:18)
12 log(1 + R ) (cid:19) i + b p k +1 X i =0 ˜˜ q i,j (cid:18)
12 log(1 + R ) (cid:19) i . Again we treat the error term by term. e k − contains one set of terms having afactor of b , b or b , which belong to the right hand side of (2.4) by construction,and another set coming from subtracting the leading terms in the expansion in R near R = ∞ . To control the latter, notice that after subtracting the first twoleading terms, the next term grows more slowly (decays faster) by a factor of R , so in view of (2.4) it suffices to show that IS ( R − (log R ) q k , Q ′ ) ⊆ IS ( R − (log R ) q k , Q ) + b IS ( R (log R ) q k , Q ′ ) . (2.10)For this, we decompose an element w ∈ IS ( R − (log R ) q k , Q ′ ) as w = (1 − a ) w + a w = (1 − a ) w + R ( tλ ) w, implying w ∈ IS ( R − (log R ) q k , Q ) + b IS ( R (log R ) q k , Q ′ )as desired.We consider t (cid:16) e k − ,j − ( − ∂ t + ∂ r + r ∂ r ) v k,j (cid:17) next. This would be zero if wereplaced R √ R by R and log(1 + R ) by log R. This means that the so this errorinvolves terms where at least one t or r derivative falls on R h R i , or on the differenceof the logarithmic terms. The analysis is similar for the two cases, and as the log R terms are already treated in [11] we focus on the former. In view of (2.9), for thecontribution of ∂ r r we need to look at t t ˜ β − ν W i k,j (cid:0) log(1 + R ) (cid:1) i ∂ r (cid:16) R √ R (cid:17) r = t ˜ β − ν a − W i k,j (cid:0) log(1 + R ) (cid:1) i (1 + R ) / ∈ t ˜ β R (cid:0) log(1 + R ) (cid:1) i (1 + R ) / Q , where we have used (2.8) for the last step. Using (2.10) this can be placed in theright hand side of (2.4). For the contribution of ∂ r , we consider the cases whenonly one or both r derivatives fall on R h R i . The former is taken care of just as thecase of ∂ r above. For the latter note that ∂ r (cid:0) R √ R (cid:1) = − λ R (1 + R ) / = − R r (1 + R ) / , which is treated as before. The contribution of ∂ t can be dealt with similarly.To control t K ( v k,j ) = t κr∂ r v k,j we again use the representation (2.9). Not-ing that a∂ a sends Q to Q ′ we can place this contribution in b IS ( R (log R ) q k , Q ′ ) , which is consistent with (2.4).It remains to consider N k ( v k ) . First note that u k − − u ∈ λ / ( tλ ) IS ( R (log R ) n , Q ) , LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 13 for some positive integer n, so u k − ∈ λ / W ( R ) + λ / ( tλ ) IS ( R (log R ) n , Q ) ⊆ λ / S ( R − ) + λ / ( tλ ) IS ( R (log R ) n , Q ) . The contribution of the quintic term can be computed as t v k ∈ k − X j ,...,j =0 t t ˜ β ( k +1 ,j )+ ··· + ˜ β ( k +1 ,j ) IS ( R (log R ) p k , Q ) ⊆ k − X j,j ,...,j =0 t ˜ β ( k,j ) (1 + R ) / IS ( R − (log R ) q k , Q )( tλ ) b − k − j + ··· + j )1 b − ( j + ··· + j )3 ⊆ k − X j =0 t ˜ β ( k,j ) IS ( R − (log R ) q k , Q ) . For the linear term we have t u k − v k ∈ ( tλ ) (cid:16) S ( R − ) + a IS ( R − (log R ) n , Q ) (cid:17) × k − X j =0 t ˜ β ( k +1 ,j ) IS ( R (log R ) p k , Q ) ⊆ k − X j =0 λ / ( tλ ) ( t ν λ ) j ( tλ ) k − j )+2 IS ( R − (log R ) n , Q ) IS ( R (log R ) p k , Q ) ⊆ k − X j =0 λ / ( t ν λ ) j ( tλ ) k − j ) IS ( R − (log R ) q k , Q ) . The quadratic, cubic, and quartic terms can be treated similarly. (cid:3) Perturbation step
After the renormalization step we must now solve the equation ∂ tt ε − r ∂ r ( r ∂ r ε ) − λ ( t ) W ( λ ( t ) r ) ε = N k − ( ε ) + K ( ε ) + e k − . (3.1)We change variables several times in order to work with coordinates better adaptedto the self similar nature of our solution. Let ε ( t, x ) = v ( τ ( t ) , λ ( t ) x ), y = λx , τ bea time coordinate that satisfies ˙ λ = dλdτ , and ∂ t ε ( t, r ) = τ ′ ( t )( v τ + ˙ λλ − y∂ y v ) . Letting Rv = ˜ ε , and L = − ∂ R − W ( R ), we obtain the equation( ∂ τ + ˙ λλ − R∂ R ) ˜ ε − ˙ λλ − ( ∂ τ + ˙ λλ − R∂ R )˜ ε + L ˜ ε = λ − R [ N k − ( R − ˜ ε ) + K ( R − ˜ ε ) + e k − ] . (3.2)By transforming this equation so that L is diagonal, (3.2) will become somethingclose enough to a transport equation to solve. We now recall some of the spectralproperties of L .On the domainDom( L ) = { f ∈ L ((0 , ∞ )) : f, f ′ ∈ AC([0 , R ]) ∀ R, f (0) = 0 , f ′′ ∈ L ((0 , ∞ )) } the operator L is self-adjoint. Therefore there exists a unitary operator F thatdiagonalizes L . The spectrum of L is continuous for ξ ≥
0, and has a lone discrete negative eigenvalue, which we will denote by ξ − . This discrete negative eigenvaluenecessitates some comments about notation. The derivative with respect to thespectral variable ∂ ξ will be understood to be 0 for this discrete part. The continuouspart of the spectral measure we denote by ρ . From the WKB ansatz, one expectsthe eigenfunctions f ξ ( R ) corresponding to ξ > f ξ ( R ) ≃ e − iR √ ξ . Define(3.4) K = F R∂ R F − − ξ∂ ξ . Upon applying F to (3.2), and using (3.4) several times, we obtain (cid:18) ∂ τ + ξ −
00 ( ∂ τ − β ( τ ) ξ∂ ξ ) + ξ (cid:19) F ˜ ε = β ( I − K )( ∂ τ − β ( τ ) ξ∂ ξ ) F ˜ ε − β ( K − K + 2[ ξ∂ ξ , K ]) F ˜ ε + λ − F R ( N k − ( R − ˜ ε ) + λ − F R ( K ( R − ˜ ε ) + e k − ) . (3.5)We will solve (3.5) with a contraction mapping argument with the norm(3.6) k u k L ∞ ,N L | ξ | s/ ρ dξ = sup τ> τ N | u ( ξ − , τ ) | + (cid:18)Z R + | u ( ξ, τ ) | | ξ | s/ ρ dξ (cid:19) / ! . By using the definition of L , along with F , it is easy to prove the following resultfrom [14] about the equivalence of (3.6) with a weighted Sobolev norm. Lemma 3.1. k R F − u k L ∞ ,N H s . k u k L ∞ ,N L | ξ | s/ ρ dξ . k R F − u k L ∞ ,N H s . Solving for the discrete component of ˜ ε requires only elementary techniques, sowe focus on the continuous components.The operator H will denote the solution map for the equation(( ∂ τ − β ( τ ) ξ∂ ξ ) + ξ ) u = f, lim τ →∞ u ( τ ) = 0 , which has the following smoothing property (see Corollary 6.3 from [14]). Lemma 3.2.
There exists a C not depending on N such that kH b k L ∞ ,N − L , / αρ + k ( ∂ τ − β ( τ ) ξ∂ ξ ) H b k L ∞ ,N − L ,αρ ≤ C N k b k L ∞ ,N L ,αρ . This smoothing effect is sufficient to counter the loss of regularity, quantified inthe following lemmas, from the terms on the right hand side of (3.2).We would like to use Proposition 6.7 from [14] to handle the term N k − , howeverour N k − differs from that of the flat background case, due to a slight differencein the renormalization sequence { u k − } . We quote the proposition despite this,since our { u k − } obey enough of the same properties as in the flat case, that theproof for the proposition is still valid. Lemma 3.3.
Assume that N is large enough and ≤ α < ν . Then the map F ˜ ε → λ − F R ( N k − ( R − ˜ ε )) is locally Lipschitz from L ∞ ,N − L ,α +1 / ρ to L ∞ ,N L ,αρ . From (3.3), we expect that the operator K is bounded. This is proven in [14]: LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 15
Lemma 3.4. a.) The operator K maps (3.7) K : L ,αρ → L ,αρ . b.) In addition, we have the commutator bound (3.8) [ K , ξ∂ ξ ] : L ,αρ → L ,αρ . Both statements hold for all α ∈ R . The K term is specific to the curved background in our problem, which createsan upper bound on the blow up rates. Lemma 3.5.
Let ν ∈ (0 , . Then the map F ˜ ε → λ − F RK ( R − ˜ ε ) is locally Lipschitz from L ∞ ,N − L ,α +1 / ρ to L ∞ ,N L ,αρ .Proof. This term becomes problematic, the intuition being as follows. The argu-ments to produce these blow up solutions heavily rely on the scale invariance of theenergy of the equation. The κ term breaks the scale since it is a curvature term(see Remark 1.2). By Lemma 3.1, it suffices to prove the result for the weightedSobolev spaces, and we further change to cartesian coordinates which facilitates theuse of these spaces. We also recall the estimate from [21] that(3.9) k (1 − ∆) s/ ( f h ) − f (1 − ∆) s/ ( h ) − h (1 − ∆) s/ ( f ) k ≤ k f k ∞ k (1 − ∆) s/ h k . Let R = qP i =1 X i . Using (3.9), and the support of κ , k λ − Rκ ( Rλ ) λ∂ R ( R − ˜ ε ) k L N − τ H α (3.10) = k λ − κ ( Rλ ) X i =1 ϕ ( R/ (2 λ )) X i ∂ X i ( R − ˜ ε ) k L N − τ H α . X i =1 k λ − k κ ( Rλ ) k ∞ k ϕ ( R/ (2 λ )) X i ∂ X i ( R − ˜ ε ) k H α k L N − τ + X i =1 k λ − k (1 − ∆) α κ ( Rλ ) k ∞ k ϕ ( R/ (2 λ )) X i ∂ X i ( R − ˜ ε ) k L k L N − τ . (3.11)By the Hausdorf inequality, k (1 − ∆) α κ ( Rλ ) k ∞ = k (1 − ∆) α κ ( Rλ ) k ∞ . k (1 + | ζ | α ) λ ˆ κ ( λζ ) k . (1 + λ − α ) . (3.12)For the other terms, we use the support of ϕ : k ϕ ( R/ (2 λ )) X i ∂ X i ( R − ˜ ε ) k H α = k (1 − ∆) α ϕ ( R/ (2 λ )) X i ∂ X i ( R − ˜ ε ) k L = k [(1 − ∆) α , ϕ ( R/ (2 λ )) X i ] ∂ X i ( R − ˜ ε ) k L + k ϕ ( R/ (2 λ )) X i (1 − ∆) α ∂ X i ( R − ˜ ε ) k L . k ∂ X i ( R − ˜ ε ) k L + λ k (1 − ∆) α ∂ X i ( R − ˜ ε ) k L . k R − ˜ ε k H α + λ k R − ˜ ε k H α . (3.13)With (3.11), (3.12), and (3.13), and writing t in terms of τ we find that k λ − Rκ ( Rλ ) λ∂ R ( R − ˜ ε ) k L N − τ H α . k τ − − /ν k R − ˜ ε k H α k L N − τ . The result follows since ν ∈ (0 , (cid:3) Remark 3.1.
For the renormalization step, all of the computations were local. Theperturbation step is a global computation, because it involves the nonlocal operator F . We must take steps to reconcile these two complementary tools in the proof ofTheorem 1.1 below. We can now use Theorem 2.1 followed by Lemma 3.5 to produce the requiredsolution to (1.1). The steps are nearly identical to the conclusion section of [14],with some small modifications due to the curved background.
Proof of Theorem 1.1.
From Theorem 2.1 we obtain an approximate solution u k +1 and an error term e k +1 , but only within the light cone. As per Remark 3.1, wemust extend u k +1 , and e k +1 before solving (3.1). We extend them to compactlysupported functions ˜ u k +1 , ˜ e k +1 of the same regularity in r ≤ t . It follows fromlemmas 3.4, 3.3, 3.5, 3.2, taking N large enough, then k large enough, and thecontraction mapping theorem that there exists a solution ε to (3.1). The function u ( t, r ) = ˜ u k +1 ( t, r ) + ε ( t, r ) is a solution to (1.5) within the light cone. Sincelim t → Z K t ( |∇ ( u − u ) | + | u − u | ) g ΩΩ ( r ) dr = 0 , and lim t → Z K ct ( |∇ u | + | u | ) g ΩΩ ( r ) dr = 0 , it is possible to choose a t so that the set with r ≤ t is contained in U ′ , and Z r>t ( |∇ (˜ u k +1 ( r, t ) + ϕ ( r ) ε ( r, t )) | + | ˜ u k +1 ( r, t ) + ϕ ( r ) ε ( r, t ) | ) g ΩΩ ( r ) dr . δ. We now solve the initial value problem for (1.1) with initial data (˜ u k +1 ( r, t ) + ϕ ( r ) ε ( r, t ) , ∂ t ˜ u k +1 ( r, t ) + ϕ ( r ) ∂ t ε ( r, t )). Call this solution u . By the transportproperties of the wave equation, u is equal to u k +1 + ε within the light cone, so itblows up at the origin. Because of global well posedness for small energies, showingthe energy outside the lightcone remains small will imply the solution must blowup only at the origin. Observe that E ( u ) ≈ E ( W ) + O ( δ ) . Therefore the energy outside the lightcone obeys Z K Ct ( 12 | ∂ t u | + 12 | ∂ r u | − | u | ) g ΩΩ ( r ) dr . δ. Since supp u ⊂ U ′ , we also have the Sobolev estimate Z r ≥ t | u | g ΩΩ ( r ) dr . (cid:18)Z r ≥ t | ∂ r u | g ΩΩ ( r ) dr (cid:19) . Therefore either Z K Ct ( | ∂ t u | + | ∂ r u | ) g ΩΩ ( r ) dr . δ, or Z K Ct ( | ∂ t u | + | ∂ r u | ) g ΩΩ ( r ) dr & . The first alternative holds for the initial data. By continuity, it holds for all times. (cid:3)
Acknowledgments:
We would like to thank Joachim Krieger for suggesting theproblem and Roland Donninger for helpful discussions.
LOW UP FOR CRITICAL WAVE EQUATIONS ON CURVED BACKGROUNDS 17
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M. E. Taylor , Tools for PDE , vol. 81 of Mathematical Surveys and Monographs, AmericanMathematical Society, Providence, RI, 2000. Pseudodifferential operators, paradifferentialoperators, and layer potentials. ´Ecole Polytechnique F´ed´erale de Lausanne, MA B1 487, CH-1015 Lausanne
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