Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime
aa r X i v : . [ m a t h . A P ] D ec Blow-up and lifespan estimate to a nonlinear waveequation in Schwarzschild spacetime
Ning-An Lai a,b, ∗ a Institute of Nonlinear Analysis and Department of Mathematics,Lishui University, Lishui 323000, China b School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Abstract
In [13], Luk proved global existence for semilinear wave equations in Kerrspacetime with small angular momentum( a ≪ M ) (cid:3) g K φ = F ( ∂φ ) , when the quadratic nonlinear term satisfies the null condition. In this work, wewill show that if the null condition does not hold, at most we can have almostglobal existence for semilinear wave equations with quadratic nonlinear term inSchwarzschild spacetime, which is the special case of Kerr with a = 0 (cid:3) g S φ = ( ∂ t φ ) , where (cid:3) g S denotes the D’Alembert operator associated with Schwarzschild met-ric. What is more, if the power of the nonlinear term is replaced with p satisfying3 / ≤ p <
2, we still can show blow-up, no matter how small the initial dataare. We do not have to assume that the support of the data should be far awayfrom the event horizon.
Keywords:
Nonlinear wave equations, blow-up, Schwarzschild spacetime,lifespan ∗ Corresponding Author
Email address: [email protected] (Ning-An Lai)
Preprint submitted to Journal of L A TEX Templates December 18, 2018 . Introduction
In this work we are going to study the nonlinear wave equation in Schwarzschildspacetime (cid:3) g S u = | u t | p , in R + t × Σ , (1)where g S denotes the Schwarzschild metric g S = F ( r ) dt − F ( r ) − dr − r dω (2)with F ( r ) = 1 − Mr , the constant M > dω is the standard metric on the unit sphere S . The D’Alembertoperator associated with the Schwarzschild metric g has the form (cid:3) g S = 1 F ( r ) (cid:16) ∂ t − Fr ∂ r ( r F ) ∂ r − Fr ∆ S (cid:17) , (3)where ∆ S denotes the standard Laplace-Beltrami operator on S . R + t × Σ iscalled the exterior of the black hole: R + t × Σ = R + t × (2 M, ∞ ) × S . Such Cauchy problem of semilinear wave equations with derivatives in thenonlinear term was first proposed by John [6], in which he studied the followingquasilinear wave equation in R ∂ t u ( t, x ) − ∆ u ( t, x ) = ∂ t | au ( t, x ) + b∂ t u ( t, x ) | p (4)and proved that the solutions blow up at a finite time when p = 2 and thecompact supported data satisfy some conditions. His method is also applicableto the following Cauchy problem ∂ t u ( t, x ) − ∆ u ( t, x ) = | u t | p (5)with 1 < p ≤
2. Assuming the data is radial symmetric and p >
2, Sideris[17] showed that the Cauchy problem (5) in R admits global small amplitudesolutions. Klainerman [8] and [9] proved global existence of small data solutions2o nonlinear wave equations of a more general form ∂ t u ( t, x ) − ∆ u ( t, x ) = F ( ∂ t u, ∇ x u ) , | D α F ( w ) | ≤ A | w | p − α , p > α, | w | ≤ p > n +1 n − . Hence it is easy to see that problem (5) in R admits a criticalpower p = 2. In R , Schaeffer [16] showed blow-up result for problem (5)with p = 3 and conjectured that it is the critical exponent(see also John [7]).Later, Agemi [1] proved that problem (5) in R with 1 < p ≤ n = 1 for problem (5) is essentially due to Masuda [15],in which he established blow-up result in R n ( n = 1 , ,
3) with power p = 2.Hidano and Tsutaya [4] and Tzvetkov [18] proved global existence for p > p c ( n )and n = 2 , R n with 1 < p ≤ n +1 n − . Recently, Hidano, Wang andYokoyama [5] established global existence for p > p c ( n ) and n ≥
4, by assumingthe data are radially symmetrical. We also refer the reader to a recent paper[11] for semilinear wave equations related to Glassey’s conjecture with scatteringdamping.Recently, Zhou and Han [21] established blow-up result to the initial bound-ary value problem (5) in exterior domain with 1 < p ≤ n +1 n − ( n ≥ R n ( n ≥ (cid:3) g S u = | u | p . In the Regge-Wheeler coordinates, by choosing special initial data they provedthe blow-up results in two cases: (I) 1 < p < √ < p < √ p > √
2, which also holds for Kerr black hole with small angular3omentum if the small data have compact support. One can also find the globalexistence result for p > p = 5 in [14].For a black hole metric g , one of the key ingredients to prove asymptoticstability is to understand the nonlinear toy problem (cid:3) g φ = ( ∂φ ) , (7)especially when the quadratic nonlinear term satisfies some special structure,i.e., null condition, due to the reason that null condition has served as a goodmodel problem for the study of stability of the flat Euclidean spacetime. Luk [13]studied the semilinear wave equations on a Kerr spacetime with small angularmomentum( a ≪ M ) (cid:3) g K φ = F ( ∂φ ) , where g K denotes the Kerr metric, which is parametrized by two parameters( M, a ), representing the mass and angular momentum of a black hole, respec-tively. And the nonlinear term F is at least quadratic. Assuming the nonlinearterm F satisfies the null condition, then he can prove global existence for any ini-tial data that are sufficiently small. In this work we are devoted to studying theCauchy problem of semiliner wave equation with derivatives in Schwarzschildspacetime, a special example of Kerr black hole with a = 0, as stated in (1). Wewill show that if the quadratic nonlinear term does not satisfy the null condition,the the solution will blow up in a finite time, no matter how small the initial dataare. Also, the lifespan estimate of exponential type will be established. Whatis more, if ≤ p <
2, we then show that the lifespan is of polynomial type.The first key step of our proof is to rewrite the equation in the Regge-Wheelercoordinates, and then choose the test function which was introduced in [3]. Thesecond one is to construct an auxiliary functional which satisfies an ordinarydifferential inequality of Riccati type, following the idea of [21]. We mentionthat the approach in [3] is based on applying variants of the classical Kato’slemma to an auxiliary quantity, constructed from the corresponding solutionsand test function. 4 emark 1.1.
We do not have to assume that the support of the initial datashould be far away from the event horizon.
Remark 1.2.
It is interesting to verify that p = 2 is the critical exponent forthe Cauchy problem (1) , thus, to show global existence for p > . Remark 1.3.
Our method does not work for the smaller power < p < , dueto the singularity when approaching to the event horizon. As stated above, our main goal is to show blow-up to the Cauchy problemof semilinear wave equations in Schwarzschild spacetime for ≤ p ≤
2. We firstintroduce the Regge-Wheeler coordinate s ( r ) = r + 2 M ln( r − M ) , (8)then the equation in (1) can be written as ∂ t u − ∂ s u − Fr ∂ s u − Fr ∆ S u = F | u t | p , (9)where F = F ( s ) = 1 − Mr ( s )and r = r ( s ) is the inverse function of (8).Without loss of generality, in what follows we consider the radial solutions,thus solutions of the form: u = u ( t, s ). Hence equation (9) is simplified to ∂ t u − ∂ s u − Fr ∂ s u = F | u t | p . (10)Set v ( t, s ) = r ( s ) u ( t, s ), we then can rewrite equation (10) in the following form ∂ t v ( t, s ) − ∂ s v ( t, s ) + 2 M Fr v ( t, s ) = F r − p | v t | p . (11)Noting that s varies from −∞ to + ∞ as r varies from 2 M to + ∞ , thenif we restrict ourselves to symmetrically radial solutions, we may focus on thefollowing Cauchy problem ∂ t v ( t, s ) − ∂ s v ( t, s ) + W ( s ) v ( t, s ) = h ( s ) | v t | p , ( t, s ) ∈ [0 , ∞ ) × R ,v (0 , s ) = εf ( s ) , v t (0 , s ) = εg ( s ) , s ∈ R , (12)5here h ( s ) = F ( s ) r ( s ) − p and W ( s ) = MF ( s ) r ( s ) with F ( s ) = 1 − Mr ( s ) . The initialdata f ( s ) , g ( s ) are nonnegative satisfying that g ( s ) does not vanish and supp f, g ⊂ { s : (cid:12)(cid:12) | s | ≤ R } with some constant R >
Theorem 1.4.
Let ≤ p ≤ , then the solutions of problem (12) blow up in afinite time T ( ε ) . Furthermore, we obtain the upper bound of lifespan estimateas follows.(I) If ≤ p < , then there exists a positive constant C which is independentof ε such that T ( ε ) ≤ C ε − p − − p . (13) (II) If p = 2 , then there exists a positive constant C which is independent of ε such that T ( ε ) ≤ exp (cid:0) C ε − (cid:1) . (14)We arrange the rest of the paper as follows. In section 2 we give somepreliminary lemmas. In section 3 we demonstrate the proof of Theorem 1.4.
2. Preliminaries
According to Birkhoff’s theorem, the Schwarzschild metric is the most gen-eral spherically symmetric, vacuum solution of the Einstein’s field equations. Werefer the reader to [10] for a general introduction of the Einstein’s field equa-tions and its exact solutions. The sphere { r = 2 M } , referred to as the eventhorizon, is merely a coordinate singularity, while the origin { r = 0 } denotes atrue curvature singularity.Set h ( s ) = F ( s ) r ( s ) − p , (15)then from the definition of (8) we have the following asymptotic behavior.6 emma 2.1. Let h ( s ) be as in (15) , then it holds h ( s ) ∼ s − p , s ≥ M + e,e s M , − ∞ < s ≤ M + e, (16) where the notation X ∼ Y means C − Y ≤ X ≤ CY with some positive constant C , and e denotes the Euler’s number. P roof. By definition (8), it is easy to get s ≥ M + e ⇔ r ≥ M + e, which yields e M + e ≤ − Mr ≤ . (17)Also, if r ≥ M + e , then there exists a positive constant such that r ≤ s ≤ Cr. (18)And hence the first part of (16) follows from (17) and (18).On the other hand, one has s ≤ M + e ⇔ M ≤ r ≤ M + e, which means that h ( s ) = ( r − M ) r − p ∼ C ( r − M ) . (19)By (8), we know that for some positive constant C M ln( r − M ) = s − r ∼ s − C, which yields r − M ∼ e s − C M = e − C M e s M . (20)Therefor, the second part of (16) comes from (19) and (20). Lemma 2.2. ([3], Lemma 5.3) Let W ( s ) = MF ( s ) r ( s ) with F ( s ) = 1 − Mr ( s ) , thengiven any A > , the equation (cid:0) − ∂ s + W ( s ) + A (cid:1) ϕ ( s ) = 0 , s ∈ R (21)7 dmits a positive solution ϕ ∈ C ( R ) such that ϕ ( s ) ∼ e As , | s | → ∞ . (22) Remark 2.3.
Noting that if the term W ( s ) disappears in (21) , then e As is anexact solution.
3. Proof of Theorem 1.4
We are now in a position to show the proof of Theorem 1.4. First we in-troduce a lemma which will be used in our proof. In the following C denotes apositive constant which may change from line to line. Lemma 3.1.
Given any α ≥ , β > and L > , there exists a positive constant C such that Z
1, thus: p ≤
2. We finish the blow-up part of Theorem 1.4.Next we will establish the upper bound of the lifespan estimates. It is easyto see that F ( t ) satisfies the following problem F ′ ( t ) ≥ C | F ( t ) | p ( t + R ) p − ,F (0) = N ε, (49)with N = 12 Z R φ ( s ) g ( s ) ds > . If we consider the Riccati equation H ′ ( t ) = C | H ( t ) | p ( t + R ) p − ,H (0) = N ε, (50)then F ( t ) blows up before H ( t ), which means that we can estimate the upperbound of lifespan of F ( t ) through that of H ( t ).In the case ≤ p <
2, we solve problem (50) and get H ( t ) = (cid:16) ( N ε ) − p + e CR − p − e C ( t + R ) − p (cid:17) − p − (51)with e C = C ( p − − p . Therefore we can estimate the lifespan of F ( t ) as T ( ε ) ≤ C ε − p − − p (52)with C = (cid:16) − pC ( p − N − p (cid:17) − p > , which is independent of ε .In the case p = 2, we solve problem (50) and get H ( t ) = (cid:16) ( N ε ) − − C ln t + RR (cid:17) − , (53)from which we derive the lifespan from above of F ( t ) as T ( ε ) ≤ Ce C ε − (54)13ith C = 1 CN > , which is independent of ε . And hence we finish the proof of Theorem 1.4.
4. Acknowledgement
The author is supported by Natural Science Foundation of Zhejiang Province(LY18A010008),NSFC(11501273, 11726612), Postdoctoral Research Foundation of China(2017M620128,2018T110332), the Scientific Research Foundation of the First-Class Disciplineof Zhejiang Province (B)(201601).
ReferencesReferences [1] R. Agemi, Blow-up of solutions to nonlinear wave equations in two spacedimensions, Manuscripta Math., 73 (1991), 153-162.[2] P. Blue and J. Sterbenz, Uniform decay of local energy and the semi-linearwave equation on Schwarzschild space, Comm. Math. Phys., 268 (2006),481-504.[3] D. Catania and V. Georgiev, Blow-up for the semilinear wave equation inthe Schwarzschild metric, Differential Integral Equations, 19 (2006), 799-830.[4] K.Hidano and K.Tsutaya, Global existence and asymptotic behavior ofsolutions for nonlinear wave equations, Indiana Univ. Math. J., 44 (1995),1273-1305.[5] K.Hidano, C.Wang and K.Yokoyam, The Glassey conjecture with radiallysymmetric data, J. Math. Pures Appl., 98(9) (2012), 518-541.[6] F. John, Blow-up for quasilinear wave equations in three space dimensions,Comm. Pure Appl. Math., 34 (1981), 29-51.147] F. John, Non-existence of global solutions of (cid:3) u = ∂∂ t F ( u t ) in two andthree space dimensions, Rend. Circ. Mat. Palermo (2) Suppl., 8 (1985),229-249.[8] S. Klainerman, Remarks on the global Sobolev inequalities in theMinkowski space R n +1 , Comm. Pure Appl. Math., 37 (1984), 443-455.[9] S. Klainerman, Uniform decay extimates and the Lorentz invariance of theclassical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332.[10] D. C. Kong and K. F. Liu, Time-periodic solutions of the Einstein’s fieldequations I: general framework, Sci. China Math., 53 (2010), 1213-1230.[11] A. N. Lai and H. Takamura, Nonexistence of global solutions of nonlinearwave equations with weak time-dependent damping related to Glassey’sconjecture, to appear in Differential Integral Equations.[12] H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and C. B. Wang, TheStrauss conjecture on Kerr black hole backgrounds, Math. Ann., 359(3-4)(2014), 637-661.[13] J. Luk, The null condition and global existence for nonlinear wave equationson slowly rotating Kerr spacetimes, Journal Eur. Math. Soc., 15(5) (2013),1629-1700.[14] J. Marzuola, J. Metcalfe, D. Tataru and M. Tohaneanu, Strichartz esti-mates on Schwarzschild black hole backgrounds, Comm. Math. Phys., 293(2010), 37-83.[15] K. Masuda, Blow-up solutions for quasi-linear wave equations in two spacedimensions, Lect. Notes Num. Appl. Anal., 6 (1983), 87-91.[16] J. Schaeffer, Finite-time blow up for u tt − ∆ u = H ( u r , u tt