DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME

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1 Electronic Journal of Differential Equations, Vol ), No. 17, pp ISSN: URL: or DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME MUHAMMET YAZICI Abstract. We consider the initial-value problem for the Klein-Gordon equation in de Sitter spacetime. We derive L decay estimates for the solution to the linear Klein-Gordon equation in de Sitter spacetime with and without source term. 1. Introduction In this article, we consider the following initial value problem for the Klein- Gordon equation in de Sitter spacetime, 2 t Φ + nφ t e 2t Φ + m 2 Φ = fx, t), x, t) R n R, Φx, ) = ϕ x), t Φx, ) = ϕ 1 x), x R n, 1.1) where f C R n+1 ), ϕ, ϕ 1 are in Sobolev space W [n/2]+1,1 R n ), and m >. In Minkowski spacetime, the initial value problem for the semilinear Klein- Gordon equation u tt u + m 2 u = u α u, has been extensively investigated. The existence of global weak solutions has been obtained by Jörgens [6], Pecher [8], Brenner [3], Ginibre and Velo [4, 5]. In order for the total energy is well-defined in the energy space, one needs the assumption α < 4/n 1). On the other hand, the initial value problem for so-called Higgs boson equation in Minkowski spacetime, and u tt u m 2 u = u α u, 2 t Φ + nhφ t e 2Ht Φ m 2 Φ = Φ α Φ, x, t) R n R x, t) R n R in de Sitter spacetime are studied by Yagdjian [11], and some qualitative property of the solution revealed if the global solution exists. In addition, it was shown by Baskin [1] that the initial value problem for ht t 2 u + n t u + t t u + e 2t ht u + λu + u α u =, y, t) Y R ht 21 Mathematics Subject Classification. 35L15, 35C15, 35Q75. Key words and phrases. De sitter spacetime; Klein-Gordon eqution; fundamental solutions, L estimates. c 218 Texas State University. Submitted June 16, 217. Published January 13,

2 2 M. YAZICI EJDE-218/17 admits a small amplitude global solution in the energy space H 1 L 2, provided λ > n 2 /4 and α = 4/n 1). Here h t is a smooth family of Riemannian metrices on compact n-dimensional manifold Y, which is characterized as an asymptotically de Sitter spacetime. In Nakamura [7], the assumption on the regularity of the initial data is weakened in the case of m n/2. Turning back to the initial value problem 1.1), the following theorem obtained by Yagdjian [12] states the estimate in the Sobolev space H s R n ). Theorem 1.1 [12]). Let Φ = Φx, t) be the solution of the initial value problem Φ tt + nφ t + e 2t Φ + m 2 Φ = f, Φx, ) = ϕ x), Φ t x, ) = ϕ 1 x) for x, t) R n, ), where ϕ, ϕ 1 C R n ) and f C R n+1 ). Let l be a nonnegative integer, m < n 2 1/2 and n 2. Then there exists a constant C > such that ) s Φ, t) W l,q R n ) Ce M n 2 )t 1 e t ) 2s n 1 p 1 q )) { ϕ W l,p R n ) + 1 e t } ) ϕ 1 W l,p R n ) + Ce n 2 M)t e n 2 M)b e b2s n 1 p 1 q )) f, b) W l,p R )db n for all t >, provided that 1 < p 2, 1 p + 1 q = 1, 1.2) 1 2 n + 1)1 p 1 q ) 2s n1 p 1 ) < 2s + 1. q n Here we have set M = 2 4 m2. Moreover, Galstian and Yagdjian [13] showed similar estimates to the initial value problem for Φ tt + nφ t e 2t Ax, x )Φ + m 2 Φ = f, t >, x R n, 1.3) in the Besov space Bp s,q, where Ax, x ) = α 2 a αx) x α is a second-order negative elliptic differential operator with real coefficients a α B and m in the set, n 2 1/2) [n/2, ). Here, B denotes the space of all C functions with uniformly bounded derivatives of all orders. The case m n 2 1/2, n/2) is also considered by Yagdjian [14] in the Besov space. In this article, we are interested in the case of < m < n 2 1/2. Decay estimate is an important tool to prove the global existence for nonlinear partial differential equations. The limiting case q = i.e. p = 1) for the decay estimate is excluded in Theorem 1.1. We remark that the decay rate for the L decay estimate is faster than the decay rate for the L 2 decay estimate. Therefore, by using the L decay estimate, we prove the following theorem. Theorem 1.2. Let Φ = Φx, t) be the solution of the initial value problem Φ tt + nφ t + e 2t Φ + m 2 Φ = f, Φx, ) = ϕ x), Φ t x, ) = ϕ 1 x) for x, t) R n, ), where ϕ, ϕ 1 C R n ) and f C R n+1 ). Let l be a nonnegative integer, m < n 2 1/2 and n 2. Then there exists a constant

3 EJDE-218/17 KLEIN-GORDON EQUATION IN CURVED SPACETIME 3 C > such that Φ, t) L R n ) Ce M n 2 )t { } ϕ W [n/2]+1,1 R n ) + ϕ 1 W [n/2]+1,1 R n ) + Ce n 2 M)t e n 2 M)b f, b) W [n/2]+1,1 R )db, n for all t >. Here we have set M = n 2 4 m2. 1.4) Here, W k,p R n ) = {u L p R n ) : D α u L p R n ), α k}, denotes a Sobolev space with the norm ) 1/p, u W k,p R n ) = D α u p 1 p < ), R n α k u W k, R n ) = α k 2. Preliminaries ess sup R n D α u. Throughout this article, the positive constants which may change, are denoted by the same letters C. We prepare some inequalities for proving Theorem 1.2. First of all, we introduce the hypergeometric function F a, b; c; ζ) and study its property. It is defined by the power series F a, b; c; ζ) = n= a) n b) n c) n ζ n, ζ < 1, n! where a, b, c C with c, 1, 2,..., and we denote a) = 1, a) n = Γa + n)/γa) = aa + 1)... a + n 1), n = 1, 2, 3,.... Here Γ is the gamma function see, e.g. [2]). We remark that there exists a constant C > such that F a, b; c; ζ) C 2.1) for all ζ [, 1] if Rec b a) > for a, b, c C with c, 1, 2,... see e.g. [9] and references therein). 3. Fundamental solutions of the linear Klein-Gordon equation We separate the initial value problem 1.1) into two parts. First, we consider the Klein-Gordon equation without source term: 2 t Φ + nφ t e 2t Φ + m 2 Φ =, x, t) R n R, Φx, ) = ϕ x), t Φx, ) = ϕ 1 x), x R n, 3.1) where Φx, ) = ϕ, Φx, ) = ϕ 1 C R n ). Next, we consider the Klein-Gordon equation with source term, 2 t Φ + nφ t e 2t Φ + m 2 Φ = fx, t), x, t) R n R, Φx, ) =, t Φx, ) =, 3.2)

4 4 M. YAZICI EJDE-218/17 where f C R n+1 ). For x, t ) R n+1, the forward and backward light cones are defined as D + x, t ) := { x, t) R n+1 : t t, x x e t e t}, D x, t ) := { x, t) R n+1 : t t, x x e t e t}. The function introduced by Yagdjian [9], [12] is Ex, t; x, t ; M) := 4e t t ) M e t + e t ) 2 x x 2) 1 2 +M 1 F 2 M, 1 2 M; 1; e t e t ) 2 x x 2 e t + e t ) 2 x x 2 for x, t) D + x, t ) D x, t ), where M = ), n 2 4 m2 and x x ) 2 = x x ).x x ) for x, x R n. The kernels K z, t; M) and K 1 z, t; M) are given by Yagdjian [9], [12] as follows and K z, t; M) [ ] : = b Ez, t;, b; M) b= = 4e t ) M 1 + e t ) 2 z 2) M e t ) 2 z 2) [ 1 e t 1 + Me 2t 1 z 2 ) ) 1 F 2 M, 1 2 M; 1; 1 e t ) 2 z 2 ) 1 + e t ) 2 z e 2t + z 2 ) M) F 1 2 M, 1 2 M; 1; 1 e t ) 2 z 2 )] 1 + e t ) 2 z 2 K 1 z, t; M) : = Ez, t;, ; M) = 4e t ) M 1 + e t ) 2 z 2) 1 2 +M 1 F 2 M, 1 2 M; 1; 1 e t ) 2 z 2 ) 1 + e t ) 2 z 2, where z 1 e t. The solution Φ = Φx, t) of the initial value problem Φ tt + nφ t e 2t Φ + m 2 Φ =, Φx, ) = ϕ x), Φ t x, ) = ϕ 1 x), 3.3) with ϕ, ϕ 1 C R n ) is given by Yagdjian-Galstian [9, 1] as follows Φx, t) = e n 1 2 t v ϕ x, φt)) + e nt/2 v ϕ x, φt)s) 2K φt)s, t; M) + nk 1 φt)s, t; M)) φt)ds + e nt/2 v ϕ1 x, φt)s)2k 1 φt)s, t; M))φt)ds, 3.4) where φt) := 1 e t with t >. Here, for ϕ C R n ), v ϕ x, t) denotes the solution of v tt v =, vx, ) = ϕx), v t x, ) =, x, t) R n, ). 3.5) Moreover, the solution Φ = Φx, t) of the initial value problem Φ tt + nφ t e 2t Φ + m 2 Φ = f, Φx, ) =, Φ t x, ) =, 3.6)

5 EJDE-218/17 KLEIN-GORDON EQUATION IN CURVED SPACETIME 5 with f C R n+1 ) is given by Yagdjian-Galstian [9, 1] as follows Φx, t) = 2e nt/2 e b e t db dre n 2 b vx, r; b)er, t;, b; M), 3.7) where vx, t; b) is the solution to the following initial value problem for the wave equation v tt v =, vx, ; b) = fx, b), v t x, ; b) =, x, t) R n, ), 3.8) where b >. 4. Proof of Theorem 1.2 We derive L estimates for the linear Klein-Gordon equation in de Sitter spacetime. We apply the following two lemmas to prove the theorem. Lemma 4.1. Let M > 1/2 and φt) = 1 e t. Then for all t >. 1 + φt)s) n 1 2 K1 φt)s, t; M) φt)ds C M e Mt 4.1) Proof. Changing the variable by 1+φt)s = r and using the definition of the kernel K 1, we obtain 1 + φt)s) n 1 2 K1 φt)s, t; M) φt)ds t 2 e = 4 M e Mt r n e t ) 2 r 1) 2 ) 1 2 +M 1 1 F 2 M, 1 2 M; 1; 1 e t ) 2 r 1) 2 ) 1 + e t ) 2 r 1) 2 dr e t 1 Ce Mt e t + 1) 2 y 2 ) 1 2 +M F 1 2 M, 1 2 M; 1; et 1) 2 y 2 ) dy, e t + 1) 2 y 2 where we have changed the variable by e t r 1) = y in the last inequality. Since M > 1/2, by 2.1) the hypergeometric function in the last inequality is bounded, and hence e t φt)s) n 1 2 K1 φt)s, t; M) φt)ds Ce Mt e t + 1) 2 y 2 ) 1 2 +M dy C M e Mt e t + 1) 2M 1 e t 1), which leads to 4.1). Lemma 4.2. Let M > 1/2 and φt) = 1 e t. Then for all t >. 1 + φt)s) n 1 2 K φt)s, t; M) φt)ds C M e Mt 4.2)

6 6 M. YAZICI EJDE-218/17 Proof. Similarly to the proof of Lemma 4.1, we obtain 1 + φt)s) n 1 2 K φt)s, t; M) φt)ds C M e Mt e t 1 e t + 1) 2 y 2 ) M 1 2 e t 1) 2 y 2) 1 [ 1 e t e 2t + M1 e 2t y 2 ))F 2 M, 1 2 M; 1; et 1) 2 y 2 ) e t + 1) 2 y 2 + e 2t 1 + y 2 ) M)F 1 2 M, 1 2 M; 1; et 1) 2 y 2 e t + 1) 2 y 2 )] dy. From [12], we have z 1 z + 1) 2 y 2 ) M 1 2 z 1) 2 y 2) 1 [z z 2 + M1 z 2 y 2 ))F 1 2 M, 1 2 M; 1; z 1)2 y 2 z + 1) 2 y 2 ) + z y 2 ) M)F 1 2 M, 1 2 M; 1; z 1)2 y 2 z + 1) 2 y 2 ) ] dy C M z + 1) 2M for all z [1, ). Hence 4.3) leads to 4.2). This completes the proof. 4.3) Proof of Theorem 1.2. First we consider the solution of the initial value problem 3.1). In the case of ϕ 1 =, from 3.4), we have Φx, t) = e n 1 2 t v ϕ x, φt)) + e nt/2 v ϕ x, φt)s)2k φt)s, t; M) Then, we obtain + nk 1 φt)s, t; M))φt)ds. Φ, t) L R n ) e n 1 2 t v ϕ, φt)) L R n ) + e nt/2 v ϕ, φt)s) L R n ) 2K φt)s, t; M) + nk 1 φt)s, t; M)) φt)ds. 4.4) As is well known, the solution vx, t) of the initial value problem 3.5) satisfies v, t) L R n ) C1 + t) n 1 2 ϕ W [n/2]+1,1 R n ) 4.5) for t, if n 2 see e.g. [15]). For all t, we have e n 1 2 t v ϕ, φt) L R n ) Ce n 1 2 t 1 + φt)) n 1 2 ϕ W [n/2]+1,1 R n ) Hence, we obtain Ce n 1 2 t ϕ W [n/2]+1,1 R n ). e n 1 2 t v ϕ, φt) L R n ) Ce n 1 2 t ϕ W [n/2]+1,1 R n ). 4.6)

7 EJDE-218/17 KLEIN-GORDON EQUATION IN CURVED SPACETIME 7 On the other hand, we obtain e nt/2 v ϕ, φt)s) L R n ) 2K φt)s, t; M) + nk 1 φt)s, t; M)) φt)ds C ϕ W [n/2]+1,1 R )e nt/2 1 + φt)s) n 1 n 2 2K φt)s, t; M) + nk 1 φt)s, t; M)) φt)ds. From Lemma 4.1 and Lemma 4.2, we have 4.7) e nt/2 1 + φt)s) n 1 2 2K φt)s, t; M) φt)ds Ce M n 2 )t, 4.8) e nt/2 1 + φt)s) n 1 2 nk1 φt)s, t; M) φt)ds Ce M n 2 )t. 4.9) Hence, from 4.6), 4.8) and 4.9) we obtain Φ, t) L R n ) Ce M n 2 )t ϕ W [n/2]+1,1 R n ) 4.1) when ϕ 1 =. For the case ϕ =, we have Φ, t) L R n ) Ce M n 2 )t ϕ 1 W [n/2]+1,1 R n ) 4.11) in a similar way. Next, we consider the solution of the initial value problem 3.2). From 3.7) and the definition of Ex, t; x, t ; M) we have Φx, t) e b e t = 2e nt/2 db dre n 2 b vx, r; b)4 M e Mb+t) e t + e b ) 2 r 2 ) 1 2 +M F 1 2 M, 1 2 M; 1; e b e t ) 2 r 2 e b + e t ) 2 r 2 )dr, where v is the solution of 3.8). From 4.5), we obtain for all r >. Hence, v, r; b) L R n ) C1 + r) n 1 2 f, b) W [n/2]+1,1 R n ) Φ, t) L R n ) C M e nt/2 e Mt e n 2 b e Mb f, b) W [n/2]+1,1 R )db n e b e t 1 + r) n 1 2 e t + e b ) 2 r 2 ) 1 2 +M F 1 2 M, 1 2 M; 1; e b e t ) 2 r 2 e b + e t ) 2 r 2 ) dr C M e nt/2 e Mt e n 2 b e Mb f, b) W [n/2]+1,1 R )db n e b e t e t + e b ) 2 r 2 ) 1 2 +M F 1 2 M, 1 2 M; 1; e b e t ) 2 r 2 e b + e t ) 2 r 2 ) dr.

8 8 M. YAZICI EJDE-218/17 If we change the variable by r = e t y, then we obtain Φ, t) L R n ) C M e nt/2 e Mt e n 2 b e Mb f, b) W [n/2]+1,1 R )db n e t b 1 e t b + 1) 2 y 2) 1 2 +M F 1 2 M, 1 2 M; 1; et b 1) 2 y 2 e t b + 1) 2 y 2 ) dy. 4.12) Since M > 1/2, by 2.1), we have the following estimate for the second integral of 4.12), e t b 1 C M e e t b + 1) 2 y 2) 1 2 +M F 1 2 M, 1 2 M; 1; et b 1) 2 y 2 e t b + 1) 2 y 2 ) dy t b 1 e t b + 1) 2 y 2) 1 2 +M dy C M e t b + 1 ) 2M 1 e t b 1) C M e t b + 1 ) 2M C M e 2Mt b), for b < t. Thus, we have Φ, t) L R n ) C M e n 2 M)t e n 2 M)b f, b) W [n/2]+1,1 R )db. 4.13) n Hence 4.1), 4.11) and 4.13) lead to 1.4). This completes the proof. Acknowledgments. The author would like to express sincerely thanks to the ananymous referee for several comments to revise the paper. References [1] D. Baskin; Strichartz Estimates on Asymptotically de Sitter Spaces, Annales Henri Poincarè, ), Issue 2, [2] H. Bateman, A. Erdelyi; Higher Transcendental Functions, 1,2, McGraw-Hill, New York, [3] P. Brenner; On the existence of global smooth solutions of certain semilinear hyperbolic equations, Math. Z., 167 2) 1979), [4] J. Ginibre, G. Velo; The global Cauchy problem for the nonlinear Klein-Gordon equation, Math Z., 1894) 1985), [5] J. Ginibre, G. Velo,; The global Cauchy problem for the nonlinear Klein-Gordon equation II, Ann. Inst. H. Poincarè Anal. Linèaire, 61) 1989), [6] K. Jörgens; Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen, Math. Z., ), [7] M. Nakamura; The Cauchy problem for semi-linear Klein-Gordon equations in de Sitter spacetime, J. Math. Anal. Appl., ), [8] H. Pecher; L p -Abschützungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z., ), [9] K. Yagdjian, A. Galstian; Fundamental solutions for the KleinGordon equation in de Sitter spacetime, Comm. Math. Phys., ), [1] K. Yagdjian; The semilinear Klein-Gordon equation in de Sitter spacetime, Discrete Contin. Dyn. Syst. Ser. S 2 3) 29), [11] K. Yagdjian; On the global solutions of the Higgs boson equation, Comm. Partial Differential Equations, ),

9 EJDE-218/17 KLEIN-GORDON EQUATION IN CURVED SPACETIME 9 [12] K. Yagdjian; Global existence of the scalar field in de Sitter spacetime, J. Math. Anal. Appl., ), [13] A. Galstian, K. Yagdjian; Global in time existence of self-interacting scalar field in de Sitter spacetimes, Nonlinear Anal. Real World Appl., ), [14] K. Yagdjian; Global existence of the self-interacting scalar field in the de Sitter universe, 217. arxiv: v2. [15] W. von Wahl; L p -decay rates for homogeneous wave-equations, Math. Z., ), Muhammet Yazıcı Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, Trabzon, 618, Turkey address:

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