Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime

Muhammet Yazıcı

doi:10.1515/dema-2025-0136

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Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime

Muhammet Yazıcı

Published/Copyright: September 20, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this article, we consider the initial value problem for a system of Klein-Gordon equations in anti-de Sitter spacetime. We assume that the mass matrix is diagonalizable with positive eigenvalues. Under the assumption for the small initial data, we prove the global existence of solutions to the initial value problem. The existence result is derived based on the assumption of power-type nonlinearity.

Keywords: semilinear Klein-Gordon system; anti-de Sitter spacetime; global existence

MSC 2010: 35L52; 35L71; 81T20

1 Introduction

The Klein-Gordon equation in curved spacetime is an important field of study in modern theoretical physics due to its connection with various fields such as astrophysics, cosmology, and quantum field theory. In curved spacetime, positive curvature corresponds to de Sitter spacetime, which describes an exponentially expanding universe with a positive cosmological constant. On the other hand, negative curvature corresponds to anti-de Sitter spacetime, characterized by a hyperbolic geometry and a negative cosmological constant.

The derivation of the Klein-Gordon equation in de Sitter spacetime is provided in [1]. For the sake of completeness, we present a brief summary of how the equation is obtained. In de Sitter spacetime, the line element has the following form:

(1) d s 2 = − 1 − r 2 R 2 d t 2 + 1 − r 2 R 2 − 1 d r 2 + r 2 ( d α 2 + sin 2 α d β 2 ) .

Here R denotes the universe radius. Using the Lemaitre-Robertson transformation and spherical coordinates, as described in [2], we obtain the following form for the line element:

(2) d s 2 = − d t 2 + e 2 H t ( d x 1 2 + d x 2 2 + d x 3 2 ) ,

where H = 1 ⁄ R . Here we take H = 1 for simplicity. In the case of n spatial dimensions, equation (2) can be rewritten as

(3) d s 2 = − d t 2 + e 2 H t ( d x 1 2 + … + d x n 2 ) .

From (3), we deduce the following diagonal matrix: ( u i k ) 0 ≤ i , k ≤ n ≔ diag ( − 1 , e 2 t , … , e 2 t ) for the line element. If we apply the determinant u ≔ det ( u i k ) 0 ≤ i , k ≤ n , and the inverse matrix ( u i k ) 0 ≤ i , k ≤ n to the following equation:

1 ∣ u ∣ ∂ ∂ x i ∣ u ∣ u i k ∂ ψ ∂ x k = m 2 ψ ,

we obtain linear Klein-Gordon equation in de Sitter spacetime

(4) ∂ t t ψ + n ∂ t ψ − e − 2 t Δ ψ = m 2 ψ .

The transformation of time t → − t converts equation (4) to the following equation:

(5) ∂ t t ψ + n ∂ t ψ − e 2 t Δ ψ = m 2 ψ ,

which can be considered as an equation in anti-de Sitter spacetime.

The following initial value problem

(6) ∂ t 2 ψ + n ψ t − e − 2 t Δ ψ + m 2 ψ = F ( ψ ) , ( x , t ) ∈ R n × R , ψ ( x , 0 ) = φ 0 ( x ) , ∂ t ψ ( x , 0 ) = φ 1 ( x ) , x ∈ R n ,

in de Sitter spacetime has been extensively investigated. Yagdjian [3] proved the global existence of the solution to the initial value problem using the L p − L q estimates when the nonlinear term F is assumed to satisfy the Lipschitz continuous condition and m ∈ ( 0 , n 2 − 1 ⁄ 2 ) ∪ [ n ⁄ 2 , ∞ ) . Nakamura [4] proved the existence of global solutions to the initial value problem in the energy space under some restrictions on the power for the power-type nonlinear term, provided m ≥ n ⁄ 2 . In [5], the necessary conditions are revealed between the spatial dimension n , the power of the nonlinear term p , and the mass m for the existence of the solutions. In addition, Baskin [6] showed that the initial value problem for

∂ t t v + n ∂ t v + ∂ t h t h t ∂ t v + e − 2 t Δ h t v + δ v = − ∣ v ∣ γ v , ( y , t ) ∈ Y × R

has a global solution with sufficiently small initial data in the energy space H 1 ⊕ L 2 , for δ > n 2 ⁄ 4 and γ = 4 ⁄ ( n − 1 ) . Y is a compact n -dimensional manifold described as an asymptotically de Sitter spacetime, and h t represents a smooth family of Riemannian metrics on Y . In [7], Yazici showed the L ∞ estimates for the fundamental solutions of linear part of equation (6) both with and without source term. Moreover, Galstian and Yagdjian [8] proved the existence of the small data global solutions to the initial value problem for

(7) ∂ t t ψ + n ∂ t ψ − e − 2 t A ( x , ∂ x ) ψ + m 2 ψ = F ( ψ ) , t > 0 , x ∈ R n ,

where A ( x , ∂ x ) = ∑ ∣ α ∣ ≤ 2 a α ( x ) ∂ x α is a second-order negative elliptic differential operator with real coefficients a α ∈ ℬ ∞ and m ∈ ( 0 , n 2 − 1 ⁄ 2 ) ∪ [ n ⁄ 2 , ∞ ) . Here, ℬ ∞ denotes the space of all C ∞ functions with uniformly bounded derivatives of all orders and F is Lipschitz continuous. It is also proved in [8] that if the small source term f = f ( x , t ) is added to equation (7) with vanishing initial data, the global solution also exists for m > 0 . The existence of small data global solvability of the initial value problem for equation (7) was proved by Yagdjian [9] in the case of m ∈ ( n 2 − 1 ⁄ 2 , n ⁄ 2 ) . There are also many studies in the literature related to hyperbolic equations in expanding model of the universe (refer, for instance [10–14] and references therein). Moreover, Yagdjian [15] proved the global existence of solutions to the system of semilinear Klein-Gordon equations, provided that the mass matrix is diagonalizable with positive eigenvalues.

Turning our attention to the anti-de Sitter model of the universe, Yagdjian and Galstian [1] demonstrated the fundamental solutions to the linear Klein-Gordon equations, both with and without source term. Galstian and Yagdjian [16] proved the global existence of the solutions to the initial value problem for the following equation in Friedmann-Lamaitre-Robertson-Walker spacetime

∂ t t ψ + n ∂ t ψ − e 2 t A ( x , ∂ x ) ψ + m 2 ψ = e − Γ t F ( ψ ) , t > 0 , x ∈ R n ,

with the same assumptions on initial data in equation (7), where m 2 ∈ C and Γ ∈ R .

In this article, we focus on the system of semilinear Klein-Gordon equations in anti-de Sitter spacetime,

(8) ∂ t t Φ + n ∂ t Φ − e 2 t Δ Φ + M Φ = F ( Φ ) , ( x , t ) ∈ R n × R ,

where F is a vector-valued function of the vector valued Φ , where Φ = [ ϕ 1 , ϕ 2 , … , ϕ j ] T for j ∈ Z + .

We assume that the matrix M is diagonalizable by a real-valued matrix O , and it has eigenvalues m 1 2 , m 2 2 , … , m j 2 . By the transformation O , the mass matrix M can be diagonalized; therefore, the unkonown function Φ is changed as follows:

Ω = O Φ , Φ = O − 1 Ω ,

and we obtain the semilinear system of Klein-Gordon equations

(9) ∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + M ω i = F ( ω i ) , i = 1 , 2 , … , j ,

where

M = O M O − 1 = m 1 2 0 0 … 0 0 m 2 2 0 … 0 0 0 ⋱ … 0 0 0 0 … m j 2 , Ω = ω 1 ω 2 ⋮ ω j , F ( Ω ) = O F ( O − 1 Φ ) .

Decay estimates are crucial in the analysis of partial differential equations, particularly in proving the global existence of solutions for nonlinear partial differential equations. Therefore, our main goal is to prove the following theorem, utilizing the L ∞ decay estimate.

Theorem 1.1

Let N ≔ 2 [ n ⁄ 2 ] + 3 and k = [ n ⁄ 2 ] + 2 , where [ ⋅ ] denotes the integer part and let n ≥ 2 . Then, there is a constant ε 0 > 0 such that if ‖ ρ 0 i ‖ W N , 2 ( R n ) + ‖ ρ 1 i ‖ W N , 2 ( R n ) ≤ ε for 0 < ε ≤ ε 0 , the initial value problem

(10) ∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = F ( ω i ) , ( x , t ) ∈ R n × R , ω i ( x , 0 ) = ρ 0 i ( x ) , ∂ t ω i ( x , 0 ) = ρ 1 i ( x ) , x ∈ R n ,

has a solution ω i ∈ C ( [ 0 , ∞ ) ; W k , ∞ ( R n ) ) satisfying e β t ‖ ω i ( ⋅ , t ) ‖ W k , ∞ ( R n ) ≤ 2 ε , where β = n ⁄ 2 for m i > n ⁄ 2 and β = 0 for m i = n ⁄ 2 with i = 1 , 2 , … , j . Here W k , ∞ ( R n ) denotes the Sobolev spaces.

This article is organized as follows. In Section 2, we present necessary decay estimates for the initial value problem of the system of linear Klein-Gordon equations both with and without source terms in Theorem 2.2. Section 3 is dedicated to the proof of Theorem 1.1. The conclusions are discussed in the last section.

Throughout the study, unless otherwise stated, the index i takes all values from 1 to j , and the positive constant C which may change line by line is written by the same letter.

2 System of Klein-Gordon equations

In this section, we consider the solution of the linear part of the system of equations in (10) which were given by Yagdjian and Galstian [1]. For ( z 0 , t 0 ) ∈ R n + 1 , the backward and forward light cones are denoted by

X − ( x 0 , t 0 ) ≔ { ( x , t ) ∈ R n + 1 : ∣ x − x 0 ∣ ≤ e t 0 − e t } , X + ( x 0 , t 0 ) ≔ { ( x , t ) ∈ R n + 1 : ∣ x − x 0 ∣ ≤ e t − e t 0 } ,

respectively. The function, introduced by Yagdjian and Galstian [1], is given as

κ i ( x , t ; x 0 , t 0 ) ≔ ( 4 e t 0 + t ) i σ i ( ( e t 0 + e t ) 2 − ∣ x − x 0 ∣ 2 ) − 1 2 − i σ i F 1 2 1 2 + i σ i , 1 2 + i σ i ; 1 ; ( e t 0 − e t ) 2 − ∣ x − x 0 ∣ 2 ( e t + e t 0 ) 2 − ∣ x − x 0 ∣ 2 ,

for ( x , t ) ∈ X − ∪ X + , where σ i = m 2 − n 2 4 and ( x − x 0 ) 2 is the inner product for x , x 0 ∈ R n . Here F 1 2 describes the hypergeometric function denoted by the power series. The next theorem provides the inequality below for the solution to the system of linear Klein-Gordon equations. The proof of the following theorem can be obtained directly from [17].

Theorem 2.1

Let ω i = ω i ( x , t ) be the solution of the initial value problem

(11) ∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = f i , ω i ( x , 0 ) = ρ 0 i , ∂ t ω i ( x , 0 ) = ρ 1 i

for ( x , t ) ∈ R n × ( 0 , ∞ ) , where ρ 0 i , ρ 1 i ∈ C 0 ∞ ( R n ) , and f i ∈ C ∞ ( R n + 1 ) . Let m ≥ n ⁄ 2 , l ∈ Z + ∪ { 0 } , and n ≥ 2 . Then, there exists a constant C > 0 such that

(12) ∥ ( − Δ ) − s ω i ( ⋅ , t ) ∥ W l , q ( R n ) ≤ C e − n 2 t ( e t − 1 ) 2 s − n 1 p − 1 q { ∥ ρ 0 i ∥ W l , p ( R n ) + ( 1 − e − t ) ( 1 + t ) 1 − sgn σ i ∥ ρ 1 i ∥ W l , p ( R n ) } + C e − n 2 t e t 2 s − n 1 p − 1 q ∫ 0 t e n 2 b ∥ f i ( ⋅ , b ) ∥ W l , p ( R n ) ( 1 + t − b ) 1 − sgn σ i d b

for all t > 0 and n 2 ≤ m when 1 < p ≤ 2 , 1 p + 1 q = 1 , 1 2 ( n + 1 ) ( 1 p − 1 q ) ≤ 2 s ≤ n ( 1 p − 1 q ) < 2 s + 1 . Here we have set σ i = m i 2 − n 2 4 .

The decay estimate is necessary for showing the global existence of nonlinear partial differential equations. Therefore, we consider the L ∞ decay estimate for the solution of system of linear Klein-Gordon equations in anti- de Sitter spacetime. In this article, we deal with the case of m i ≥ n ⁄ 2 and prove the following theorem.

Theorem 2.2

Let ω i = ω i ( x , t ) be the solution of the initial value problem

∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = f i , ω i ( x , 0 ) = ρ 0 i ( x ) , ∂ t ω i ( x , 0 ) = ρ 1 i ( x )

for ( x , t ) ∈ R n × ( 0 , ∞ ) , where ρ 0 i , ρ 1 i ∈ C 0 ∞ ( R n ) , and f i ∈ C ∞ ( R n + 1 ) . Then, there exists a constant C > 0 such that

(13) ∥ ω i ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ C e − n 2 t { ∥ ρ 0 i ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) + ( 1 + t ) 1 − sgn σ i ( 1 − e − t ) ∥ ρ 1 i ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) } + C e − n 2 t ∫ 0 t e n 2 b ∥ f i ( ⋅ , b ) ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) ( 1 + t − b ) 1 − sgn σ i d b

for all t > 0 . Here we have set σ i = m i 2 − n 2 4 .

Proof

We need to separate the initial value problem into two parts in order to prove the theorem. From [18], the solution ω i = ω i ( x , t ) of the initial value problem

∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = 0 , ω i ( x , 0 ) = ρ 0 i ( x ) , ∂ i ω i ( x , 0 ) = ρ 1 i ( x )

has the following decay estimate:

(14) ∥ ω i ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ C e − n 2 t { ∥ ω 0 i ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) + ( 1 + t ) 1 − sgn σ i ( 1 − e − t ) ∥ ω 1 i ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) } .

Next we consider the initial value problem with nonzero initial data,

∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = f i , ω i ( x , 0 ) = 0 , ∂ t ω i ( x , 0 ) = 0 ,

with f i ∈ C ∞ ( R n ) . The solution ω i = ω i ( x , t ) is

(15) ω i ( x , t ) = 2 e − n 2 t ∫ 0 t d b ∫ 0 e t − e b d r e n 2 b ν i ( x , r ; b ) κ i ( r , t ; 0 , b ) ,

where ν i ( x , t ; b ) is the solution to the following initial value problem for the wave equation:

(16) ∂ t t ν i − Δ ν i = 0 , ν i ( x , 0 ; b ) = f i ( x , b ) , ν i t ( x , 0 ; b ) = 0 , ( x , t ) ∈ R n × ( 0 , ∞ ) ,

where b > 0 . The solution ν i ( x , t ) of the initial value problem (16) satisfies

(17) ‖ ν i ( ⋅ , r ; b ) ‖ L ∞ ( R n ) ≤ C ( 1 + r ) − n − 1 2 ‖ f i ( ⋅ , b ) ‖ W [ n ⁄ 2 ] + 1,1 ( R n ) ,

for all r > 0 , if n ≥ 2 (e.g., [19]). By using (17), we have

‖ ω i ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ‖ f i ( ⋅ , b ) ‖ W [ n ⁄ 2 ] + 1,1 ( R n ) d b × ∫ 0 e t − e b ( 1 + r ) − n − 1 2 ( ( e t + e b ) 2 − r 2 ) − 1 2 F 1 2 1 2 + i σ i , 1 2 + i σ i ; 1 ; ( e b − e t ) 2 − r 2 ( e b + e t ) 2 − r 2 d r ≤ C e − n 2 t ∫ 0 t e n 2 b ‖ f i ( ⋅ , b ) ‖ W [ n ⁄ 2 ] + 1,1 ( R n ) d b × ∫ 0 e t − e b ( ( e t + e b ) 2 − r 2 ) − 1 2 F 1 2 1 2 + i σ i , 1 2 + i σ i ; 1 ; ( e b − e t ) 2 − r 2 ( e b + e t ) 2 − r 2 d r .

If we change the variable by r = e b s , then we obtain

(18) ‖ ω i ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ‖ f i ( ⋅ , b ) ‖ W [ n ⁄ 2 ] + 1,1 ( R n ) d b × ∫ 0 e t − b − 1 ( ( e t − b + 1 ) 2 − s 2 ) − 1 2 F 1 2 1 2 + i σ i , 1 2 + i σ i ; 1 ; ( e t − b − 1 ) 2 − s 2 ( e t − b + 1 ) 2 − s 2 d s

from [17], we apply z = e t − b to the second integrand of the right-hand side of inequality (18);

‖ ω i ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ‖ f i ( ⋅ , b ) ‖ W [ n ⁄ 2 ] + 1,1 ( R n ) ( e t − b − 1 ) ( e t − b + 1 ) − 1 ( 1 + t − b ) 1 − sgn σ i d b ,

and we obtain

(19) ∥ ω i ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ∥ f i ( ⋅ , b ) ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) ( 1 + t − b ) 1 − sgn σ i d b .

This completes the proof of Theorem 2.2.□

3 Small data global existence

We multiply (10) with ∂ x η where η is a multi-index, it follows that

∂ t t ∂ x η ω i + n ∂ t ∂ x η ∂ t ω i − e 2 t Δ ∂ x η ω i + m i 2 ∂ x η ω i = ∂ x η F ( ω i ) , ( x , t ) ∈ R n × ( 0 , T ) , ∂ x η ω i ( x , 0 ) = ∂ x η ρ 0 i ( x ) , ∂ x η ( ∂ t ω i ( x , 0 ) ) = ∂ x η ρ 1 i ( x ) , x ∈ R n .

Therefore,

(20) ∂ x η ω i ( x , t ) = ω 0 i ( x , t ) + L [ ∂ x η F ( ω i ) ] ( x , t ) for ( x , t ) ∈ R n × [ 0 , T ) ,

where ω 0 i is the solution of

∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = 0 , ( x , t ) ∈ R n × ( 0 , T ) , ω i ( x , 0 ) = ∂ x η ρ 0 i ( x ) , ∂ t ω i ( x , 0 ) = ∂ x η ϕ 1 ( x ) , x ∈ R n ,

and for a smooth function f i , we have set

L [ f i ] ( x , t ) = 2 e − n 2 t ∫ 0 t d b ∫ 0 e t − e b d r e n 2 b ν i ( x , r ; b ) κ i ( r , t ; 0 , b ) for ( x , t ) ∈ R n × ( 0 , T ) .

Here ν i ( x , t ; b ) is the solution to the problem

∂ t t ν i − Δ ν i = 0 , ν i ( x , 0 ; b ) = f i ( x , b ) , ∂ t ν i ( x , 0 ; b ) = 0 , ( x , t ) ∈ R n × ( 0 , T ) .

In other words, L [ f i ] is the solution of

∂ t t ω i + n ∂ t ω i − e 2 t Δ ω i + m i 2 ω i = f i , ( x , t ) ∈ R n × ( 0 , T ) , ω i ( x , 0 ) = 0 , ∂ t ω i ( x , 0 ) = 0 , x ∈ R n .

Let ∣ η ∣ ≤ k = [ n ⁄ 2 ] + 2 . Then, from (19) we obtain

(21) ∥ L [ ∂ x η F ( ω i ) ] ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ( 1 + t − b ) 1 − sgn σ i ∥ F ( ω i ) ( ⋅ , b ) ∥ W N , 1 ( R n ) d b

for t ∈ [ 0 , T ) , where we put N = 2 [ n ⁄ 2 ] + 3 . From (14), we also have

∥ ω 0 i ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ C e − n 2 t ( 1 + t ) 1 − sgn σ i { ∥ ρ 0 i ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) + ∥ ρ 1 i ∥ W [ n ⁄ 2 ] + 1,1 ( R n ) } t ∈ [ 0 , T )

and by using the Hölder’s inequality, the following inequality holds for the compact support initial data,

(22) ∥ ω 0 i ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ C e − n 2 t ( 1 + t ) 1 − sgn σ i ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) , t ∈ [ 0 , T ) .

Therefore, from (21) and (22), we obtain

(23) ∥ ω i ( ⋅ , t ) ∥ W k , ∞ ( R n ) ≤ C e − n 2 t ( 1 + t ) 1 − sgn σ i ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + e − n 2 t ∫ 0 t e n 2 b ( 1 + t − b ) 1 − sgn σ i ∥ F ( ω i ) ( ⋅ , b ) ∥ W N , 1 ( R n ) d b .

We need to use the following lemma to estimate the nonlinear part of (23).

Lemma 3.1

Let F ( ω i ) = ∣ ω i ∣ α ω i with an even integer α > 0 , N = 2 [ n ⁄ 2 ] + 3 , and k = [ n ⁄ 2 ] + 2 . If ω i is a solution of (10) with compactly supported initial data, then we have

(24) ∥ F ( ω i ) ∥ W N , 1 ( R n ) ≤ C ∥ ω i ∥ W k , ∞ ( R n ) α ∥ ω i ∥ W N , 2 ( R n ) ,

where C is a positive constant.

Proof

Since F ( 0 ) = 0 , the finite speed propagation property implies

(25) ∥ F ( ω i ) ∥ W N , 1 ( R n ) ≤ C ∥ F ( ω i ) ∥ W N , 2 ( R n )

from Hölder’s inequality, where C is a positive constant independent of t . It follows that

∥ F ( ω i ) ∥ W N , 2 ( R n ) = ∑ ∣ ρ ∣ ≤ N ∫ R n ∣ ∂ ρ ( ( ω i ω i ¯ ) α ⁄ 2 ω i ) ∣ 2 d x 1 ⁄ 2 = ∑ ∣ ρ ∣ ≤ N ∫ R n ∑ ρ 1 + … + ρ α + 1 = ρ C ρ α ( ∂ ρ 1 ω i ∂ ρ 2 ω i ¯ … ∂ ρ α − 1 ω i ∂ ρ α ω i ¯ ∂ ρ α + 1 ω i ) 2 d x 1 ⁄ 2 ,

where C ρ α is a suitable constant. Without loss of generality, we may assume ∣ ρ 1 ∣ , … , ∣ ρ α ∣ ≤ [ N ⁄ 2 ] + 1 . Hence, we have

∥ F ( ω i ) ∥ W N , 2 ( R n ) ≤ C ‖ ω i ‖ W N , 2 ( R n ) ∑ ∣ v ∣ ≤ [ N ⁄ 2 ] + 1 ∥ ∂ v ω i ∥ L ∞ ( R n ) α .

This completes the proof.□

Proof of Theorem 1.1

Since the local smooth solution of (10) exists, we need to derive a suitable a priori estimate for proving the global solvability of (10). Let N = 2 [ n ⁄ 2 ] + 3 and k = [ N ⁄ 2 ] + 1 . We assume that the solution of (10) satisfies

(26) e β t ‖ ω i ( ⋅ , t ) ‖ W k , ∞ ( R n ) ≤ 2 ε for t ∈ [ 0 , T ) ,

where β = n 2 if n ⁄ 2 < m and β = 0 if m = n ⁄ 2 for T > 0 and ε > 0 .

First, we consider the case m i > n ⁄ 2 . From (23) and (24), we deduce

(27) e n 2 t ∥ ω i ( ⋅ , t ) ∥ W k , ∞ ( R n ) ≤ C ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + C ∫ 0 t e n 2 b ∥ ω i ( ⋅ , b ) ∥ W k , ∞ ( R n ) α ∥ ω i ( ⋅ , b ) ∥ W N , 2 ( R n ) d b .

Therefore, we need to evaluate ∥ ω i ( ⋅ , b ) ∥ W N , 2 ( R n ) .

Let ∣ η ∣ ≤ N in (20). Then, from (12) with φ 0 i ≡ φ 1 i ≡ 0 , p = q = 2 and s = l = 0 we have

∥ L [ ∂ x η F ( ω i ) ] ( ⋅ , t ) ∥ L 2 ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ∥ F ( ω i ) ( ⋅ , b ) ∥ W N , 2 ( R n ) d b .

In view of the proof of Lemma 3.1, we obtain

∥ F ( ω i ( ⋅ , b ) ) ∥ W N , 2 ( R n ) ≤ C ‖ ω i ( ⋅ , b ) ‖ W k , ∞ ( R n ) α ‖ ω i ( ⋅ , b ) ‖ W N , 2 ( R n ) .

On the other hand, from (12) with f ≡ 0 , s = 0 , l = N , and p = q = 2 , we obtain

‖ ω 0 i ( ⋅ , t ) ‖ W N , 2 ( R n ) ≤ C e − n 2 t ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) )

for t ∈ [ 0 , T ) . Summing up, we obtain

‖ ∂ x η ω i ( ⋅ , t ) ‖ L 2 ( R n ) ≤ C e − n 2 t ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + C e − n 2 t ∫ 0 t e n 2 b ‖ ω i ( ⋅ , b ) ‖ W k , ∞ ( R n ) α ‖ ω i ( ⋅ , b ) ‖ W N , 2 ( R n ) d b ,

so that (26) yields

e n 2 t ‖ ω i ( ⋅ , t ) ‖ W N , 2 ( R n ) ≤ C ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + C ∫ 0 t e − α n 2 b ( 2 ε ) α e n 2 b ‖ ω i ( ⋅ , b ) ‖ W N , 2 ( R n ) d b

for t ∈ [ 0 , T ) . Since we assumed ‖ ρ 0 i ‖ W N , 2 ( R n ) + ‖ ρ 1 i ‖ W N , 2 ( R n ) ≤ ε , we see from Gronwall’s inequality that for t ∈ [ 0 , T )

e n 2 t ‖ ω i ( ⋅ , t ) ‖ W N , 2 ( R n ) ≤ C ε e ∫ 0 t C ε α e − α ( n 2 ) b d b ≤ C ε ,

for n ≥ 2 . Using these bounds in (27), we obtain

(28) e n 2 t ‖ ω i ( ⋅ , t ) ‖ W k , ∞ ( R n ) ≤ C C 1 ε + C ∫ 0 t ε α + 1 e − α ( n 2 ) b d b ≤ C 1 ε + C C 2 ε α + 1

for t ∈ [ 0 , T ) , where C 1 , C 2 are positive constants.

Next we consider the case, m i = n ⁄ 2 . From (23) and (24), we deduce

(29) ∥ ω i ( ⋅ , t ) ∥ W k , ∞ ( R n ) ≤ C ( 1 + t ) e − n 2 t ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + C e − n 2 t ∫ 0 t e n 2 b ( 1 + t − b ) ∥ ω i ( ⋅ , b ) ∥ W k , ∞ ( R n ) α ∥ ω i ( ⋅ , b ) ∥ W N , 2 ( R n ) d b .

Therefore, we need to evaluate ∥ ω i ( ⋅ , b ) ∥ W N , 2 ( R n ) .

Let ∣ η ∣ ≤ N in (20). Then, from (12) with φ 0 i ≡ φ 1 i ≡ 0 , p = q = 2 , and s = l = 0 , we have

∥ L [ ∂ x η F ( ω i ) ] ( ⋅ , t ) ∥ L 2 ( R n ) ≤ C e − n 2 t ∫ 0 t e n 2 b ( 1 + t − b ) ∥ F ( ω i ) ( ⋅ , b ) ∥ W N , 2 ( R n ) d b .

In view of the proof of Lemma 3.1, we obtain

∥ F ( ω i ( ⋅ , b ) ) ∥ W N , 2 ( R n ) ≤ C ‖ ω i ( ⋅ , b ) ‖ W k , ∞ ( R n ) α ‖ ω i ( ⋅ , b ) ‖ W N , 2 ( R n ) .

On the other hand, from (12) with f ≡ 0 , s = 0 , l = N and p = q = 2 , we obtain

‖ ω 0 i ( ⋅ , t ) ‖ W N , 2 ( R n ) ≤ C ( 1 + t ) e − n 2 t ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) ≤ C ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) )

for t ∈ [ 0 , T ) . Summing up, we obtain

‖ ∂ x η ω i ( ⋅ , t ) ‖ L 2 ( R n ) ≤ C ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + C e − n 2 t ∫ 0 t ( 1 + t − b ) e n 2 b ‖ ω i ( ⋅ , b ) ‖ W k , ∞ ( R n ) α ‖ ω i ( ⋅ , b ) ‖ W N , 2 ( R n ) d b ,

so that (26) yields

‖ ω i ( ⋅ , t ) ‖ W N , 2 ( R n ) ≤ C ( ∥ ρ 0 i ∥ W N , 2 ( R n ) + ∥ ρ 1 i ∥ W N , 2 ( R n ) ) + C e − n 2 t ∫ 0 t ( 1 + t − b ) e n 2 b ( 2 ε ) α ‖ ω i ( ⋅ , b ) ‖ W N , 2 ( R n ) d b

for t ∈ [ 0 , T ) . Since we assumed ‖ ρ 0 i ‖ W N , 2 ( R n ) + ‖ ρ 1 i ‖ W N , 2 ( R n ) ≤ ε , we see from Gronwall’s inequality that for t ∈ [ 0 , T )

‖ ω i ( ⋅ , t ) ‖ W N , 2 ( R n ) ≤ C ε e e − n 2 t ∫ 0 t C R α ε α ( 1 + t − b ) e n 2 b d b ≤ C ε ,

for n ≥ 2 . Using these bounds in (29), we obtain

(30) ‖ ω i ( ⋅ , t ) ‖ W k , ∞ ( R n ) ≤ C C 1 ε + C e − n 2 t ∫ 0 t ε α + 1 ( 1 + t − b ) e n 2 b d b ≤ C 1 ε + C C 2 ε α + 1

for t ∈ [ 0 , T ) , where C 1 , C 2 are positive constants.

If we choose ε and C 1 such that C C 2 ε α ≤ C 1 3 in (28) and (30), then we obtain

e γ t ‖ ω i ( ⋅ , t ) ‖ W k , ∞ ( R n ) ≤ 4 C 1 ε 3 ,

where γ = 0 if m = n ⁄ 2 and γ = n 2 if n ⁄ 2 < m . By combining the existence of the local solution, we conclude that the initial value problem (10) admits a global solution.□

4 Conclusion

In this article, we consider the system of semilinear Klein-Gordon equations with small initial conditions in anti-de Sitter spacetime. We use L ∞ decay estimate for linear part of the system of Klein-Gordon equations for m ≥ n ⁄ 2 . The existence result is proved under the condition that the nonlinear vector function is F ( ω i ) = ∣ ω i ∣ α ω i with a positive even integer α > 0 . The existence can also be shown using the energy estimate under different conditions for nonlinearity in the energy space. Numerical applications for the initial and boundary value problem of these equations will be presented in a forthcoming paper.

Acknowledgement

I would like to express my sincere thanks to the referees and the editor for their valuable comments and constructive suggestions.

Funding information: The author states that no funding was involved.
Author contribution: The author has accepted responsibility for the entire content of this article and approved its submission.
Conflict of interest: The author states that there is no conflict of interest.

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Received: 2024-12-11

Revised: 2025-02-28

Accepted: 2025-04-17

Published Online: 2025-09-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0136

Keywords for this article

semilinear Klein-Gordon system; anti-de Sitter spacetime; global existence

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