The semilinear Klein-Gordon equation in de Sitter spacetime

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\Box_g \phi-m^2 \phi = -|\phi |^p $ with the small mass $m \le n/2$ in de Sitter space-time with the metric $g$. We prove that for every $p>1$ the large energy solution blows up, while for the small energy solutions we give a borderline $p=p(m,n)$ for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of the Kato's lemma.


Introduction
In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation 2 g φ − m 2 φ = −|φ| p with the small mass m ≤ n/2 in de Sitter space-time.
In the model of the universe proposed by de Sitter the line element has the form The constant M bh may have a meaning of the "mass of the black hole". The corresponding metric with this line element is called the Schwarzschild -de Sitter metric. The Cauchy problem for the semilinear Klein-Gordon equation in Minkowski spacetime (M bh = Λ = 0) is well investigated. (See, e.g., [7] and references therein.) In particular, Keel and Tao [7] for the semilinear equation u tt − ∆u = F (u), u(0, x) = εϕ 0 (x), u t (0, x) = εϕ 1 (x) proved that if n = 1, 2, 3 and 1 < p < 1 + 2n, then there exists a (non-Hamiltonian) nonlinearity F satisfying |D α F (u)| ≤ C|u| p−|α| for 0 ≤ α ≤ [p] and such that there is no finite energy global solution supported in the forward light cone, for any nontrivial smooth compactly supported ϕ 0 and ϕ 1 and for any ε > 0. There is an interesting question of instability of the ground state standing solutions e iωt φ ω (x) for nonlinear Klein-Gordon equation ∂ 2 t u − ∆u + u = |u| p−1 u. Here φ ω is a ground state of the equation −∆φ + (1 − ω 2 )φ = |φ| p−1 φ, while 0 < p − 1 < 4/(N − 2) and 0 ≤ |ω| < 1. Ohta and Todorova [9] showed that instability occurs in the very strong sense that an arbitrarily small perturbation of the initial data can make the perturbed solution blow up in finite time.
The Cauchy problem for the linear wave equation without source term on the maximally extended Schwarzschild -de Sitter spacetime in the case of non-extremal black-hole corresponding to parameter values 0 < M bh < 1 They established that the problem in the Regge-Wheeler coordinates is locally well-posed in H σ for any σ ∈ [1, p + 1). Then for the special choice of the initial data they proved the blow-up of the solution in two cases: (a) p ∈ (1, 1 + √ 2) and small initial data supported far away from the black hole; (b) p ∈ (2, 1 + √ 2) and large data supported near the black hole. In both cases, they also gave an estimate from above for the lifespan of the solution.
In the present paper we focus on the another limit case as M bh → 0 in 0 < M bh < 1 3 √ Λ , namely, we set M bh = 0 to ignore completely influence of the black hole. Thus, the line element in de Sitter spacetime has the form The Lamaître-Robertson transformation [8] leads to the following form for the line element: By defining coordinates x ′ , y ′ , z ′ connected with r ′ , θ ′ , φ ′ by the usual equations connecting Cartesian coordinates and polar coordinates in a Euclidean space, the line element may be written [8,Sec.134] The new coordinates r ′ , θ ′ , φ ′ , t ′ can take all values from −∞ to ∞. Here R is the "radius" of the universe.
In this paper we study blow-up phenomena for semilinear equation by applying the Lamaître-Robertson transformation and by employing the fundamental solutions for some model linear hyperbolic equation with variable speed of propagation. In [16] the Klein-Gordon operator in Robertson-Walker spacetime, that is , with a support in the forward light cone D + (x 0 , t 0 ), x 0 ∈ R n , t 0 ∈ R, and the fundamental solution with a support in the backward light cone D − (x 0 , t 0 ), x 0 ∈ R n , t 0 ∈ R, defined by D ± (x 0 , t 0 ) := (x, t) ∈ R n+1 ; |x − x 0 | ≤ ±(e −t0 − e −t ) , are constructed. These fundamental solutions have been used to represent solutions of the Cauchy problem and to prove L p − L q estimates for the solutions of the equation with and without a source term that provide with some necessary tools for the studying semilinear equations.
In the Robertson-Walker spacetime [5], one can choose coordinates so that the metric has the form In particular, the metric in de Sitter and anti-de Sitter spacetime in the Lamaître-Robertson coordinates [8] has this form with S(t) = e t and S(t) = e −t , respectively. The matter waves in the de Sitter spacetime are described by the function φ, which satisfies equations of motion. In the de Sitter universe the equation for the scalar field with mass m and potential function V is the covariant Klein-Gordon equation with the usual summation convention. Written explicitly in coordinates in the de Sitter spacetime it, in particular, for V ′ (φ) = −|φ| p has the form In this paper we restrict ourselves with consideration of the semilinear equation for particle with small mass m, that is 0 ≤ m ≤ n/2. If we introduce the new unknown function u = e n 2 t φ, then it takes the form of the semilinier Klein-Gordon equation for u on de Sitter spacetime where non-negative curved mass M ≥ 0 is defined as follows: The equation (2) can be regarded as Klein-Gordon equation with imaginary mass. Equations with imaginary mass appear in several physical models such as φ 4 field model, tachion (super-light) fields, Landau-Ginzburg-Higgs equation and others. To solve the Cauchy problem for semilinear equation we use fundamental solution of the corresponding linear operator. We denote by G the resolving operator of the problem Thus, u = G[f ]. The equation of (3) is strictly hyperbolic. This implies the well-posedness of the Cauchy problem (3) in the different functional spaces. Consequently, the operator is well-defined in those functional spaces. Then, the speed of propagation is variable, namely, it is equal to e −t . The second-order strictly hyperbolic equation (3) possesses two fundamental solutions resolving the Cauchy problem without source term f . They can be written in terms of the Fourier integral operators, which give complete description of the wave front sets of the solutions. Moreover, the integrability of the characteristic roots, ∞ 0 |λ i (t, ξ)|dt < ∞, i = 1, 2, leads to the existence of the so-called "horizon" for that equation. More precisely, any signal emitted from the spatial point x 0 ∈ R n at time t 0 ∈ R remains inside the ball B n t0 (x 0 ) := {x ∈ R n | |x − x 0 | < e −t0 } for all time t ∈ (t 0 , ∞). In particular, it can cause a nonexistence of the L p − L q decay for the solutions. In [13] this phenomenon is illustrated by a model equation with permanently bounded domain of influence, power decay of characteristic roots, and without L p − L q decay. The above mentioned L p − L q decay estimates are one of the important tools for studying nonlinear problems (see, e.g. [11]). In this paper we show that this phenomenon causes the blow up of the solution. The equation (3) is neither Lorentz invariant nor invariant with respect to usual scaling and that creates additional difficulties.
Operator G is constructed in [16] for the case of the large mass m ≥ n/2. The analytic continuation of this operator in parameter M into C allows us to use G also in the case of small mass 0 ≤ m ≤ n/2. More precisely, we define the operator G acting on f (x, t) ∈ C ∞ (R × [0, ∞)) by where F a, b; c; ζ is the hypergeometric function.(See, e.g., [1]. For analytic continuation see , e.g., [12,Sec. 1.8] .) If n is odd, n = 2m + 1, m ∈ N, then for f ∈ C ∞ (R n × [0, ∞)), we define . Constant ω n−1 is the area of the unit sphere S n−1 ⊂ R n . If n is even, n = 2m, m ∈ N, then for f ∈ C ∞ (R n ×[0, ∞)), the operator G is given by the next expression . Thus, in both cases, of even and odd n, one can write where the function v(x, t; b) is a solution to the Cauchy problem for the wave equation It can be proved that if n 1 − 2 q ≤ 1, then for every given T > 0 the operator G can be extended to the bounded operator: Consequently the operator G maps in the corresponding topologies. Moreover, .
Then any solution u = u(x, t) of the equation (2) which takes initial value u(x, 0) = ϕ 0 (x), ∂ t u(x, 0) = ϕ 1 (x), solves also integral equation Let Γ ∈ C([0, ∞)). For every given function u 0 ∈ C([0, T ]; L q ′ (R n )) we consider integral equation (5) u( for the function u ∈ C([0, T ]; L q (R n )) ∩ C([0, T ]; L p (R n )). Here q ′ ≥ q > 1, p ≥ 1. The last integral equation corresponds to the slightly more general equation than (2), namely, to the nonlocal equation If u 0 is generated by the Cauchy problem (4), then the solution u = u(x, t) of (6) is said to be a weak solution of the Cauchy problem with the initial conditions for the equation (7). In the present paper we are looking for the conditions on the function Γ, on constants M , n, p, and β that guarantee a non-existence of global in time weak solution, namely, the blow-up phenomena.
We are especially interested in the scale of functions The function e − n(p−1)

)) is either non-decreasing or non-increasing, and if
with the numbers ε > 0 and c > 0, while for M = 0 it satisfies Then, for every p > 1, N , and ε there exists u 0 ∈ C ∞ (R n × [0, ∞)) which for any given slice of constant (6) with permanently bounded support does not exist for all q ∈ [2, ∞) and β > 1/p − 1. More precisely, there is T > 0 such that This theorem shows that instability of the trivial solution occurs in the very strong sense, that is, an arbitrarily small perturbation of the initial data can make the perturbed solution blowing up in finite time.
If we allow large initial data, then according to the next theorem, for every d 0 ∈ R and M > 0 the solution blows up in finite time.
Theorem 1.2 Suppose that function Γ(t) = e γt , where γ ∈ R and that the curved mass is positive, M > 0. Then, for every p > 1 and n there exists (6) with permanently bounded support does not exist for all Thus, for every p > 1 the large energy classical solution of the Cauchy for equation (1) blows up. We will prove global existence of the small energy solution in a forthcoming paper.
The remaining part of this paper is organized as follows. In Section 2 we prove some auxiliary integral representations for the function sinh(t) and the linear function via Gauss's hypergeometric function and multidimensional integrals involving also fundamental solution of the Cauchy problem for wave equation in Minkowski spacetime. In Section 3 we suggest two simple generalizations of Kato's lemma, which allow us to handle the case of differential inequalities with exponentially decreasing kernels. In Section 4 we complete the proofs of both theorems.  [14].
can be represented as follows: (ii) If n is odd, n = 2m + 1, m ∈ N, then with c Here the constant ω n−1 is the area of the unit sphere S n−1 ⊂ R n .
Proof. First we consider case (i). According to Theorem 0.3 [16] for every function f ∈ C ∞ (R × [0, ∞)), which for any given slice of constant time t = const ≥ 0 has a compact support in x, the function is a unique C ∞ -solution to the Cauchy problem with n = 1. It follows On the other hand, from the linear Klein-Gordon equation (9) and the vanishing initial data, we obtain then (11) and (12) imply We easily find Then (11) implies On the other hand Thus, for the arbitrary function f ∈ C ∞ (R × [0, ∞)) for all t due to (10) one has

It follows (8). Thus (i) is proved.
To prove case (ii) with n is odd, n = 2m + 1, m ∈ N, we use the identity and take into consideration that the kernel is an even function of r 1 . In the case of (iii) when n is even, n = 2m, m ∈ N, we apply the identity The proposition is proven. 2 If we set in the above integrals b = 0 then we get integral representations of the function sinh(M t) depending on parameter M > 0. We also note that both sides of these formulas are translation invariant in t. By passing to the limit as M → 0 we arrive at the following corollary.
(ii) If n is odd, n = 2m + 1, m ∈ N, then with c

The second order differential inequalities
The second order differential inequality with the power decreasing kernel play key role in proving blow-up of the solutions of the semilinear equations. Kato's lemma [6] allows us to derive from inequalitÿ w ≥ bt −1−p w p , p > 1, b > 0, t large a boundedness of the life-span of solution with property w t ≥ a > 0. For the equation in de Sitter spacetime the kernel of the corresponding ordinary differential inequality decreases exponentially: There is a global solution to the last inequality. Hence, in order to reach exact conditions on the involving functions we have to generalize Kato's lemma. It is done in two following lemmas. ([a, b)), and where Γ ∈ C 1 ([a, ∞)) is non-negative function, Γ(a) > 0, and p > 1. Assume that for all t ∈ [a, b) eitheṙ If there exists a 1 ∈ (a, b) such that then b must be finite unless lim t→∞ F (t) is finite.
Proof. First we consider the case ofΓ ≤ 0. The conditions of the lemma imply that derivative of the energy density function is non-negative, We integrate the last inequality and obtain for all t ∈ [a, b) .
In fact, according to the second inequality of the condition (13) we have Hence, It follows d dt According to the first inequality of the condition (13) there exists a 1 > a such that a 1 , b). Thus, for large t we get contradiction. The case of uniformly positive function Γ follows from Kato's Lemma [6]. Lemma is proven.
2 Next we turn to the case of the small energy and exponentially decreasing Γ(t).
where A, γ ∈ C 1 ([a, ∞)) are non-negative functions and p > 1, c 0 > 0. Assume that and that If there exist ε > 0 and c > 0 such that then b must be finite.
Proof. There is a point a 1 ≥ a such that F t (a 1 ) > 0. Then F t (t) ≥ F t (a 1 ) for all t ≥ a 1 and consequently for sufficiently large a 2 . Furthermore, according to (16) for the energy density function we have The last inequality implies for all t ∈ [a 1 , b). For sufficiently large a 2 ≥ a 1 using conditions (14), (15), and (17) we derive for all t ∈ [a 2 , b). But with sufficiently large a 2 ≥ a 1 we obtain The last nonlinear differential inequality does not have global solution with F > 0. Lemma is proven. 2 Remark 3. 3 We note here that the equation a > 0, and γ(t) = c γ e (pa−d)t . The condition (17) implies a > d/(p − 1). On the other hand, the first inequality of (14) holds only if a ≤ d/(p − 1).

Nonexistence of the global solution for the integral equation associated with the Klein-Gordon equation
Since G is a fundamental solution of the strictly hyperbolic operator, for every given function u 0 ∈ C([0, T ]; L q (R n )) ∩ C ∞ ([0, T ] × R n ) there exist T 0 > 0 and solution u ∈ C([0, T 0 ]; L q (R n )). Moreover, for every given T one can prove existence of the solution u ∈ C([0, T ]; L q (R n )) provided that sup t∈[0,T ] u 0 (·, t) L q (R n ) is small enough. Theorem 1.1 shows that the set of such T , in general, is bounded.
Proof of Theorem 1.1. Let u 0 ∈ C ∞ ([0, ∞)×R n ) be a function with the permanently bounded support, that is supp u 0 (·, t) ⊂ { x ∈ R n ; |x| ≤ constant } for all t ≥ 0. We denote ϕ 0 (x) := u 0 (x, 0) and ϕ 1 (x) := ∂ t u 0 (x, 0). One can find u 0 such that where The solution of the problem (4) with the data ϕ 0 (x), ϕ 1 (x) ∈ C ∞ 0 (R n ) exemplifies such function. Indeed, this unique smooth solution obeys finite propagation speed property that implies supp u 0 (·, t) ⊂ { x ∈ R n ; |x| ≤ In order to check (18) for that solution u 0 we integrate (4) with respect to x over R n and then solve the initial problem with data (19) for the obtained ordinary differential equation.
Suppose that u ∈ C([0, ∞); L q (R n )) with permanently bounded support is a solution to (6) generated by u 0 . According to the definition of the solution, for every given T > 0 we have and u(x, 0) = ϕ 0 (x) , u t (x, 0) = ϕ 1 (x) . Then u ∈ C([0, ∞); L 1 (R n )) and we can integrate equation (6): In particular, To evaluate the last term of (20) we apply Proposition 2.1. Consider the case of odd n ≥ 3. Then, for the smooth function u = u(x, t) we obtain Therefore, We obtain, Thus, for the solution u = u(x, t) we have proven
In the case of M > 0 we obtain then the function F (t) is It follows F ∈ C 2 ([0, ∞)). More precisely, In particular, since Γ(t) ≥ 0, we obtain On the other hand, since the solution u = u(x, t) has permanently bounded support, then supp u(·, t) ⊂ { x ∈ R n ; |x| ≤ R } for some positive number R. Using the compact support of u(·, t) and Hölder's inequality we get with τ n the volume of the unit ball in R n , Here we assume Γ(t) > 0. Thus By means of the inequality M C 0 + C 1 > 0 we conclude that F (t) ≥ 0 and thaẗ Hence, for appropriate C 0 and C 1 the last inequality together with (21) to (23) implies the following system of the ordinary differential inequalities The Lemma 3.1 shows that if F (t) ∈ C 2 ([0, b)) and the energy of particle is large, then b must be finite. The conditions of the Lemma 3.1 are fulfilled on (0, b) for the function with γ > 0 without any condition on the energy. They are fulfilled with γ < 0 if the kinetic energy and the potential energy are sufficiently large, that is C 0 > 0, C 1 > 0, and Next we turn to the case of the small energy and exponentially decreasing Γ(t). We apply Lemma 3.2 with A(t) = e Mt and p replaced with p(β + 1). More precisely, if we set then the conditions of the last lemma read: The last inequality follows from the monotonicity of Γ(t). By the condition of the theorem, there exist ε > 0 and c > 0 such that that coincides with (17). The case of M > 0 is proved. Now consider the case of M = 0. Let Then Corollary 2.2 allows us to write Hence (20) reads: . Now we choose a function u 0 ∈ C ∞ ([0, ∞) × R n ) such that R n u 0 (x, t)dx = C 0 + C 1 t .
The solution of the problem (4) with M = 0 exemplifies such functions. Thus It follows F ∈ C 2 ([0, ∞)). More precisely, In particular, On the other hand according to (24) we obtain for all t in [0, ∞). By means of the condition C 1 > 0 we concludë F (t) ≥ CΓ(t)F (t) p(β+1) for large t with C > 0 .
But for appropriate C 0 and C 1 one has F (t) > 0 and the last inequality together with (25) implies for all t ∈ [a, b).
The Kato's Lemma 2 [6] shows that if F (t) ∈ C 2 ([0, b)) and Γ(t) ≥ t −1−p(β+1) with p(β + 1) > 1, then b must be finite. Theorem is proven. Proof of Theorem 1.2. The case of γ ≥ 0 is covered by Theorem 1.1 and implies a blow-up even for the small data. Therefore, we restrict ourselves to the case of γ < 0. Then, with a special choice of C 0 and C 1 after arguments have been used in the proof of Theorem 1.1 we arrive at the following system of the ordinary differential inequalities The first inequality is fulfilled if C 0 , that is the initial potential energy, is sufficiently large, while the second one is fulfilled if C 1 , that is the initial kinetic energy, is large enough. Theorem is proven.