The semilinear Klein-Gordon equation in de Sitter spacetime

Abstract

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation ✷gφ − m 2 φ = −|φ| p with the small mass m ≤ 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.3 √ Λ , is considered by Dafermos and Rodnianski [3]. They proved that in the region bounded by a set of black/white hole horizons and cosmological horizons, solutions converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. The bounds on decay rates for solutions to the wave equation in the Schwarzschild -de Sitter spacetime is a first step to a mathematical understanding of non-linear stability problems for spacetimes containing black holes.Catania and Georgiev [2] studied the Cauchy problem for the semilinear wave equation ✷ g φ = |φ| p in the Schwarzschild metric (3 + 1)-dimensional space-time, that is the case of Λ = 0 in 0 < M bh < 1 3 √ Λ .

Global existence of the scalar field in de Sitter spacetime

“…We note here that any classical solution to Eq. (0.5) solves also the integral equation [28]. For the case of nonlinearity F (Φ) = c|Φ| α+1 , c ̸ = 0, the results of [28] imply the nonexistence of the global solution even for arbitrary small function Φ 0 (x, 0) under some conditions on n, α, and M.…”

“…If we allow large initial data, then, according to Theorem 1.2 [28], the concentration of the mass, due to the non-dispersion property of the de Sitter spacetime, leads to the nonexistence of the global solution, which cannot be recovered even by adding an exponentially decaying factor in the nonlinear term. More precisely, the next theorem states that the solution blows up in finite time.…”

“…(0.3) was studied in [28]. For the case of nonlinearity F (Φ) = c|Φ| α+1 , c ̸ = 0, Theorem 1.1 [28], implies the nonexistence of the global solution even for arbitrary small initial functions ϕ 0 (x) and ϕ 1 (x) under some conditions on n, α, and M. By means of the evident transformation one can apply the conclusion of Theorem 1.1 [28] to the equation with imaginary physical mass (see (3.1)) and derive the following blowup result. …”

Global in time existence of self-interacting scalar field in de Sitter spacetimes

“…Then, according to Theorem 1.1 [30], if c = 0, α > 0, and m = 0, then for every positive numbers ε and s there exist functions ψ 0 ,…”

Strichartz Estimates on Asymptotically de Sitter Spaces

“…The assumptions on λ and f k imply that solutions obey a global energy bound and that the energy is positive definite, while the assumption on h Yagdjian [26] studied a similar equation (with an exponentially decaying function multiplying the nonlinearity) on de Sitter space. In that work, the author also considers only the "large mass" setting (λ ≥ n 2 4 ) and obtains a global existence result for a family of nonlinearities.…”