Strichartz Estimates on Asymptotically de Sitter Spaces

Baskin, Dean

doi:10.1007/s00023-012-0187-7

Cited by 29 publications

(36 citation statements)

References 33 publications

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“…It is also assumed that α = 4 n−1 . In Theorem 1.3 [18] the existence of the global solution for small energy data is stated. (For more references on the asymptotically de Sitter spaces, see the bibliography in [19,20].)…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Global existence of the scalar field in de Sitter spacetime

Yagdjian¹

2012

Journal of Mathematical Analysis and Applications

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L q estimates Higgs boson equation a b s t r a c tIn this article we prove the global existence of the small data solutions of the Cauchy problem for the semilinear Klein-Gordon equation in the de Sitter spacetime. Unlike the same problem in the Minkowski spacetime, we have no restriction on the order of nonlinearity and structure of the nonlinear term, provided that a physical mass of the field is outside of some bounded interval.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Baskin [18] discussed the small data global solutions for the scalar Klein-Gordon equation on asymptotically de Sitter spaces, which is a compact manifold with a boundary. More precisely, in [18] the Cauchy problem is considered for the semilinear equation…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Global existence of the scalar field in de Sitter spacetime

Yagdjian¹

2012

Journal of Mathematical Analysis and Applications

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“…Although recently the equations in the de Sitter and anti-de Sitter space-times became the focus of interest for an increasing number of authors (see, e.g., [1,4,6,11,14,15,19,23,24,28] and the bibliography therein) which investigate those equations from a wide spectrum of perspectives, there are very few papers on the semilinear Klein-Gordon equation in the de Sitter space-time. Here we mention some of them closely related to our main result.…”

Section: Condition (L)mentioning

confidence: 99%

“…In Theorem 1.3 [6] the existence of the global solution for small energy data is stated. (For more references on the asymptotically de Sitter spaces, see the bibliography in [5], [28].)…”

Section: Condition (L)mentioning

confidence: 99%

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

Galstian

Yagdjian

2017

Nonlinear Analysis: Real World Applications

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We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space-time with the time slices being Riemannian manifolds. IntroductionIn this paper we prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space-time with the time slices being Riemannian manifolds. In the spatially flat de Sitter model, these slices are R 3 , while in the spatially closed and spatially open cases these slices can be the three-sphere S 3 and the three-hyperboloid H 3 , respectively (see, e.g., [12, p.113]).The metric g in the de Sitter space-time is defined as follows, g 00 = g 00 = −1, g 0j = g 0j = 0, g ij (x, t) = e 2t σ ij (x), i, j = 1, 2, . . . , n, where n j=1 σ ij (x)σ jk (x) = δ ik , and δ ij is Kronecker's delta. The metric σ ij (x) describes the time slices. In the quantum field theory the matter fields are described by a function ψ that must satisfy equations of motion. In the case of a massive scalar field, the equation of motion is the semilinear Klein-Gordon equation generated by the metric g:Here m is a physical mass of the particle. In physical terms this equation describes a local selfinteraction for a scalar particle. A typical example of a potential function would be V (ψ) = ψ 4 . The semilinear equations are also commonly used models for general nonlinear problems.The covariant Klein-Gordon equation in the de Sitter space-time in the coordinates is

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Integral transform approach to solving Klein-Gordon equation with variable coefficients

Yagdjian¹

2015

Math. Nachr.

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In this paper we describe the integral transform that allows to write solutions of one partial differential equation via solution of another one. This transform was suggested by the author in the case when the last equation is a wave equation, and then it was used to investigate several well-known equations such as Tricomi-type equation, the Klein-Gordon equation in the de Sitter and Einstein-de Sitter spacetimes. A generalization given in this paper allows us to consider also the Klein-Gordon equations with coefficients depending on the spatial variables. Keywords: Klein-Gordon equation; Curved spacetime; Representation of solution Introduction and Statement of ResultsIn this paper we give some generalization of the approach suggested in [28], which is aimed to reduce equations with variable coefficients to more simple ones. This transform was used in a series of papers [10, 11], [28]-[33] to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes. The listed equations play an important role in the gas dynamics, elementary particle physics, quantum field theory in curved spaces, and cosmology.Consider for the smooth function f = f (x, t) the solution w = w A,f (x, t; b) to the problemwith the parameter b ∈ I = [t 0 , T ] ⊆ R, t 0 < T ≤ ∞, and with 0 < T 1 ≤ ∞.Here Ω is a domain in R n , while A(x, ∂ x ) is the partial differential operator A(x, ∂ x ) = |α|≤m a α (x)D α x . We are going to present the integral operator In fact, the function u = u(x, t) takes initial values as follows u(x, t 0 ) = 0, u t (x, t 0 ) = 0, x ∈ Ω .

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Strichartz Estimates on Asymptotically de Sitter Spaces

Cited by 29 publications

References 33 publications

Global existence of the scalar field in de Sitter spacetime

Global existence of the scalar field in de Sitter spacetime

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

Integral transform approach to solving Klein-Gordon equation with variable coefficients

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