A note on wave equation in Einstein and de Sitter space-time

Abstract

We consider the wave propagating in the Einstein and de Sitter space-time. The covariant d’Alembert’s operator in the Einstein and de Sitter space-time belongs to the family of the non-Fuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the Lp−Lq estimates for solutions.

Integral transform approach to generalized Tricomi equations

“…In order to keep down the length of this paper, we postpone applications of Theorem 1.5 to the derivation of the Strichartz estimates and to global well-posedness of the nonlinear generalized Tricomi equation in the metric (17). We note here that γ = −1 for the metric (17) (see [13]) that makes the hypergeometric function polynomial. Moreover, the hypergeometric function is polynomial for γ = −1, −2, .…”

“…The case of m > 2 covers the beam equation and hyperbolic in the sense of Petrowski (p-evolution ) equations. On the other hand, the Cauchy-Kowalewski theorem guarantees solvability of the problem in the real analytic functions category for the partial differential equation (13) with any positive ℓ and m = 2. Furthermore, the operator A(x, ∂ x ) = |α|≤2 a α (x)∂ α x can be replaced with an abstract operator A acting on some linear topological space of functions.…”

“…The case of negative ℓ requires some modifications in the setting of the initial conditions at t = 0. For the EdeS spacetime these modifications are suggested in [13]; they are the so-called weighted initial conditions. For the negative ℓ operators K 0 and K 1 and corresponding weighted initial conditions will be discussed in the forthcoming paper.…”

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We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14], [41]-[46] the particular version of this transform in order 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. MSC: 35C15; 35Q75; 35Q05; 83F05; 76H05 Definition 1.2 Let φ ∈ C(R + ) be non-negative increasing function and Ω be a backward time connected set.

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Wave Asymptotics at a Cosmological Time-Singularity: Classical and Quantum Scalar Fields

“…There are also numerous works on the field equations in the smooth (without singulatity) FLRW space-times, in particular in the De Sitter universe (see the fundamental work of H. Friedrich [26], and also e.g. [4], [21], [29], [30], [31], [36], [40], [47], [49], [52], [58], [62]). In contrast, surprisingly enough, few mathematical papers deal with the simpler issue of the behaviour of the linear fields near the cosmological singularities.…”