The wave equation in the Einstein and de Sitter spacetime

Abstract

We consider the wave propagating in the Einstein & de Sitter spacetime. The covariant d'Alembert's operator in the Einstein & de Sitter spacetime 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 L p − L q estimates for solutions.

Decay of solutions of the wave equation in expanding cosmological spacetimes

“…This was used in [1] to show that, in the flat case, solutions with initial data of compact support decay as t −1 , that is, as a(t) − 3 2 (but without decay estimates for the time derivative). The same problem was studied further in [11,12], including L p -L q decay estimates and paramatrices. The wave equation in the de Sitter spacetime with flat 3-dimensional spatial sections was analyzed by Rendall [16]; he proved that the time derivative decays at least as e −Ht (with H = Λ/3 being the Hubble constant, where Λ > 0 is the cosmological constant), that is, as a(t) −1 , and conjectured a decay of order e −2Ht , that is, a(t) −2 .…”

Solutions of the wave equation bounded at the Big Bang

“…Since φ τ0 − φ τ1 is a solution of the wave equation for τ ≥ τ 1 , the energy inequality (7) and the a priori bound (9) imply…”

“…To prove uniqueness, we note that by using initial data in C ∞ (Σ) approaching φ(τ 0 , · ) in H 1 (Σ) and ∂ τ φ(τ 0 , · ) in L 2 (Σ), it is easy to extend the energy inequality (7) to weak solutions in…”

“…A first example was provided by Klainerman and Sarnak [9], who gave the explicit solution for the wave equation for two particular FLRW models (the so-called Einstein-de Sitter universe and its hyperbolic analogue): their work was used in [1] to prove the existence of solutions bounded at the Big Bang in these two models. The problem of prescribing asymptotic data at the Big Bang for the Einstein-de Sitter universe was further analyzed in [7,8]. More recently, Ringström [11] studied linear systems of wave equations on general cosmological backgrounds with convergent asymptotics, including the problem of imposing asymptotic data.…”

On t-dependent hyperbolic systems. Part 2

“…Equations with time-dependent coefficients arise in a natural way when studying the physics of an expanding universe. This has attracted considerable interest over the recent years, just to mention a few references we refer to the work of K. Yagdjian, A. Galstian and T. Kinoshita [8,30,9] and sketch the relation to our approach. Especially for the family of Friedmann-Lemaître-Robertson-Walker spacetimes on R 1+n with metrics of the form ds 2 = − dt 2 + 1 a 2 (t) n j=1 dx 2 j , a(t) > 0, (5.39) in an expanding / shrinking universe the covariant Klein-Gordon equation for scalar fields takes the form u tt − a 2 (t)∆u − nȧ with variable mass and dissipation.…”