The Cauchy problem for the Klein–Gordon equation under the quartic potential in the de Sitter spacetime

Abstract

The Cauchy problem for the Klein–Gordon equation under the quartic potential is considered in the de Sitter spacetime. The existence of global solutions for small rough initial data is shown based on the mechanism of the spontaneous symmetry breaking for the small positive Hubble constant. The effects of the spatial expansion and contraction on the problem are considered.

A note on blow-up results for semilinear wave equations in de Sitter and anti-de Sitter spacetimes

“…The nonlinear term in (1.2) has been already considered in the literature for the Klein-Gordon equation in de Sitter spacetime by Yagdjian [21] when (1.3) is satisfied and by Nakamura [13] for a pure imaginary mass (i.e. for m 2 < 0 with our notations) both for de Sitter and anti-de Sitter spacetimes.…”

Numerical accuracy and stability of semilinear Klein–Gordon equation in de Sitter spacetime

“…The Klein-Gordon equation is one of the relativistic wave equations. There have been some analytical investigations of this equation (e.g., [1][2][3]). However, it is difficult to quantitatively evaluate the solutions analytically; therefore, we carry out numerical simulations to investigate the solutions.…”

On a semilinear wave equation in anti-de Sitter spacetime: the critical case

“…We recall the integral representation formulas (and their applications) established by Yagdjian and Yagdjian-Galstian in [28,21,22,23,24,25,26] and the global existence results for semilinear wave models in [9] and [1]. Concerning blow-up results, we recall the blow-up result with a pure imaginary mass term in (1.1), namely, when we replace m with im, both for de Sitter and anti-de Sitter spacetime in [10,Proposition 1.1]. Moreover, in [20] a blow-up result is proved in a de Sitter-type spacetime when m = 0.…”