Blow up of solutions of semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

Abstract

Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann-Lemaître-Robertson-Walker spacetimes. For the case of accelerated expansion, we show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper bounds of the lifespan of blow-up solutions. Comparing to the case of the Minkowski spacetime, we discuss how the scale factor affects the lifespan of blow-up solutions of the equation.

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

“…The upper bound for p in the blow-up range (1.9) is a shift of magnitude 2 of the Glassey exponent that appears as upper bound in (1.11). This kind of phenomenon has already been observed in the semilinear model with power nonlinearity in [19,15] for the wave equation in the generalized Einstein-de Sitter spacetime.…”

“…In recent years, many papers have been devoted to the study of blow-up results and lifespan estimates for the semilinear wave equation in the generalized Einstein -de Sitter (EdS) spacetime with power nonlinearities [3,14] and generalizations [19,20,15]. More specifically, it has been conjectured that the critical exponent for the semilinear Cauchy problem with power nonlinearity |u| p…”

“…Let us emphasize that the support conditions in the statements of Theorems 1.1 and 1.2 can be proved extending the representation formulae from Section 2 to the multidimensional case. However, this is beyond the purposes of the present paper and we refer to the appendix in [19] for the proof of the property of finite speed of propagation.…”

On Glassey’s conjecture for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

“…The spatially flat FLRW metric is given by g : ds 2 = -dt 2 + a(t) 2 dσ 2 , where the speed of light is equal to 1, dσ 2 is the line element of an n-dimensional Euclidean space, and a(t) is the scale factor, which describes expansion or contraction of the spatial metric. As in our earlier work [1][2][3], we treat the scale factor as…”

“…The constant w appears in the equation of state relating the pressure to the density for the perfect fluid. See [1].…”