Finite lifespan of solutions of the semilinear wave equation in the Einstein–de Sitter spacetime

Abstract

We examine the solutions of the semilinear wave equation, and, in particular, of the ϕ p model of quantum field theory in the curved space-time. More exactly, for 1 < p < 4 we prove that solution of the massless self-interacting scalar field equation in the Einstein-de Sitter universe has finite lifespan.

On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case

“…Their work also showed the existence of global classical solutions for small ε > 0, if p > p c (n) and either n = 2 or n = 3 and the data are radially symmetric. Finally, we mention that our original equations (1.3) is related to semilinear wave equations in the Einstein-de Sitter spacetime considered by Galstian & Yagdjian [4]. We recall that p 0 (n) in (1.5) is the critical exponent for the semilinear wave equation conjectured by Strauss [11].…”

On heatlike lifespan of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime

“…If α = 2/3 and µ = 2, our blow-up range of p is the same as 1 < p < p cr (n) in [3]. We note that these upper bounds stated so far are sharp if the power p is dominated by the Strauss exponent or p c (n, α, µ).…”

“…For the case α = 2/3, µ = 2 and n ≥ 1 in (1.3), it is proved by Galstian and Yagdjian [3] that finite time blow-up occurs if 1 < p < p F (n/3) or 1 < p < p cr (n), where p cr (n) is the positive root of…”

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

“…In the related literature (see, for example, [3]), the differential operator with timedependent coefficients on the left-hand side of (1.1) is called the wave operator on the generalized Einstein-de Sitter spacetime. This nomenclature is due to the fact that for k = 2 3 and n = 3 the operator ∂ 2 t − t − 4 3 ∆ + 2t −1 ∂ t coincides with the d'Alembertian operator in Einstein-de Sitter's Lorentzian metric.…”

“…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…”