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

TL;DR

This paper proves that solutions to a specific semilinear wave equation in generalized Einstein-de Sitter spacetime cannot exist globally, identifying critical exponents that depend on space dimension and spacetime geometry.

Contribution

It introduces a new blow-up result for derivative-type nonlinear wave equations in this spacetime, using Yagdjian's integral transform to determine solution behavior.

Findings

Blow-up results for the wave equation in Einstein-de Sitter spacetime.

Identification of a Glassey-type critical exponent.

Dependence of the exponent on space dimension and spacetime metric.

Abstract

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.