Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime

TL;DR

This paper investigates the lifespan of solutions to the semilinear wave equation in Einstein-de Sitter spacetime, establishing blow-up results and lifespan estimates, especially in critical cases, using iterative and analytical techniques.

Contribution

It introduces a novel approach combining iteration and special function theory to analyze critical lifespan estimates in a cosmological spacetime setting.

Findings

Derived upper bounds for solution lifespan

Established blow-up results in critical cases

Developed a new method for analyzing non-autonomous linear ODEs

Abstract

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions.