Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime

TL;DR

This paper establishes blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime, proposing a generalized critical exponent and analyzing solutions via special functions.

Contribution

It introduces new blow-up criteria for damped wave equations in a cosmological spacetime and conjectures a generalized critical exponent extending Strauss's classical result.

Findings

Proved blow-up results using iteration methods.

Conjectured a generalized critical exponent.

Analyzed solutions of a linear ODE with special functions.

Abstract

We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for the critical exponent due to the main blow-up results, which is consistent with many special cases of the considered model and provides a natural generalization of Strauss exponent. In the critical case, we consider a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solutions are determined by means of special functions' theory.