Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type

TL;DR

This paper studies the blow-up behavior and lifespan of solutions to a damped wave equation in Einstein-de Sitter spacetime with derivative nonlinearity, extending previous results to any positive damping coefficient.

Contribution

It generalizes previous blow-up results to any positive damping coefficient in a semilinear wave equation within Einstein-de Sitter spacetime.

Findings

Blow-up occurs under certain conditions on the nonlinear exponent.

Lifespan estimates are derived for solutions with critical decay rates.

Results extend previous work limited to specific damping coefficients.

Abstract

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.