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

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. We show the blow-up in finite time of local in time solutions and we establish upper bound estimates for the lifespan, provided that the exponent in the nonlinear term is below a suitable threshold and that the Cauchy data are nonnegative and compactly supported.