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

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

“…Remark 1. The upper bound for p in the blow-up range ( 9) is a shift of magnitude 2 of the Glassey exponent that appears as upper bound in (11). This kind of phenomenon has already been observed in the semilinear model with power nonlinearity in [31,24].…”

“…We point out that in the general case µ > 0, the blow-up argument from Theorems 1.1 and 1.2 does not work sharply, especially for µ in the interval (k, 2 − k). In the forthcoming paper [11], we will study systematically the blow-up dynamic for (48) via a completely different approach.…”

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

“…• we expect global (in time) existence of solutions with the range γ > 1 so that C Gla = 0, which stands for the supercritical nonlinearity case. [12], the wave model with scale-invariant damping (and mass) [12,17], the damped wave model in the Einstein-de Sitter spacetime [6], and the generalized Tricomi equations [13].…”

Blow-up result for a weakly coupled system of wave equations with a scale-invariant damping, mass term and time derivative nonlinearity

“…In the context of combined nonlinearities, similar results are obtained in [3,12]. Now, in a more general context, namely m, µ, ν > 0, the damped wave equation with Tricomi and mass terms is considered in [1] which somehow constitutes an improvement of [13,14] for −1 < m < 0 and without the mass term. More precisely, it is proven in [1] that the blow-up happens for all p ≤ p T (N + µ m+1 , m).…”

“…Hence, in the same spirit of use as e.g. in [10,12,13,14], we define the positive test function ψ η i (x, t) as the solution of the conjugate equation associated with the linear problem of (1.1), namely (3.1) ) x + e −η (1+m) x for N = 1.…”