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

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.

On heatlike lifespan of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime

“…In our results, however, another exponent appears as a blow-up condition in some case. This is different from the results by [6]. We emphasize that the generalized exponent of p G (n + µ) cannot always be the critical exponent for the global existence of solutions.…”

“…If α = 0, our upper bounds of the lifespan coincide with the results above by [5]. Similar results are independently shown by [6] where energy solutions are treated. In our results, however, another exponent appears as a blow-up condition in some case.…”

On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity

“…We emphasize that the role of r 1 and r 2 is fully interchangeable in the previous identity. Combining (14) and (15), after some straightforward steps we arrive at…”

“…Moreover, under the same conditions for the parameters as above, also the case with derivative type nonlinearity |∂ t u| p is studied in [13,14,36]. Furthermore, we point out that in [35,36] even the case ℓ −1 with ν 2 = 0 is studied for different semilinear terms.…”

On a semilinear wave equation in anti-de Sitter spacetime: the critical case

“…Finally, for the wave equation in Einstein-de Sitter spacetime (that is, for the d'Alember operator EdS = ∂ 2 t − t −2k ∆ + bt −1 ∂ t with k ∈ (0, 1) and b 0) we cite the papers [3,4] for the linear model, [5,16,11,12,17,19] for the semilinear model with power nonlinearity |v| p and [6,7,18] for the semilinear model with nonlinearity of derivative type |∂ t v| p . It is interesting to compare our approach in this paper to deal with a critical case in comparison to those in [16,11,12] for the treatment of the corresponding critical cases in Einsten-de Sitter spacetime.…”