Lifespan estimates for local solutions to the semilinear wave equation in Einstein–de Sitter spacetime

Abstract

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument, and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions.

Keywords:

• Semilinear wave equation
• Einstein-de Sitter spacetime
• power nonlinearity
• critical case
• lifespan estimates

AMS CLASSIFICATIONS:

• Primary: 35B44
• 35L05
• 35L71
• Secondary: 33C10

Acknowledgments

The author thanks Karen Yagdjian (UTRGV) and Hiroyuki Takamura (Tohoku Univ.) for valuable discussions on the model considered in this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

A. Palmieri is supported by the Japan Society for the Promotion of Science (JSPS) – JSPS Postdoctoral Fellowship for Research in Japan (PE20003) and by the GNAMPA project ‘Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari dispersive’.