×

The lifespan of solutions to nonlinear Schrödinger and Klein-Gordon equations. (English) Zbl 1178.35087

Summary: Precise information on the lifespan sometimes tells us how the nonlinearity affects large time behavior of solutions to nonlinear evolution equations. As pointed out by John and Hörmander in 1987, there are surprising connections between the lifespan and the null condition in the wave equation case. In this paper we give a review of analogous lifespan estimates for nonlinear Schrödinger and Klein-Gordon equations. We also discuss how this viewpoint could give a unified understanding of many previous results.

MSC:

35B44 Blow-up in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L67 Shocks and singularities for hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI