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) |