Norm inflation for solutions of semi-linear one-dimensional Klein–Gordon equations

In space dimension larger than or equal to 2, the nonlinear Klein–Gordon equation with small, smooth, decaying initial data has global-in-time solutions. This no longer holds true in one space dimension, where examples of blowing-up solutions are known. On the other hand, it has been proved that if the nonlinearity satisfies a convenient compatibility condition, the “null condition”, one recovers global existence and that the solutions satisfy the same dispersive bounds as linear solutions. The goal of this paper is to show that, in the case of cubic semi-linear nonlinearities, this null condition is optimal, in the sense that, when it does not hold, one may construct small, smooth, decaying initial data giving rise to solutions that display inflation of their $L^{\infty }$ and $L^2$-norms in finite time.