Lifespan estimates for the semi-linear Klein–Gordon equation with a quadratic potential in dimension one

We show for almost every m>0m>0, the solution to the semi-linear Klein–Gordon equation with a quadratic potential in dimension one, exists over a longer time interval than the one given by local existence theory, using the normal form method. By using an Lp−LqLp−Lq estimate for eigenfunctions of the harmonic oscillator and by carefully analysis on the nonlinearity, we improve the result obtained by the author before.