Almost global existence for the semi-linear Klein-Gordon equation on the circle

We show that the solution to the semi-linear Klein-Gordon equation on the circle with a nonlinearity satisfying a convenient condition, exists almost globally, for almost every positive mass, provided that it is either even or odd as a function of the space variable. We also show, that if the nonlinearity is independent of the space derivative of the unknown and if it is also even as a function of the time derivative, then the solution exists almost globally, for almost every positive mass. The results are based on the method of normal forms. The difficulty is to find a structure of the nonlinearity so that the process of normal forms can be performed up to any order.