Mathematical Physics Seminar

We study the 1D Klein-Gordon equation with quadratic and variable coefficient
cubic nonlinearity. This problem exhibits a striking resonant
interaction between
the spatial frequencies of the nonlinear coefficients and the temporal
oscillations of the solutions. We prove global existence and (in
L-infinity) decay, as well as some
smoothness of the solution at time-like infinity, with the help of
several new classes of
normal-forms transformations. The analysis also shows the limited
smoothness of the solution, in the presence of the resonances.