Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation

We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals.