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

Abstract We investigate the long-time behaviour 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. A key difficulty arises from the viscous damping, which introduces a nonzero real part in the coefficient of the remaining critical resonant term. By carefully decomposing the solution into a Gaussian profile and a zero-mean remainder, we isolate the leading-order resonant dynamics. We identify a sign condition under which these dynamics are of absorption type. In this case, we prove global-in-time existence and enhanced diffusive decay for solutions with small initial data, and we characterize their leading-order asymptotics. Conversely, when this condition fails, the reduced resonant dynamics exhibit finite-time blow-up. In this regime, global existence is not expected, but solutions with small initial data persist and decay diffusively over exponentially long time intervals.