Exponential Energy Decay for Damped Klein-Gordon Equation with Nonlinearities of Arbitrary Growth

We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in N or in the exterior of a star-shaped obstacle. Such a result was first proved by Zuazua [37, 38] for defocusing nonlinearity with moderate growth, and later extended to the energy subcritical case by Dehman et al. [7], using linear approximation and unique continuation arguments. We propose a different approach based solely on Morawetz-type a priori estimates, which applies to defocusing nonlinearity of arbitrary growth, including the energy critical case, the supercritical case and exponential nonlinearities in any dimensions. One advantage of our proof, even in the case of moderate growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than the one of the ground state, once we get control of the nonlinear part in Morawetz-type estimates. In particular this can be achieved when we have the scattering for the undamped equation.