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We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in R N or in the exterior of a star-shaped obstacle. Such a result was first proved by Zuazua [40,41] for defocusing nonlinearity with moderate growth, and later extended to the energy subcritical case by Dehman-Lebeau-Zuazua [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.

We consider the Cauchy problem for the inhomogeneous nonlinear SchrödingerOnly partial results are known for the local existence in the subcritical case α < (4 − 2b)/(N − 2s) and much more less in the critical case α = (4 − 2b)/(N − 2s). In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for b = 0 and b > 0.

No abstract

We prove smoothing effect for solutions of a regularized Schrödinger equation on compact manifolds under the hypothesis of the Geometric control.