2020 Volume 63 Issue 3 Pages 293-322
In this paper, we investigate the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity and localized damping. First, by using the semigroup method, we prove that the global existence and uniqueness of the solution to the linear problem. To overcome some difficulties, such as the presence of the perturbed effect and smooth effect, benefited from the ideas of M. M. Cavalcanti et al. [8], we derive smooth estimates and establish the exponential decay by using the multiplier techniques and the so-called compactness-uniqueness technique. On the other hand, we prove the existence, uniqueness of a local solution to nonlinear problem by using the semigroup theory and fixed point argument. Secondly, we extend the local solution to the global solution by using some priori estimates. Finally, we establish the exponential decay of the solution of nonlinear problem by using Sobolev inequality and Unique Continuation Principle.
