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.

It is proved that the unique viscosity solution of degenerate/singular elliptic partial differential equations (PDE for short) under gradient constraint coincides with that of the PDE with suitably selected bilateral obstacles. To this end, it is necessary to establish the Lipschitz estimates on viscosity solutions of bilateral obstacle problems. In the case of singular elliptic PDE, the notion of -solutions is adapted.

In this paper we investigate the Cauchy problem for hyperbolic operators with double characteristics in the framework of the space of C^∞ functions. In the case where the coefficients of their principal parts depend only on the time variable and are real analytic, we give a sufficient condition for C^∞ well-posedness, which is also a necessary one when the space dimension is less than 3 or the coefficients of the principal parts are semi-algebraic functions (e.g., polynomials) of the time variable.