We study the asymptotic behavior in time of solutions to the Cauchy problems for the nonlinear Schrödinger equation with a critical power nonlinearity and the Hartree equation. We prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the Cauchy problem with small initial data. This estimate is very important for the proof of the existence of modified scattering states to the nonlinear Schrödinger equations with a critical nonlinearity and the Hartree equation. In order to derive the desired estimates we introduce a certain phase function since the previous methods, based solely on a priori estimates of the operator x + it ∇ acting on the solution without specifying any phase function, do not work for the critical case under consideration. The well-known nonexistence of the usual L 2 scattering states shows that our result is sharp.

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We study the initial value problem for the cubic nonlinear Klein-Gordon equationwhere µ ∈ R and the initial data are real-valued functions. We obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works. (2000). 35Q53, 35Q55. Mathematics Subject Classification

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equationin the sub critical case ∈ 2 − ε 3 , 2 . We assume that the initial data v 0 , 1 + * xAlso we suppose that the mean value of initial dataThen there exists a positive value ε such that the Cauchy problem (1) has a unique global solution v (t, x) ∈ C [0, ∞) ; L ∞ ∩ L 1,a , satisfying the following time decay estimate:v (t) L ∞ Cε t − 1 for large t > 0, here 2 − ε 3 < < 2.

We study the final value problem for the nonlinear Schrödinger equations in one or two space dimensions with the gauge-invariant nonlinearity of the critical long-range power and nonresonant polynomial nonlinearities of the same power, which are not gauge invariant. We construct a unique global solution (for positive time) which is asymptotic for large time to a given profile with sufficiently small norm in lower order Sobolev spaces with decay at the spatial infinity and at the zero frequency.