We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: ut + (|u|ρ−1u)x + Huxx = 0, where H is the Hilbert transform, x, t ∈… (More)

which suggests dissipativity if Im λ < 0. In fact, it is proved in [17] that the solution decays like O((t log t)−1/2) in Lx as t → +∞ if Im λ < 0 and u0 is small enough. Since the non-trivial free… (More)

We study the asymptotic behavior of solutions, in particular the scattering theory, for the nonlinear Schrödinger equations with cubic and quadratic nonlinearities in one or two space dimensions. The… (More)

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂2 t u + ∂tu − ∆u + λu 2 n = 0, x ∈ Rn, t > 0, u(0, x) =… (More)

We study a global existence in time of small solutions to the quadratic nonlinear Schrödinger equation in two space dimensions, { i∂tu+ 1 2 ∆u = N (u), (t, x) ∈ R × R2, u(0, x) = u0(x), x ∈ R2, (0.1)

We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces i∂tu+ (1/2)∆u = (u), (t,x) ∈ R×R2; u(0,x)=φ(x),x ∈R2, where… (More)

We study large-time asymptotic behavior of solutions to the Cauchy problem for a model of nonlinear dissipative evolution equation. The linear part is a pseudodifferential operator and the… (More)

We consider the nonlinear Schrödinger systems −i∂tu1 + 1 2 ∆u1 = F (u1, u2), i∂tu2 + 1 2 ∆u2 = F (u1, u2) in n space dimensions, where F is a p-th order local or nonlocal nonlinearity smooth up to… (More)