Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

Abstract

We study the time-asymptotic behavior of solutions of the Schrödinger equation with nonlinear dissipationWe give a precise description of the behavior of the solutions (including decay rates in L 2 and L ∞ , and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that α is sufficiently close to 2 N .

Lower bound estimate for the dissipative nonlinear Schrödinger equation

“…[9,12,27]). For the dissipative case Im λ < 0, there are several works that study the precise large time behavior when p ≤ 1 + 2/n (see [3,4,6,14]) and the exponent p = 1 + 2/n plays a critical role not for the scattering but the large time behavior. In recent work [5] (cf [16]), it is shown that there exists a scattering state…”

On global attractors for 2D damped driven nonlinear Schrödinger equations

“…Recently Cazenave and Han established the long-time behavior for nonlinear autonomous Schrödinger equation in R n with a nonlinear subcritical dissipation [17].…”

Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term

“…and gave the characterizations of finite time blow up, boundedness and convergence to the ground state for the initial boundary value problem of Equation (5) with…”

“…Lian and Xu [17] proved the asymptotic behavior for the nonlinear wave equation with weak and strong damping terms and logarithmic source term. Cazenave and Han [5] studied asymptotic behavior Schrödinger equation with nonlinear dissipation. For the fourth-order wave equation with strong damping, Yang et al [29] derived the existence of strong and weak uniform attractors with non-compact external forces.…”