Advances in Analysis

Download PDF (667.1 KB) PP. 40 - 60 Pub. Date: July 8, 2016

DOI: 10.22606/aan.2016.11005

• Chaosheng Zhu^*
  School of mathematics and statistics, Southwest University, Chongqing, 400715, P. R. China.

This paper is devoted to the large time behavior and especially to the regularity of the global attractor of the dissipative 1D nonlinear Schrödinger equation with nonlocal integral term on R. We first prove that the existence of the global attractor A_r in the strong topology of H¹(R) and the existence of the exponential attractor M which contains the global attractor A , are still finite dimensional, and attract the trajectories exponentially fast. We also show that the global attractor A_r is regular, i.e., A_r is included, bounded and compact in H²(R) assuming that the forcing term f(x) is of class H²(R). Furthermore we estimate the number of the determining modes for this equation. Moreover, we show that the solution trajectories and the global attractor of the nonlocal Schrödinger equation converge to those of the usual Schrödinger equation, as the coefficient of the nonlocal integral term goes to zero.

Nonlinear Schrödinger equation, Nonlocal integral term, Global attractor, Exponential attractor, Regularity, Determining modes, Trajectory convergence, Attractor convergence.

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