No abstract

This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on R 1 . Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H 1 (R 1 ). Then we prove that this system possesses a global attractor Aτ in H 1 (R 1 ). In addition, we show that the global attractor Aτ is regular, i.e., Aτ is actually included, bounded and compact in H 2 (R 1 ). Finally, we estimate the finite fractal dimensions of Aτ .

No abstract

No abstract

In this paper, we investigate the asymptotic behavior of the solutions for the Kuramoto-Sivashinsky equation with a time delay. We prove the global existence of solutions and energy decay. By using the Liapunov function method, we shall show that the solution is exponentially decay if the delay parameter τ is sufficiently small.