Attractors For Suspension Bridge Equations Research Articles

Longtime dynamics for a type of suspension bridge equation with past history and time delay

In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain \begin{document}$ \Omega $\end{document} of \begin{document}$ \mathbb{R}^N $\end{document} . Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.

Long-time behavior for suspension bridge equations with time delay

In this paper, we consider suspension bridge equations with time delay of the form $$\begin{aligned} u_{tt}(x,t) + \Delta ^2 u (x,t) + k u^+ (x,t) + a_0 u_t (x,t) + a_1 u_t (x, t- \tau ) + f(u(x,t)) = g(x). \end{aligned}$$ Many researchers have studied well-posedness, decay rates of energy, and existence of attractors for suspension bridge equations without delay effects. But, as far as we know, there is no work about suspension equations with time delay. In addition, there are not many studies on attractors for other delayed systems. Thus we first provide well-posedness for suspension equations with time delay. And then show the existence of global attractors and the finite dimensionality of the attractors by establishing energy functionals which are related to the norm of the phase space to our problem.

Random attractors for the extensible suspension bridge equation with white noise

This paper is devoted to the dynamical behavior of a stochastic extensible suspension bridge equation with white noise. For a deterministic suspension bridge equation, there are many classical results such as existence and uniqueness of a solution and long-term behavior of solutions. To the best of our knowledge, the existence of attractors for the suspension bridge equation with white noise was not yet considered. We intend to investigate these problems. We first obtain the dissipativeness of a solution in higher-energy spaces H3(Ω)×(H2(Ω)∩H01(Ω)). This implies that the random dynamical system generated by the equation has a random attractor in (H2(Ω)∩H01(Ω))×L2(Ω), which is a tempered random set in the space in H3(Ω)×(H2(Ω)∩H01(Ω)).