Long-time behavior for suspension bridge equations with time delay

Abstract

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.