Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent
Fengjuan Meng Chengkui Zhong
In this paper, we are concerned with some properties of the global attractor of weakly damped wave equations. We get the existence of multiple stationary solutions for wave equations with weakly damping. Furthermore, we provide some approaches to verify the small neighborhood of the origin is an attracting domain which is important to obtain the multiple equilibrium points in global attractor.
keywords: wave equations. global attractor Lyapunov functional $Z_2$ index equilibrium points
Attractors for wave equations with nonlinear damping on time-dependent space
Fengjuan Meng Meihua Yang Chengkui Zhong
In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation \begin{eqnarray*} \varepsilon(t) u_{tt}+g(u_{t})-\Delta u+\varphi (u)=f \end{eqnarray*} with Dirichlet boundary condition, in which, the coefficient $\varepsilon$ depends explicitly on time, the damping $g$ is nonlinear and the nonlinearity $\varphi$ has a critical growth. Spirited by this concrete problem, we establish a sufficient and necessary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process via contractive functions. Finally, by the new framework, we obtain the existence of the time-dependent attractors for the wave equations with nonlinear damping.
keywords: nonlinear damping wave equation Time-dependent attractor critical exponent.

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