In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation ε(t)utt + g(ut) − ∆u + ϕ(u) = f with Dirichlet boundary condition, in which, the coefficient ε depends explicitly on time, the damping g is nonlinear and the nonlinearity ϕ 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 timedependent attractors for the wave equations with nonlinear damping.
In this paper, we are concerned with infinite dimensional dynamical systems in time-dependent space. First, we characterize some necessary and sufficient conditions for the existence of the time-dependent global attractor by using a measure of noncompactness. Then, we give a new method to verify the sufficient condition. As a simple application, we prove the existence of the time-dependent global attractor for the damped equation in strong topological space.
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