Attractors for wave equations with nonlinear damping on time-dependent space

Abstract

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.

Time-dependent asymptotic behavior of the solution for plate equations with linear memory

“…Recently, Meng et al investigated the long-time behavior of the solution for the wave equation with nonlinear damping g(u t ) on the time-dependent space, in which they found a new technical method verifying compactness of the process via defining the contractive functions, see [16]. In [17], Meng and Liu gave the necessary and sufficient conditions of the existence of time-dependent global attractor borrowed from the ideas in [15].…”

“…It is worth mentioning that the dissipative condition (1.5) is weaker than one in [16], because the authors made use of the dissipative condition, i.e., lim inf…”

The global attractor for a class of extensible beams with nonlocal weak damping

“…• We will obtain the asymptotic smoothness by taking advantage of the energy reconstruction method given by Chueshov and Lasiecka [10], which was used to prove the existence of global attractors in (1.5) with nonlinear dissipation g(u t ) and a subcritical nonlinear term F . The main reason is that when the velocity u t is very small, the nonlocal damping u t p u t is weaker than the linear damping u t , it is more difficult to obtain the asymptotic smoothness by utilizing the decomposition of semigroup or contractive functions method than in the case of linear damping u t (see [34,33,38,23] ).…”

Dynamics of wave equations with moving boundary

“…In this section we recall a well-known compactness criterion established in Chueshov and Lasiecka [6,Proposition 3.2] and [7,Proposition 2.10], for autonomous systems. Nonautonomous versions of that result were presented in [24,26], with X t = X, and in [22] with time-dependent spaces. To our purpose, we consider this compactness criterion in a D-universe framework.…”

“…The second author was supported by Ministerio de Educación-DGPU and Ministerio de Ciencia e Innovación (Spain) under projects PHB2010-0002-PC and MTM2015-63723-P. The authors thank the referee for his/her constructive remarks and for bringing to their attention the references [8,22].…”