Uniform decay rate estimates for the beam equation with locally distributed nonlinear damping

Abstract

In this paper, we study the semilinear beam equation with a locally distributed nonlinear damping on a smooth bounded domain. We first construct approximate solutions, and we show that the aforementioned approximate solutions decay uniformly in the weak phase space by using an observability inequality associated to the linear problem and a unique continuation property. Then, we prove the global existence as well as the uniform decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively.

Global attractor of the Euler-Bernoulli equations with a localized nonlinear damping

“…There are also some achievements in the global attractor for the plate/beam equations with fully interior/boundary dissipation (see [6,20,22,26,28,15]) and the long-time behavior of solutions for the beam/plate equations with a localized damping (see [2,4,11,12,13,14,16,18,24,27,30]). In particular, as the locally distributed and unbounded nature of the damping, the authors [18,24] established the exponential decay of solutions by using a frequency domain method and a contradiction argument.…”

“…The stabilization of the viscoelastic Euler-Bernoulli type equation with a local nonlinear dissipation has been proved in [16] by using multiplier methods and a unique continuation result from [23]. The authors [4] have proved the uniform decay rate estimates for the beam equation with locally distributed nonlinear damping by using an observability inequality associated to the linear problem and a unique continuation property. The energy decay estimates for the Bernoulli-Euler type equation with a local degenerate dissipation was proved in [27] by multiplier methods and a contradiction argument.…”