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

Abstract

The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping utt + ∆ 2 u − m(∇u 2)∆u + ut p ut + f (u) = h, in Ω × R + , p ≥ 0 on a bounded smooth domain Ω ⊂ R n with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient m(∇u 2) and the nonlinear term f (u), the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.

The global attractor for the wave equation with nonlocal strong damping

“…Compared with nonlocal weak damping [38], [36] and [37], we found that the main differences and difficulties with these are mainly reflected in the following points.…”

“…where the nonlocal energy damping Ω (|∆u| 2 + |u t | 2 )dx q ∆u t was first proposed by Balakrishnan and Taylor [6]. Moreover, in [36], the authors studied the long-time behavior of a class of extensible beams equation with the nonlocal weak damping…”

“…(1) For well-posedness, due to the unboundedness of the damping in L 2 (Ω), we can not employ proposition 1.12 in [9] (as in [38], [36], [37]) to verify the generalized solution of problem (1.1) is also a weak solution. Thus, we use the regularity ∇u t (•) p+2 ∈ L 1 (0, T ) to deal with the well-posedness of Eq.…”

Long-time dynamics of a class of nonlocal extensible beams with time delay

“…For nonlocal damping, Jorge Silva and Narciso [18,19] studied well-posedness and long-time dynamics for equation (1) with nonlocal nonlinear damping terms; N( ∇u 2 )(− ) θ (u t ) and N( ∇u 2 )g(u t ), respectively. Zhao et al [20,21] investigated the damping k( u t…”

Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping

“…Thus as a preliminary, we give several lemmas concerning the repeated limit inferior in section 2. To handle the difficulty that nonlocal damping coefficient u t p brings in energy estimate, we use the strong monotone inequality for the general inner product space established in our another paper [50].…”