Long‐time behavior of a nonlinear viscoelastic beam equation with past history

Abstract

This paper is concerned with the existence of a global attractor for the nonlinear viscoelastic beam equation with past history memory uttMathClass-bin+kΔ2uMathClass-bin−MathClass-op∫MathClass-bin−MathClass-rel∞tμ(tMathClass-bin−τ)Δ2u(τ)dτMathClass-bin+f(u)MathClass-bin+g(ut)MathClass-rel=hMathClass-punc,where g(ut) is a damping like | ut | rut and f(u) is a source term like | u | αu − | u | βu, by considering 0 ≤ β < α and r > 0. Copyright © 2014 John Wiley & Sons, Ltd.

The global attractor for a suspension bridge with memory and partially hinged boundary conditions

“…Then we have P r o o f . We follow the method of Nakao in [15] (see also [16]). Let B ⊂ H be bounded and positively invariant for S(t) and (u 0 , u 1 , ϕ 0 ), (ũ 0 ,ũ 1 ,φ 0 ) ∈ B.…”

“…(1.2) 1 . In [16], Narciso considered (1.2) 1 , in a bounded domain of R n , n ≥ 1, subject to clamped boundary conditions, with a nonlinear damping term k(u t ) and source term h(u) satisfying some growth assumptions. He proved the existence of a global attractor.…”

“…He proved the existence of a global attractor. In [6], Conti and Pelin improved the results of [16], in a bounded domain of R 3 , by removing the growth condition on h(u) and weakening conditions on both functions k(u t ) and h(u). They also allowed the kernel function g to blow up at zero and established the existence of a regular global attractor.…”