Abstract
We discuss the long-time dynamical behavior of the non-autonomous suspension bridge-type equation, where the nonlinearity is translation compact and the time-dependent external forces only satisfy Condition ( ) instead of being translation compact. By applying some new results and the energy estimate technique, the existence of uniform attractors is obtained. The result improves and extends some known results. MSC:34Q35, 35B40, 35B41.
Highlights
Consider the following equations: ⎧⎨utt + uxxxx + δut + ku+ = l + h(x, t), in (, L) × R,⎩u(, t) = u(L, t) = uxx(, t) = uxx(L, t) =, t ∈ R. ( . )Suspension bridge equations ( . ) have been posed as a new problem in the field of nonlinear analysis [ ] by Lazer and McKenna in
In this paper, we will discuss the following non-autonomous suspension bridge-type equation: Let be an open bounded subset of R with smooth boundary, Rτ = [τ, +∞], and we add the nonlinear forcing term g(u, t) to ( . ) and neglect gravity, we can obtain the following initial-boundary value problem:
Where u(x, t) is an unknown function, which could represent the deflection of the road bed in the vertical plane; h(x, t) and g(u, t) are time-dependent external forces; ku+ represents the restoring force, k denotes the spring constant; αut represents the viscous damping, α is a given positive constant
Summary
In this paper, we will discuss the following non-autonomous suspension bridge-type equation: Let be an open bounded subset of R with smooth boundary, Rτ = [τ , +∞], and we add the nonlinear forcing term g(u, t) (which is dependent on the deflection u and time t) to U (x), on ∂ × Rτ , x∈ , where u(x, t) is an unknown function, which could represent the deflection of the road bed in the vertical plane; h(x, t) and g(u, t) are time-dependent external forces; ku+ represents the restoring force, k denotes the spring constant; αut represents the viscous damping, α is a given positive constant To our knowledge, this is the first time for one to consider the non-autonomous dynamics of equation In Section we prove our main result about the existence of a uniform attractor for the non-autonomous dynamical system generated by the solution of ( . )
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