Abstract
In this paper, we study a plate equation as a model for a suspension bridge with time-varying delay and time-varying weights. Under some conditions on the delay and weight functions, we establish a stability result for the associated energy functional. The present work extends and generalizes some similar results in the case of wave or plate equations.
Highlights
We consider = (0, π) × (−, ) ⊂ R2 to be a thin rectangular plate with suspension hangers and suppose the plate is partially hinged on the vertical edges and free on the horizontal edges, see [7,8,9] for details on suspension bridge models through partially hinged plates
Motivated by the result in [3], where an energy decay estimate is established for a wave equation with constant feedback (τ ≡ constant) and h ≡ 0, we establish a stability result for Problem (1.1)–(1.3)
It follows from (3.1) and (3.2) that E(t) ≤ m E(0)e−λ1(1+t)a , ∀ t ≥ 0
Summary
The works of Nicaise and Pignotti [20,21], where they considered a wave equation with boundary or internal time-varying delay and established an exponential stability result. Enyi and Mukiawa [6] considered a viscoelastic plate equati√on when δ1 and δ2 are constants and time-varying delay and proved a general decay result provided |δ2| < |δ1| 1 − d. For existence and stability results for the wave equation with time-varying weights and constant time feedback, see [2,3,12] and references therein. Motivated by the result in [3], where an energy decay estimate is established for a wave equation with constant feedback (τ ≡ constant) and h ≡ 0, we establish a stability result for Problem (1.1)–(1.3).
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