Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay

Abstract

This paper is concerned with a stability result for a Kirchhoff beam equation with variable exponents and time delay. The exponential and polynomial stability results are proved based on Komornik's inequality.

Existence of beam-equation solutions with strong damping and p(x)-biharmonic operator

“…For the p(x)-biharmonic elliptical problems, Ge, Zhou and Wu [21] studied the problem (9) ∆ 2 p(x) u = f (x, u), in Ω, u = 0, ∆u = 0, on ∂Ω, where f (x, u) = λV (x)|u| q(x)−2 u, λ is a positive real number, V is a weight function and p, q : Ω → (1, ∞) are continuous functions. Considering different situations concerning the growth rates involved in Problem (9), they proved the existence of a continuous family of eigenvalues using the mountain pass theorem and Ekeland's variational principle.…”

“…Li and Tang [22] studied Problem (9) with Navier boundary condition and for f (x, u) = λ|u| p(x)−2 u+g(x, u), where λ ≤ 0 and g(x, u) is a Carathéodory function. Using the mountain pass theorem and Fountain theorem, they established the existence of at least one solution.…”

“…Kong, in [23], considered Problem (9), where f (x, u) = λb(x)|u| γ(x)−2 u − λc(x)|u| β(x)−2 u − a(x)|u| p(x)−2 u, with λ > 0 is a parameter and a, b, c, β, γ ∈ C(Ω) are nonnegative functions. He proved the existence of weak solutions to the problem associated with Navier boundary conditions.…”

“…The author established the local existence of weak solutions and determined the finite-time blowup of solutions with nonpositive initial energy. Regarding the equations with variable exponent nonlinearities, we also refer to [4][5][6][7][8][9].…”