In this report I study the Kirchhoff equations, which are the basis for considering elastic filaments, in the context of a twisted planar ring. I apply an arc length preserving perturbation scheme that perturbs the basis used to describe the system directly, and analyse the generated normal mode solutions up to second-order. I also consider how these modes evolve in time.

In this paper we present a generalization of the Goryachev –Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics’ equations. This form of equations is recently developed and we can use it for the description of rigid body mo...

* Correspondence: [email protected] Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China Abstract In this article, we study the existence and multiplicity of positive solutions for the Neumann boundary value problems involving the p(x)-Kirchhoff of the form ⎪⎨⎪⎩ −M (∫ 1 p(x) (|∇u|p(x) + λ|u|p(x))dx ) (div (|∇u|p(x)−2∇u) − λ|u|p(x)−2u) = f (x, u) in , ∂u ∂v = 0...

In this paper‎, ‎we prove a multiplicity result for some biharmonic elliptic equation of Kirchhoff type and involving nonlinearities with critical exponential growth at infinity‎. ‎Using some variational arguments and exploiting the symmetries of the problem‎, ‎we establish a multiplicity result giving two nontrivial solutions‎.

Existence and bifurcation of positive solutions to a Kirchhoff type equation ⎧⎪⎨ ⎪⎩ − ( a + b ∫ Ω |∇u|2 ) u= νf (x,u), in Ω, u= 0, on ∂Ω are considered by using topological degree argument and variational method. Here f is a continuous function which is asymptotically linear at zero and is asymptotically 3-linear at infinity. The new results fill in a gap of recent research about the Kirchhoff ...

In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction ü = −γu̇+m(‖∇u‖)∆u− δ|u|u+ f, which is known as degenerate if m(·) ≥ 0, and non-degenerate if m(·) ≥ m0 > 0. We would like to point out that, to the author’s knowledge, exponential decay for this type of equations has been studied just for the special cases of α. Our aim ...