and many authors have studied the existence and uniqueness of global solution, the blowup of the solution (see [–] and the references therein). WhenM is not a constant function, equation (.)without the damping and source terms is often called a Kirchhoff-type wave equation; it has first been introduced by Kirchhoff [] in order to describe the nonlinear vibrations of an elastic string. When...

in this paper‎, ‎we consider the following kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{r}^{3}}|nabla u|^{2}right)delta u+v(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{r}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{r}^{3},$ ‎$uin h^{1}(mathbb{r}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎the aim of this paper is to study the existence of positive ...

Wave-equation migration is an accurate and robust alternative to Kirchhoff migration when multipathing and other complex wave phenomena occur, such as in subsalt exploration. In these situations, images obtained by wave-equation migration are often superior to images obtained by Kirchhoff methods. However, 3-D wave-equation prestack imaging is an immature technology compared with Kirchhoff imag...

The Kirchhoff analogy for elastic rods establishes the equivalence between the solutions of the classical spinning top and the stationary solutions of the Kirchhoff model for thin elastic rods with circular cross-sections. In this paper the Kirchhoff analogy is further generalized to show that the classical Kovalevskaya solution for the rigid body problem is formally equivalent to the solution ...

Firstand second-order reflection coefficients are presented for the small slope approximation. The first-order reflection coefficient is identical to the Kirchhoff, or physical optics, result, and the secondorder reflection coefficient reduces to those of perturbation theory and the Kirchhoff approximation in the appropriate limits. Numerical results are obtained for acoustic or TE-polarized el...

Let G be a simple connected graph with n ≤ 2 vertices and m edges, and let μ1 ≥ μ2 ≥...≥μn-1 >μn=0 be its Laplacian eigenvalues. The Kirchhoff index and Laplacian-energy-like invariant (LEL) of graph G are defined as Kf(G)=nΣi=1n-1<...

In this paper, we consider the coupled Kirchhoff system in subcritical case and critical case. For case, first study least energy of limit by using Nehari manifold method exclude existence semi-trivial solutions. Then improve global compactness lemma to overcome loss compactness. Combining with linking theorem topological degree, construct a Palais-Smale sequence high level prove positive Moreo...

We study a nonlocal elliptic equation of $p$-Kirchhoff type involving the critical Sobolev exponent. First we give sufficient conditions for $(\text{PS})$ condition to hold. Then prove some existence and multiplicity results using tools from Morse theory, in particular, notion cohomological local splitting eigenvalues based on Fadell-Rabinowitz index.

‎Using Nehari manifold methods and Mountain pass theorem‎, ‎the existence of nontrivial and radially symmetric solutions for a class of $p$-Kirchhoff-type system are established.

A generalized Kowalevski Hamiltonian and new integrable cases on e(3) and so(4). 1 A new integrable case for the Kirchhoff equation. The Kirchhoff equations (see for example [1]) for the motion of a rigid body in an ideal fluid read as follows M t = M × ∂H ∂M + Γ × ∂H ∂Γ , Γ t = Γ × ∂H ∂M ,