Kirchhoff Equations Research Articles

Nodal solutions for Kirchhoff equation in [formula omitted] with critical growth

In this paper, we consider the following Kirchhoff equation in R3 with critical growth. (a+b∫R3|∇u|2dx)Δu−V(x)u+u5+μ|u|q−2u=0,inR3,u(x)→0,as|x|→∞,where V is the potential function, a,b,μ>0,5<q<6 are constants. We assume V is a radial function and is bounded from below by a positive constant. We prove that for any given positive integer k, the problem has a radial solution, having k nodal domains exactly.

Generalization of the Cauchy–Poisson Method and the Construction of Timoshenko-Type Equations

We consider a generalization of the Cauchy–Poisson method to the n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. The presented investigation generalizes and supplements the results obtained earlier. In the Euclidean space, we introduce certain restrictions for the derivatives. The principle of hyperbolic degeneracy in terms of parameters is formulated and its realization in the form of necessary and sufficient conditions is presented. In a special case of fourdimensional space (in which the operators are preserved up to the sixth order), we obtain a generalized hyperbolic equation for the bending vibrations of plates with coefficients that depend only on the Poisson ratio. This equation includes, as special cases, the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As the development of Maxwell's and Einstein's investigations of the propagation of perturbations with finite velocity in continuous media, we can mention the nontrivial construction of Timoshenko’s equation for the bending vibrations of a beam.

Concentrated solution for some non-local and non-variational singularly perturbed problems

In the present paper, we prove the existence of concentrated solution for the following non-local and non-variational singularly perturbed problem, 0.1 $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\varepsilon ^2A(\varepsilon ^{-n}||u||^q_{L^q})\Delta u+ V(x)u=|u|^{p-1}u, &{}x\in \mathbb {R}^n,\\ 0<u\in H^1(\mathbb {R}^n), \lim \limits _{|x|\rightarrow \infty }u(x)=0, \end{array} \right. \end{aligned}$$ where $$n\ge 1$$ , $$1<p<2^*-1$$ , $$2\le q<2^*$$ , $$A: (0, +\infty )\rightarrow [a, +\infty )$$ , $$V(x): \mathbb {R}^n\rightarrow \mathbb {R}$$ are two continuous functions, $$a>0$$ and $$\varepsilon $$ is a positive and small parameter. Problem (0.1) can be seen as a model to study the vibration of nonlinear string or bacteria’s density balance law in $$\mathbb {R}^n$$ . The existence is based on the well-known Lyapunov–Schmidt reduction method. In order to make Lyapunov–Schmidt reduction method work well, existence and so-called non-degeneracy result of positive solutions for following problem will be needed. 0.2 $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -A(\Vert U\Vert ^q_{L^q})\Delta U(y)+ \alpha U(y)=U^{p}(y), &{}y\in \mathbb {R}^n,\\ 0<U\in H^1(\mathbb {R}^n). \end{array} \right. \end{aligned}$$ In Sect. 2, existence and non-degeneracy results for positive solutions of problem (0.2) are proved. Compared to the standard Schrodinger equation, our results imply that the non-local term $$A(\Vert U\Vert _{L^q}^q)$$ has no effect on the non-degeneracy result of positive solutions of problem (0.2) when the solution U satisfies $$\Vert U\Vert ^q_{L^q}A'(\Vert U\Vert ^q_{L^q})\ne \frac{2}{n}A(\Vert U\Vert ^q_{L^q})$$ and that the non-local term $$A(\Vert U\Vert _{L^q}^q)$$ has significant effect on the existence result. Since the linearized operator L[U] for problem (0.2) is non-self-adjoint, our non-degeneracy result is a result about the Kernel space of the adjoint operator of linearized operator L[U] for problem (0.2) rather than the Kernel space of L[U] itself, which is essentially different from common non-local and self-adjoint equations, such as Kirchhoff equation, fractional Laplace equations and Choquard equations. Moreover, in our non-degeneracy result, we only assume that the solution U satisfies $$\Vert U\Vert _{L^q}^qA'(\Vert U\Vert _{L^q}^q)\ne \frac{2}{n}A(\Vert U\Vert _{L^q}^q)$$ . In our existence result of the concentrated solution, we only assume that $$A\in C^{1,1}_{loc}$$ . These two assumptions imply that A can be a non-monotonous or unbounded function.

Sign-changing solutions for a fractional Kirchhoff equation

Using a minimization argument and a quantitative deformation lemma, we establish the existence of least energy sign-changing solutions for the following nonlinear Kirchhoff problem (a+b[u]2)(−Δ)su+V(x)u=K(x)f(u)inR3,where a,b>0 are constants, s∈(0,1), (−Δ)s is the fractional Laplacian, V,K are continuous, positive functions, allowed for vanishing behavior at infinity, and f is a continuous function satisfying suitable growth assumptions. Moreover, when the nonlinearity f is odd, we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals.

One‐bump positive solutions for the Kirchhoff equations with perturbation in

In this paper, we study the Kirchhoff equations with perturbation in . Applying the finite reduction method, we prove that the equation has one‐bump positive solutions under some suitable conditions that are given in Section 1.

Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

In this paper we address the following Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta (g(|\nabla u|_2^2) u + u^r) = a u + b u^p&{} \quad \text{ in }~\varOmega , u>0&{}\quad \text{ in }~\varOmega , u= 0&{} \quad \text{ on }~\partial \varOmega , \end{array} \right. \end{aligned}$$ in a bounded and smooth domain $$\varOmega $$ in $$\mathbb {R}^{N}$$ . By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters a, b, p, r and the non-linearity g, the existence of positive solutions.

Time periodic solutions of one-dimensional forced Kirchhoff equations with x-dependent coefficients under spatial periodic conditions

In this paper, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under the spatial periodic conditions, which models the forced vibrations of an inhomogeneous string in presence of a time periodic external forcing with period \(2\pi /\omega \) and amplitude \(\epsilon >0\). By using the Lyapunov–Schmidt reduction and the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period \(2\pi /\omega \) and order \(\epsilon \). Such results hold for parameters \((\omega ,\epsilon )\) in a positive measure Cantor set that has asymptotically full measure as the amplitude \(\epsilon \) goes to zero.

About analytical ansatz to the solving procedure for Kelvin–Kirchhoff equations

Abstract In this paper, absolutely new analytical ansatz for solving Kelvin–Kirchhoff equations has been presented. The aforesaid approach was formulated first in Ershkov (2017) for solving Poisson equations; furthermore, a new type of the solving procedure for Euler–Poissonequations (rigid body rotation over the fixed point) is implemented here for solving momentum equation of Kelvin–Kirchhoff. The system of equations of Kelvin–Kirchhoff problem has been explored with regard to the existence of an analytic way of presentation of the analytical solution. A new and elegant ansatz is suggested in this publication whereby, in solving, the momentum equation is reduced to a system of three linear ODEs of 1st order in regard to the three components of the velocity of the spherical particle (dependent on time t). In this premise, a proper elegant partial solution has been obtained due to the invariant dependence between temporary components of the solution. We conclude that the system of Kelvin–Kirchhoff equations has not the analytical presentation of solution (in quadratures) even in case of zero components of fluid force, influencing on the motion of the particle.

Approximation of abstract Cauchy problems for dissipative operators with respect to metric-like functionals

This paper discusses approximation of semigroups of locally Lipschitz operators by discrete semigroups of local quasi-contractions with respect to metric-like functionals. The main approximation result not only extends a product formula for contractive semigroups generated by maximal dissipative operators in real Hilbert spaces but also gives an algorithm for solving a mixed problem for the Kirchhoff equation to which the contractive semigroup theory or Kato's quasilinear theory cannot directly apply.

Multiplicity Results for Kirchhoff Equations with Hardy-Littlewood-Sobolev Critical Nonlinearity

In this paper, we deal with the multiplicity results for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity in $\mathbb {R}^{N}$. By using the second concentration-compactness principle and concentration-compactness principle at infinity to prove that the (PS)c condition holds locally and together with the new version of symmetric mountain pass theorem of Kajikiya, we prove that the problem admits infinitely many solutions under suitable conditions.

On a generalized Timoshenko-Kirchhoff equation with sublinear nonlinearities

In this paper we consider a generalized fourth order nonlinear Kirchhoff equation in a bounded domain in RN,N≥2 under Navier boundary conditions and with sublinear nonlinearity. We employ a change of variable which reduces the problem to a semilinear one. Then variational tools are used in order to prove the existence of a solution.

Infinitely many sign-changing solutions for Kirchhoff type equations

ABSTRACTIn this paper, we study the Kirchhoff type equation where a, b>0 are constants and is a given positive potential. The nonlinearity covers the pure power type with 2<p<4, a case in which few existence results of sign-changing solutions are known. By using the method of invariant sets of descending flow, we obtain a sign-changing solution for the given problem. Furthermore, if f is odd with respect to u, we prove that the problem admits infinitely many sign-changing solutions.

Solutions for fourth-order Kirchhoff type elliptic equations involving concave–convex nonlinearities in [formula omitted

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations Δ2u−M(‖∇u‖22)Δu+V(x)u=f(x,u),x∈RN,where M(t):R→R is the Kirchhoff function, f(x,u)=λk(x,u)+h(x,u), λ≥0, k(x,u) is of sublinear growth and h(x,u) satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for λ=0. For λ>0 small enough, we obtain at least two nontrivial solutions. Furthermore, if f(x,u) is odd in u, we show that above equations possess infinitely many solutions for all λ≥0. Our theorems generalize some known results in the literatures even for λ=0 and our proof is based on the variational methods.

Prestrike modeling in SF6 circuit breakers

Prestrike is a common phenomenon in Circuit Breakers (CB) when the moving contact approaching to the fixed contact during the closing operation. This paper focuses on prestrike modeling in SF6 circuit breakers. A multi-physics method is developed to accurately modeling the prestrikes in SF6 circuit breakers. In this study, a model is developed for a simultaneous solution of the MHD equations and the governing equations of the circuit feeding the plasma in order to consider mutual effects of the circuit and the plasma. Navier-Stoke equations, turbulent flow equations, and the equation of heat transfer in fluids, coupled with Maxwell's equations are physical equations governing the circuit breakers. Kirchhoff's voltage and current laws equations are governing equations of the external electric circuit. The Electric-Circuit-Constrained Magnetohydrodynamic (ECC-MHD) model developed in this paper is able to study the electromagnetic properties of conducting fluids fed by electrical circuits. Motion of moving contact and temperature dependency of SF6 gas properties are considered during the solving of this multi-physics problem. Finite Element Method (FEM) is used to solve the multi-physics equations governing the circuit breaker. Finally a 72 kV single pressure puffer type SF6 circuit breaker is used for numerical studies. To investigate the mutual effects of the feeding circuit parameters and generated plasma in the circuit breaker, numerical studies are performed for different values of resistance and inductance in the arc current path.

ON DECAY AND BLOW UP OF SOLUTIONS FOR A SYSTEM OF KIRCHHOFF TYPE EQUATIONS WITH DAMPING TERMS

In this paper, we investigate system of Kirchhoff type equations with bounded domain. We obtain decay of solutions by using multiplier method. Later, we will proved blow up results for negative inital energy.

Infinitely many solutions for non-local elliptic non-degenerate p-Kirchhoff equations with critical exponent

ABSTRACTIn this paper, we obtain the existence of infinitely many weak solutions with negative energy for non-degenerate elliptic Kirchhoff equation with critical exponent driven by the fractional p-Laplacian. We show this result when is small enough by using a Lusternik–Schnirelmann argument.

The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material

We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $\lambda$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $\lambda_0^*$ for which the local minimum has non-negative energy when $\lambda\ge \lambda_0^*$. Moreover we characterize the extremal parameter $\lambda^*$ for which if $\lambda>\lambda^*$, then the only solution to the Kirchhoff equation is the zero function. In fact, $\lambda^*$ can be characterized in terms of the best constant of Sobolev embeddings. We also study the asymptotic behavior of the solutions when $\lambda\downarrow 0$.

Time-dependent study of boiling heat transfer coefficient in a vertical minichannel

Purpose The purpose of this paper is to determine the time-dependent heat transfer coefficient during FC-72 flow boiling in a 1.7-mm-deep vertical and asymmetrically heated minichannel. Design/methodology/approach The temperature of the minichannel heated wall was recorded continuously with the use of thermocouples. The heat transfer coefficients for the subcooled and saturated boiling regions at the heated wall–fluid contact surface were calculated from the Robin boundary condition. Both the wall and fluid temperatures were obtained from the solution of the inverse nonstationary problems in two adjacent domains: the heated wall and flowing fluid. The FEM with Trefftz-type basis functions was applied to solve the inverse problem. Findings The obtained time-dependent heat transfer coefficient in subcooled boiling achieved rather low values, whereas in saturated boiling, the coefficient was the highest at the channel inlet. The boiling curves were plotted to illustrate the results. Practical implications The results of experiments are the best source of information for the design of minichannel cooling systems used for thermoregulation of components and heat exchangers. High-tech minichannel heat exchangers are applied in various industrial applications as microelectronics devices, gas turbines, internal combustion engines, nuclear reactors, X-ray sources and organic rankine cycle (ORC) modules. Originality/value In the study, the Trefftz functions for the nonstationary Fourier–Kirchhoff equation with the factor describing void fraction were determined and then used to construct the time-dependent basis functions in FEM.

A Nonstandard Higher-Order Variational Model for Speckle Noise Removal and Thin-Structure Detection

In this work, we propose a multiscale approach for a nonstandard higher-order PDE based on the $p(\cdot)$-Kirchhoff energy. First, we consider a topological gradient approach for a semilinear case in order to detect important object of image. Then, we consider a fully nonlinear $p(\cdot)$-Kirchhoff equation with variables exponent functions that are chosen adaptively based on the map furnished by the topological gradient in order to preserve important features of the image. Then, we consider the split Bregman method for the numerical implementation of our proposed model. We compare our model with other classical variational approaches such that the TVL and biharmonic restoration models. Finally, we present some numerical results to illustrate the effectiveness of our approach.

Liouville theorems for Kirchhoff equations in RN

This paper is devoted to the nonexistence of nontrivial weak solutions for the Kirchhoff equation −a+b∫RN|∇u|2 dxΔu=|x|qf(u) in RN. We prove that the equation has no weak solution if a ≥ 0, b &amp;gt; 0, q ≤ −2, and f is a positive, convex, nondecreasing function. If only b ≠ 0 and f is a non-negative function, we establish the nonexistence of weak solutions u satisfying lim infR→+∞R1−N2∫BR|x|qf(u) dx&amp;gt;0. This implies that the equation has no weak solution when N ≤ 2 and f is a positive function. We also show that the equation has no stable weak solution in dimension N&amp;lt;4q+143 if f(u) = eu, a ≥ 0, and b &amp;gt; 0.