Kirchhoff Equations Research Articles

Solving procedure for the Kelvin–Kirchhoff equations in case of non-stationary rotations of slim disc

We have presented in this communication a new solving procedure for Kelvin–Kirchhoff equations, considering the dynamics of the falling or ascending the rigid disc in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the falling or ascending body, I1 = I2. Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of disc’s rotation has been explored in regard to the existence of an analytic way of presentation for the approximated solution (where we consider the case of laminar flow at slow regime of deceleration of the symmetric disc’s rotation). This scenario is associated with consideration of that the Stokes boundary layer phenomenon on the boundaries of the body has also been assumed at formulation of basic Kelvin-Kirchhoff equations. It allows us to use the results of Jeffery’s fundamental work, in which analytical solution for rotations of spheroid’s particle in Euler angles was obtained. The results of calculations for the components of angular velocity {Ωi} should then be used for solving momentum equation of Kelvin–Kirchhoff system.

Existence of positive solutions for a critical fractional Kirchhoff equation with potential vanishing at infinity

AbstractIn this paper, we consider the following fractional Kirchhoff equation with critical nonlinearity where , is the fractional Laplace operator with , V vanishes at infinity and . Under appropriate assumptions on f, we prove the existence of positive solution by using the variational method.

High perturbations of critical fractional Kirchhoff equations with logarithmic nonlinearity

This paper deals with the study of combined effects of logarithmic and critical nonlinearities for the following class of fractional p-Kirchhoff equations: M([u]s,pp)(−Δ)psu=λ|u|q−2uln|u|2+|u|ps∗−2uinΩ,u=0inRN∖Ω,where Ω⊂RN is a bounded domain with Lipschitz boundary, N>sp with s∈(0,1), p≥2, ps∗=Np∕(N−ps) is the fractional critical Sobolev exponent, and λ is a positive parameter. The main result establishes the existence of nontrivial solutions in the case of high perturbations of the logarithmic nonlinearity (large values of λ). The features of this paper are the following: (i) the presence of a logarithmic nonlinearity; (ii) the lack of compactness due to the critical term; (iii) our analysis includes the degenerate case, which corresponds to the Kirchhoff term M vanishing at zero.

Stable weak solutions to weighted Kirchhoff equations of Lane\u2013Emden type

This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations: −M(∫RNξ(z)|∇Gu|2dz)divG(ξ(z)∇Gu)=η(z)|u|p−1u,z=(x,y)∈RN=RN1×RN2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& -M \\biggl( \\int_{\\mathbb{R}^{N}}\\xi(z) \\vert \\nabla_{G}u \\vert ^{2}\\,dz \\biggr){ \\operatorname{div}}_{G} \\bigl(\\xi(z) \\nabla_{G}u \\bigr) \\\\& \\quad=\\eta(z) \\vert u \\vert ^{p-1}u,\\quad z=(x,y) \\in \\mathbb{R}^{N}=\\mathbb{R}^{N_{1}}\\times\\mathbb{R}^{N_{2}}, \\end{aligned}$$ \\end{document} where M(t)=a+bt^{k}, tgeq0, with a,b,kgeq0, a+b>0, k=0 if and only if b=0. Let N=N_{1}+N_{2}geq2, p>1+2k and xi(z),eta(z)in L^{1}_{mathrm{loc}}(mathbb{R}^{N})setminus{ 0} be nonnegative functions such that xi(z)leq C|z|_{G}^{theta} and eta(z)geq C'|z|_{G}^{d} for large |z|_{G} with d>theta-2. Here alphageq0 and |z|_{G}=(|x|^{2(1+alpha)}+|y|^{2})^{frac{1}{2(1+alpha)}}. operatorname{div}_{G} (resp., nabla_{G}) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on k, θ, d, and N_{alpha}=N_{1}+(1+alpha)N_{2}, the nonexistence of stable weak solutions to the problem is obtained. A distinguished feature of this paper is that the Kirchhoff function M could be zero, which implies that the above problem is degenerate.

A New Solving Procedure for the Kelvin–Kirchhoff Equations in Case of a Falling Rotating Torus

We present a new solving procedure in this paper for Kelvin–Kirchhoff equations, considering the dynamics of a falling rigid rotating torus in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the rotating body, [Formula: see text]. The fundamental law of angular momentum conservation is used for the aforementioned solving procedure. The system of Euler equations for the dynamics of torus rotation is explored for an analytic way of presentation of the approximated solution (where we consider the case of laminar flow at slow regime of torus rotation). The second finding is that the Stokes boundary layer phenomenon on the boundaries of the torus also assumed additionally at the formulation of basic Kelvin–Kirchhoff equations (for which the analytical expressions for the components of fluid’s torque vector [Formula: see text] were obtained earlier). The results for calculating the components of angular velocity [Formula: see text] should then be used for full solving the momentum equation of Kelvin–Kirchhoff system. The trajectories of motion can be divided into, preferably, three classes: zigzagging, helical spiral motion, and the chaotic regime of oscillations.

The Family of Global Attractors for a Generalized Kirchhoff Equations

This paper deals with the existence and uniqueness of solutions of generalized Kirchhoff equations and the family of global attractors for the equation and its dimension estimation. First, the stress term of Kirchhoff equation is properly assumed. When certain conditions are met between the order m and the degree p of Banach space , the existence and uniqueness of the solution of equation are obtained by a prior estimation and Galerkin’s method; Then, the bounded absorption set is obtained by prior estimation, and it is proved that the solution semigroup generated by the equation has a family of global attractors in phase space by using Rellich-Kondrachov compact embedding theorem. Further, the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Fréchet differentiable on ; Finally, the upper bound of Hausdorff dimension and Fractal dimension of is estimated, and the Hausdorff dimension and Fractal dimension are finite.

Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti–Rabinowitz conditions

The paper focuses on the modified Kirchhoff equation \begin{align*} -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\Delta u-u\Delta (u^2)+V(x)u=\lambda f(u), \quad x\in \mathbb{R}^N, \end{align*} where $a,b>0$, $V(x)\in C(\mathbb{R}^N,\mathbb{R})$ and $\lambda<1$ is a positive parameter. We just assume that the nonlinearity $f(t)$ is continuous and superlinear in a neighborhood of $t = 0$ and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f(t) may be supercritical.

A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations

<p style='text-indent:20px;'>In this paper, we proved a fractional Kirchhoff version of Hopf lemma for anti-symmetry functions and applied it to prove the symmetry and monotonicity of solutions for fractional Kirchhoff equations in the whole space by method of moving planes. We also obtain radially symmetry and monotonicity of solutions for fractional Kirchhoff equations in the unit ball. As far as we know, this is the first time to apply direct method of moving planes to fractional Kirchhoff problems.

A Family of the Global Attractor for Higher Order Nonlinear Kirchhoff Equation

In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations with strong damping terms. We will properly assume the stress term M(s) and nonlinear term g(ut). First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.

Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth

<abstract> In this paper, we consider a Neumann problem of Kirchhoff type equation <p class="disp_formula">$ \begin{equation*} \begin{cases} -\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u = Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &amp;\rm \mathrm{in}\ \ \Omega, \\ \frac{\partial u}{\partial v} = 0, &amp;\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $ where $ \Omega $ $ \subset $ $ \mathbb{R}^3 $ is a bounded domain with a smooth boundary, $ a, b &gt; 0 $, $ 1 &lt; q &lt; 2 $, $ \lambda &gt; 0 $ is a real parameter, $ Q(x) $ and $ P(x) $ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions. </abstract>

Nonexistence of global solutions for a class of viscoelastic wave equations

&lt;p style='text-indent:20px;'&gt;We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.&lt;/p&gt;

含Hardy-Sobolev临界指数的分数阶Kirchhoff型方程多重解的存在性

含Hardy-Sobolev临界指数的分数阶Kirchhoff型方程多重解的存在性

A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.

Growth of solutions with $ L^{2(p+2)} $-norm for a coupled nonlinear viscoelastic Kirchhoff equation with degenerate damping terms

&lt;abstract&gt;&lt;p&gt;In this work, we consider a coupled nonlinear viscoelastic Kirchhoff equations with degenerate damping, dispersion and source terms. Under suitable hypothesis, we will prove that when the initial data are large enough (in the energy point of view), the energy grows exponentially and thus so the $ L^{2(p+2)} $-norm.&lt;/p&gt;&lt;/abstract&gt;

POSITIVE AND SIGN-CHANGING SOLUTIONS FOR THE FRACTIONAL KIRCHHOFF EQUATION WITH CRITICAL GROWTH

<abstract> We are interested in the existence of positive and sign-changing solutions for a fractional Kirchhoff equation. Under some mild conditions on the potentials $ V $ and $ h $, using variational methods, we prove the existence of positive ground state solutions and least energy sign-changing solutions. </abstract>

The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping

In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor Ak in space Ek. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on Ek via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor Ak.

Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; $ \begin{cases} -(a+ b\|u\|_{K}^{2})\mathcal{L}_K u+V(x)u = |u|^{2^{\ast}_{\alpha}-2}u+ k f(x,u),&amp;amp;x\in\Omega,\\ u = 0,&amp;amp;x\in\mathbb{R}^{3}\backslash\Omega, \end{cases} $ &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ k $&lt;/tex-math&gt;&lt;/inline-formula&gt; is a positive parameter, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;$ \mathcal{L}_K $&lt;/tex-math&gt;&lt;/inline-formula&gt; stands for a nonlocal fractional operator which is defined with the kernel function &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;$ K $&lt;/tex-math&gt;&lt;/inline-formula&gt;. By using the nodal Nehari manifold method, we obtain a least energy nodal solution &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;$ u $&lt;/tex-math&gt;&lt;/inline-formula&gt; and a ground state solution &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;$ v $&lt;/tex-math&gt;&lt;/inline-formula&gt; to this problem when &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;$ k\gg1 $&lt;/tex-math&gt;&lt;/inline-formula&gt;, where the nonlinear function &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;$ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R} $&lt;/tex-math&gt;&lt;/inline-formula&gt; is a Carathéodory function.&lt;/p&gt;

Blow up results for a viscoelastic Kirchhoff-type equation with logarithmic nonlinearity and strong damping

A Kirchhoff equation type with memory term competing with a logarithmic source is considered. By using potential well theory, we obtained the global existence of solution for the initial data in a stability set created from Nehari Manifold and prove blow up results for initial data in the instability set.

Existence of solutions for modified Kirchhoff-type equation without the Ambrosetti-Rabinowitz condition

This paper is devoted to studying a class of modified Kirchhoff-type equations $ \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\Big)\Delta u+V(x)u-u\Delta(u^2) = f(x,u), \quad \mbox{in} \mathbb{R}^3, \end{equation*} $ where $ a > 0, b\geq 0 $ are two constants and $ V:{\mathbb{R}}^{3}\rightarrow {\mathbb{R}} $ is a potential function. The existence of non-trivial solution to the above problem is obtained by the perturbation methods. Moreover, when $ u > 0 $ and $ f(x, u) = f(u) $, under suitable hypotheses on $ V(x) $ and $ f(u) $, we obtain the existence of a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. The character of this work is that for $ f(u)\sim|u|^{p-2}u $ we prove the existence of a positive ground state solution in the case where $ p\in(2, 3] $, which has few results for the modified Kirchhoff equation. Hence our results improve and extend the existence results in the related literatures.

Concentration phenomena for magnetic Kirchhoff equations with critical growth

&lt;p style='text-indent:20px;'&gt;In this paper, we study the following nonlinear magnetic Kirchhoff equation with critical growth&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{align*} \left\{ \begin{aligned} &amp;amp;-\Big(a\epsilon^{2}+b\epsilon\, [u]_{A/\epsilon}^{2}\Big)\Delta_{A/\epsilon} u+V(x)u = f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3, \\ &amp;amp;u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{align*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \epsilon&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a parameter, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ a, b&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are constants, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ V:\mathbb{R}^{3}\rightarrow \mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ A: \mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are continuous potentials, and &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ f: \mathbb{R}\rightarrow \mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a nonlinear term with subcritical growth. Under a local assumption on the potential &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ V $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, combining variational methods, penalization techniques and the Ljusternik-Schnirelmann theory, we establish multiplicity and concentration properties of solutions to the above problem for &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \varepsilon $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; small. A feature of this paper is that the function &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is assumed to be only continuous, which allows to consider larger classes of nonlinearities in the reaction.&lt;/p&gt;