Kirchhoff Equations Research Articles

Multiplicity of solutions for Kirchhoff type equations with critical growth in ℝ N

In this paper, we consider the following Kirchhoff type problem with critical growth: { − ( a + b ∫ R N | ∇u | 2 d x ) Δu = λk ( x ) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 , a ≥ 0 , b>0, 1<q<2, λ , μ > 0 , 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and k ( x ) ∈ L 2 ∗ / ( 2 ∗ − q ) ( R N ) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions.

Concentration behavior and local uniqueness of normalized solutions for Kirchhoff type equation

Concentration behavior and local uniqueness of normalized solutions for Kirchhoff type equation

Asymptotics for a parabolic problem of Kirchhoff type with singular critical exponential nonlinearity

AbstractThe main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as goes to infinity for the following class of parabolic Kirchhoff equations: where is a bounded domain with a Lipschitz boundary, , , , , , and is the fractional ‐Laplacian operator, .

Nontrivial solutions for indefinite Schrödinger–Poisson systems and Kirchhoff equations

In this paper we consider 4-superlinear Schrödinger–Poisson systems. The potential V here is indefinite so that the Schrödinger operator −Δ+V possesses a finite-dimensional negative space. By Morse theory, we obtain the existence of nontrivial solutions for this problem. Similar result for Kirchhoff equations on R3 is also presented.

Global solutions to the Kirchhoff equation with spectral gap data in the energy space

We prove that the classical hyperbolic Kirchhoff equation admits global-in-time solutions for some classes of initial data in the energy space. We also show that there are enough such solutions so that every initial datum in the energy space is the sum of two initial data for which a global-in-time solution exists. The proof relies on the notion of spectral gap data, namely initial data whose components vanish for large intervals of frequencies. We do not pass through the linearized equation, because it is not well-posed at this low level of regularity.

Blow up, growth, and decay of solutions for a class of coupled nonlinear viscoelastic Kirchhoff equations with distributed delay and variable exponents

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay, and variable exponents. Under a suitable hypothesis the blow-up and growth of solutions are proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case of absence of the source term f1=f2=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{1}=f_{2}=0$\\end{document}.

Impedance matching design of capacitively coupled plasma with fluid and external circuit coupled model

AbstractThis paper establishes a fully self‐consistent coupled model of fluid and external circuits. The Kirchhoff equation, the charge conservation equation, and Poisson equation are coupled via boundary conditions and integrated into the fluid model for iterative parameter solution. On the basis of this model, we investigate the influence of impedance matching on single‐frequency capacitively coupled plasma characteristics under different parameters and topological structures. The findings suggest that after several iterations the matching parameters converge. Using different initial circuit parameters, the adjustable capacitance and inductance are eventually adjusted to approximately equal values, resulting in the same optimal matching state, whereas diverse discharge parameters led to different outcomes. Under fixed parameters for two topologies, the power absorption efficiency increases, and the reflection coefficient approaches zero, and the best matching is found. This model can be extended to different fluid programs to investigate the impact of complex external circuits with impedance matching network on plasma discharge while simultaneously seeking best impedance matching.

Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case

We study the following nonlinear mass supercritical Kirchhoff equation: −(a+b∫RN|∇u|2)△u+μu=f(u)in RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ - \\biggl(a+b \\int _{\\mathbb{R}^{N}} \\vert \ abla u \\vert ^{2} \\biggr) \ riangle u+ \\mu u=f(u) \\quad \ ext{in } {\\mathbb{R}^{N}}, $$\\end{document} where a,b,m>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a ,b,m>0$\\end{document} are prescribed, and the normalized constrain ∫RN|u|2dx=m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\int _{\\mathbb{R}^{N}}|u|^{2}\\,dx =m$\\end{document} is satisfied in the case 1≤N≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1\\leq N\\leq 3$\\end{document}. The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when 1≤N≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1\\leq N\\leq 3$\\end{document} and obtain infinitely many radial solutions when 2≤N≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2\\leq N\\leq 3$\\end{document} by constructing a particular bounded Palais–Smale sequence.

Modelling Crystalline α-Mg Phase Growth in an Amorphous Alloy Mg72Zn28

A model of α-Mg grain growth in an amorphous Mg72Zn28 alloy matrix was developed together with numerical software. Its application enables tracking the growth process of the α-Mg phase in an amorphous alloy. The model was based on the diffusion-driven growth of α-Mg in an amorphous alloy under appropriate boundary conditions at an isothermal annealing temperature and taking into account the presence of a grain with an initial radius of 1 nm. The numerical model was based on a mathematical model of heat flow, described by the Fourier–Kirchhoff equation, and diffusion, described by Fick’s second law. The initial boundary conditions necessary to simulate grain growth in the amorphous phase were established. The results of the numerical simulation indicate grain growth with increasing isothermal annealing temperature and increasing isothermal annealing time.

Normalized solutions and least action solutions for Kirchhoff equation with saturable nonlinearity

In this paper, we are concerned with the existence of solutions for the following Kirchhoff type equation with saturable nonlinearity: −1+∫RN|∇u|2dxΔu+λu=μg(x)+u21+g(x)+u2uinRN,u∈Hr1(RN),where N≥3, λ∈R and μ>0 is a parameter. By imposing some suitable conditions on the function g(x), we obtain the existence of normalized solutions and least action solutions for the above problem. Some related results are greatly improved and extended.

A planar Kirchhoff equation with exponential growth and double nonlocal term

We investigate the existence of a positive solution for an elliptic problem of the Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation combined a nonlocal term with an exponential growth governed by the Trudinger-Moser inequality.

Existence and Multiplicity of Normalized Solutions with Positive Energy for the Kirchhoff Equation

Existence and Multiplicity of Normalized Solutions with Positive Energy for the Kirchhoff Equation

Normalized solutions to the Kirchhoff Equation with triple critical exponents in [formula omitted]

In this paper, we investigate the normalized solutions for the nonlinear critical Kirchhoff equations with combined nonlinearities: −a+b∫RN|∇u|2dxΔu=λu+μ|u|u+|u|2u,∫RNu2dx=c2,x∈RN,where N=4, λ, μ∈R and a, b, c>0 are constants. In R4, some interesting phenomena occur, which are, the L2-critical exponent for Δu is 2+4N=3, while the L2-critical exponent for (∫R4|∇u|2dx)Δu is equal to the Sobolev critical exponent, i.e., 2+8N=2NN−2=4. This paper investigates the case that the nonlinearity with triple critical term and proves the existence of a positive mountain-pass type normalized solution through variational methods and energy estimations.

A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

A class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs are studied. Under suitable assumptions on the exponents of nonlinear source terms and the memory kernel, the existence of global solutions in an appropriate functional space is established by a combination of the theory of potential wells and the Galerkin approximations. Furthermore, the asymptotic behavior of global solutions is obtained by a combination of the theory of potential wells and the perturbed energy method.

Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory term

This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor.

Stability and critical dimension for Kirchhoff systems in closed manifolds

Abstract The Kirchhoff equation was proposed in 1883 by Kirchhoff [Vorlesungen über Mechanik, Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and PDE’s, G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as ∂ 2 u ∂ t 2 + a + b ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) , $\frac{{\partial }^{2}u}{\partial {t}^{2}}+\left(a+b{\int }_{{\Omega}}\vert \nabla u{\vert }^{2}\mathrm{d}x\right){\Delta}u=f\left(x,u\right),$ where Δ = − ∑ ∂ 2 ∂ x i 2 ${\Delta}=-\sum \frac{{\partial }^{2}}{\partial {x}_{i}^{2}}$ is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck.

Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations

We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground state solutions exist for the fractional Kirchhoff equation with p=4.

A Note on a Mixed Pseudo-Parabolic Kirchhoff Equation with Logarithmic Damping

A Note on a Mixed Pseudo-Parabolic Kirchhoff Equation with Logarithmic Damping

Effective chaos for the Kirchhoff equation on tori

Effective chaos for the Kirchhoff equation on tori

A Conservative Finite Element Scheme for the Kirchhoff Equation

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.