Kirchhoff Equations Research Articles

The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Abstract In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of D k {{\mathcal{D}}}_{k} -absorbing sets of the stochastic dynamical system Φ k {\Phi }_{k} is obtained, and the asymptotic compactness of Φ k {\Phi }_{k} is proved by the decomposition method. Finally, the D k {{\mathcal{D}}}_{k} -stochastic attractor family of the stochastic dynamical system Φ k {\Phi }_{k} in V m + k ( Ω ) × V k ( Ω ) {V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.

On the singularly perturbation fractional Kirchhoff equations: Critical case

Abstract This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a , b > 0 a,b\gt 0 are given constants, ε \varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} with 0 < s < 1 0\lt s\lt 1 and N ≥ 4 s N\ge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 \varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N > 4 s N\gt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε \varepsilon small.

Existence and multiple solutions for the critical fractional p‐Kirchhoff type problems involving sign‐changing weight functions

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation involving the fractional p‐Laplace operator . More precisely, we consider where is an open bounded domain with Lipschitz boundary with m > 1, a > 0, b > 0, dimension is the fractional critical Sobolev exponent, and the parameters λ > 0, 0 < s < 1 < q < p < ∞. Applying the Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity of nonnegative solutions.

New multiplicity results for a class of nonlocal equation with steep potential well

In this paper, we consider a class of Kirchhoff type equation with Hartree nonlinearity: where parameters, , is the Riesz potential, in , and . Under some suitable assumptions for potential and spatial dimensions N, by using the filtration of Nehari manifold and Ekeland variational principle, we prove that above nonlocal equation admits at least two positive solutions. The effect of the steep potential well and spatial dimensions on the number of positive solutions are completely showed.

Existence and concentration of ground state solutions for Kirchhoff type equations with general nonlinearities

In this paper, we study the following singularly perturbed Kirchhoff type equation in subcritical case where ε > 0 is a parameter, and . Under some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of V in the semiclassical limit by applying some new techniques and subtle analyses. Moreover, we use Moser iteration to conquer the difficulty caused by the lack of Sobolev inequality.

Existence and decay of solutions for a parabolic type Kirchhoff equation with logarithmic nonlinearity

This paper deals with the parabolic type Kirchhoff equation with logarithmic nonlinearity in a bounded domain. Firstly, we prove the local and global existence of solutions. Later, we investigate the decay of solutions. This improves and extend some previous studies.

RESEARCH OF AUTOGENERATOR PARAMETRIC TEMPERATURE SENSORS

Autogenerator parametric temperature sensors based on transistor microelectronic structures with negative differential resistance with primary transducers such as thermistors and thermodiodes are proposed, and the primary thermosensitive elements are active elements of sensor autogenerator circuits, which simplifies their design. Based on the consideration of physical processes in primary heat-sensitive transducers and autogenerators of sensors, mathematical models of temperature sensors have been developed, on the basis of which parametric dependences of transformation and sensitivity functions are obtained. It is shown that the main contribution to the conversion function is made by temperature. This causes a change in the equivalent capacitance and negative differential resistance of the sensor autogenerators, which in turn changes the output frequency of the temperature sensors. The sensitivity of the temperature sensors varies from 1.2 kHz/0C to 2.35 kHz/0C when the temperature changes from 0 0C to 125 0C. The obtained parametric dependences of temperature sensor conversion functions show the possibility to obtain basic sensor characteristics much easier and clearly demonstrate the influence of each parameter of primary converters and autogenerator elements on sensor output frequency in comparison with calculations of conversion functions from equivalent device circuits based on Kirchhoff equations solution. Frequency output temperature sensors do not require analog-to-digital converters and amplifiers for further processing of information signals, which reduces the cost of information and measuring equipment, in addition, it is possible to transmit information over distances when operating sensors at ultrahigh frequencies.

Non-degeneracy of Positive Solutions for Fractional Kirchhoff Problems: High Dimensional Cases

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem (a+b∫RN|(-Δ)s2u|2dx)(-Δ)su+u=up,inRN,\\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}$$\\begin{aligned} \\Big (a+b{\\int _{\\mathbb {R}^{N}}}|(-\\Delta )^{\\frac{s}{2}}u|^2\\mathrm{{d}}x\\Big )(-\\Delta )^su+u=u^p,\\quad \\text {in}\\ \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}where a,b>0, 0<s<1, 1<p<frac{N+2s}{N-2s} and (-Delta )^s is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions N>4s, i.e., we show that there exist two non-degenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low dimensional fractional Kirchhoff equation. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we can derive the existence of solutions to the singularly perturbation problems.

Mathematical behavior of solutions of the Kirchhoff type equation with logarithmic nonlinearity

We consider the existence and decay estimates of solutions for Kirchhoff type equation with damping logarithmic source term. We proved global existence of solutions under suitable conditons by potential well method and the decay estimates result of the solutions for subcritical energy level.

Existence of positive solutions for fractional Kirchhoff equation

Existence of positive solutions for fractional Kirchhoff equation

Normalized ground states for Kirchhoff equations in R3 with a critical nonlinearity

This paper is concerned with the existence of ground states for a class of Kirchhoff type equations in R3 with combined power nonlinearities −a+b∫R3|∇u(x)|2Δu=λu+|u|p−2u+u5 with the restriction ∫R3u2=c2 in the Sobolev critical case 2* = 6 and proves that the problem has a ground state solution (uc,λc)∈S(c)×R for any c &amp;gt; 0, a, b &amp;gt; 0, and 143≤p&amp;lt;6, where S(c)=u∈H1(R3):∫R3u2=c2.

Linearly stable KAM tori for higher dimensional Kirchhoff equations

We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for higher dimensional Kirchhoff equations with periodic boundary conditions. The proof is based on an improved Kuksin lemma and the special form of the nonlinear term.

Nodal solutions for Kirchhoff equations with Choquard nonlinearity

Nodal solutions for Kirchhoff equations with Choquard nonlinearity

Orbital Stability of Nonlinear Schrödinger–Kirchhoff Equations

Orbital Stability of Nonlinear Schrödinger–Kirchhoff Equations

Lifespan of solutions to a hyperbolic type Kirchhoff equation with arbitrarily high initial energy

In this paper, an initial boundary value problem for a hyperbolic type Kirchhoff equation with a strong dissipation and a general nonlinearity is considered. First, local existence and uniqueness of weak solutions are obtained with the help of Banach fixed point theorem. Then, by constructing an auxiliary functional and adopting the concavity argument, we give a new finite time blow-up criterion for this problem, which in particular implies that the problem admits blow-up solutions with arbitrarily high initial energy. Meanwhile, a bound for the blow-up time is derived from above. Further, we obtain a lower bound for the blow-up time when blow-up occurs. From methods to results, we partially extend the ones obtained in earlier literature.

Well-posedness and stability for Kirchhoff equation with non-porous acoustic boundary conditions

In this paper we prove the well-posedness to the wave equation of Kirchhoff type. Under a portion of the boundary, we consider the acoustic boundary conditions. We also prove the exponential stability of the energy associated to the problem. Our result generalize the previous literature where only the Carrier model was considered (when the nonlinearity involves the L2(Ω) norm) or the Kirchhoff model with porous acoustic boundary conditions. The main tool is the Faedo-Galerkin method. Due to the presence of acoustic boundary conditions, we can not use special basis and new estimates are necessary. To prove the stability we use integral estimates.

Longer Lifespan for Many Solutions of the Kirchhoff Equation

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 9 July 2020Accepted: 15 September 2021Published online: 06 January 2022KeywordsKirchhoff equation, quasilinear wave equations, Hamiltonian PDEs, quasilinear normal forms, Cauchy problems, effective equations, long time dynamics, resonancesAMS Subject Headings35L72, 35Q74, 35A01, 37K45, 70K45, 70K65Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Normalized solutions for nonlinear Kirchhoff type equations in high dimensions

&lt;abstract&gt;&lt;p&gt;We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions $ \mathbb{R}^N(N\geqslant4) $. In particular, in dimension $ N = 4 $, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $ 2+\frac{8}{N} $ is equal to the energy critical exponent $ \frac{2N}{N-2} $, which leads to the fact that the equation no longer has a variational structure in dimensions $ N\geqslant 4 $ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.&lt;/p&gt;&lt;/abstract&gt;

Stationary Kirchhoff equations and systems with reaction terms

&lt;abstract&gt;&lt;p&gt;In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.&lt;/p&gt;&lt;/abstract&gt;

The existence and nonexistence of positive solutions for a singular Kirchhoff equation with convection term

&lt;abstract&gt;&lt;p&gt;This paper considers a singular Kirchhoff equation with convection and a parameter. By defining new sub-supersolutions, we prove a new sub-supersolution theorem. Combining method of sub-supersolution with the comparison principle, for Kirchhoff equation with convection, we get the conclusion about positive solutions when nonlinear term is singular and sign-changing.&lt;/p&gt;&lt;/abstract&gt;