Higher-order Kirchhoff-type Equations Research Articles

Existence, Decay, and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Delay Term

This article deals with the study of the higher-order Kirchhoff-type equation with delay term in a bounded domain with initial boundary conditions, where firstly, we prove the global existence result of the solution. Then, we discuss the decay of solutions by using Nakao’s technique and denote polynomially and exponentially. Furthermore, the blow-up result is established for negative initial energy under appropriate conditions.

GLOBAL NONEXISTENCE OF THE HIGHER ORDER KIRCHHOFF TYPE SYSTEM WITH LOGARITHMIC NONLINEARITIES

This paper deals with the the system a class of nonlinear higher-order Kirchhoff-type equations with logarithmic nonlinearities. Under the appropriate assumptions, the theorem of global nonexistence is established at positive initial energy levels.

Inertial Manifolds for Generalized Higher-Order Kirchhoff Type Equations

The existence of inertial manifolds for higher-order Kirchhoff type equations with strong damping terms is studied. The Hadamard graph norm conversion method is used for obtaining the existence of inertial manifolds for this kind of equations under certain spectral intervals.

A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation

In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.

The Global and Exponential Attractors for the Higher-order Kirchhoff-type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: ${u_{tt}} + {( - \Delta )^m}{u_t} + {\left( {\alpha + \beta\left\| {{\nabla ^m}u} \right\|^2} \right)^{q}}{( - \Delta )^m}u + g(u) = f(x)$. At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in $H_0^m(\Omega ) \times {L^2}(\Omega )$ according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.

Random Attractor of the Stochastic Strongly Damped for the Higher-Order Nonlinear Kirchhoff-Type Equation

In this paper, we consider the stochastic higher-order Kirchhoff-type equation with nonlinear strongly dissipation and white noise. We first deal with random term by using Ornstein-Uhlenbeck process and establish the wellness of the solution, then the existence of global random attractor are proved.

The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*

We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:2( ) ( )m mt t tu    u    D u  ( ) ( ) ( )m  u  g u  f x . Under of the properassume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkinmethod, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of theglobal attractor.

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

Existence of solutions for a higher order Kirchhoff type problem with exponential critical growth

A higher order Kirchhoff type equation ∫R2m(∣∇mu∣2+∑γ=0m−1aγ(x)∣∇γu∣2)dx((−Δ)mu+∑γ=0m−1(−1)γ∇γ⋅(aγ(x)∇γu))=f(x,u)∣x∣β+ϵh(x)in R2m is considered in this paper. We assume that the nonlinearity of the equation has exponential critical growth and prove that, for a positive ϵ which is small enough, there are two distinct nontrivial solutions to the equation. When ϵ=0, we also prove that the equation has a nontrivial mountain-pass type solution.

Global existence and asymptotic behavior of solutions for a system of higher-order Kirchhoff-type equations

This paper deals with the global existence and energy decay of solutions for some coupled system of higher-order Kirchhoff-type equations with nonlinear dissipa- tive and source terms in a bounded domain. We prove the existence of global solutions for this problem by constructing a stable set in H m1 0 (W) H m2 0 (W) and give the decay estimate of global solutions by applying a lemma of V. Komornik. with initial data u(x, 0) = u0(x), ut(x, 0) = u1(x), x2 W, (1.3) v(x, 0) = v0(x), vt(x, 0) = v1(x), x2 W,

Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations

In this work, we consider the initial boundary value problem for the Kirchhoff-type equations with damping and source terms <img src=image/13402608_001.png> in a bounded domain. We prove the blow up of the solution with positive initial energy by using the technique of [26] with a modification in the energy functional due to the different nature of problems. This improves earlier results in the literature [3, 9, 13, 21].

Global existence and energy decay estimate of solutions for a higher-order Kirchhoff type equation with damping and source term

This paper deals with the initial–boundary value problem for a class of nonlinear higher-order Kirchhoff-type equation with dissipative term utt+‖A12u‖2pAu+a|ut|q−2ut=b|u|r−2u,x∈Ω,t>0 in a bounded domain, where A=(−Δ)m, m>1 is a positive integer, a,b,p>0 and q,r>2 are constants. We obtain the global existence of solutions by constructing a stable set in H0m(Ω) and show the energy decay estimate by applying a lemma of V. Komornik.

Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation

Abstract In this paper, we consider the nonlinear Kirchhoff-type equation $$ u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u $$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.

Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation

This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipation u t t + ( ∫ Ω ׀ D m u ׀ 2 d x ) q ( − Δ ) m u + u t ׀ u t ׀ r = ׀ u ׀ p u , x ∈ Ω , t > 0 , in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while if p > max{r, 2q}, then for any initial data with negative initial energy, the solution blowsup at finite time in L p +2 norm.