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

2016 ◽  
Vol 05 (04) ◽  
pp. 203-217 ◽  
Author(s):  
Yuting Sun ◽  
Yunlong Gao ◽  
Guoguang Lin
Keyword(s):  
Type Equation ◽  
Global Attractors ◽  
Higher Order ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Strongly Damped

scholarly journals Strong Solutions and Global Attractors for Kirchhoff Type Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Xiangping Chen
Keyword(s):  
Phase Space ◽  
Type Equation ◽  
Strong Solutions ◽  
Global Attractors ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Long Time ◽  
Linear Damping

We study the long-time behavior of the Kirchhoff type equation with linear damping. We prove the existence of strong solution and the semigroup associated with the solution possesses a global attractor in the higher phase space.

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

2017 ◽  
Vol 9 (4) ◽  
pp. 145 ◽  
Author(s):  
Guoguang Lin ◽  
Yunlong Gao
Keyword(s):  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Attractors ◽  
Higher Order ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Kirchhoff Type ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value

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.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hazal Yüksekkaya ◽  
Erhan Pişkin ◽  
Salah Mahmoud Boulaaras ◽  
Bahri Belkacem Cherif
Keyword(s):  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Energy ◽  
Higher Order ◽  
Delay Term ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Decay Of Solutions ◽  
Equation With Delay

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.

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