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.

Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Shahrouzi ◽  
Firoozeh Kargarfard
Keyword(s):  
Variable Exponent ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Type Equation ◽  
Initial Energy ◽  
Kirchhoff Type ◽  
Positive Initial Energy ◽  
Kirchhoff Type Equation ◽  
Upper Bound Estimate

AbstractThis paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition. Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time. Moreover, we obtain the upper bound estimate of the blow-up time.

Global existence, nonexistence, and decay of solutions for a wave equation of p-Laplacian type with weak and p-Laplacian damping, nonlinear boundary delay and source terms

2021 ◽  
pp. 1-16
Author(s):  
Nouri Boumaza ◽  
Billel Gheraibia
Keyword(s):  
Global Existence ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Source Terms ◽  
Delay Term ◽  
Laplacian Equation ◽  
Decay Of Solutions

In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2. To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term.

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.

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