scholarly journals Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Gang Li ◽  
Linghui Hong ◽  
Wenjun Liu
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
High Energy ◽  
Dirichlet Boundary Conditions ◽  
Kirchhoff Type ◽  
Nonexistence Result ◽  
Global Nonexistence ◽  
Viscoelastic Wave ◽  
Nonexistence Of Solutions

We consider viscoelastic wave equations of the Kirchhoff typeutt-M(∥∇u∥22)Δu+∫0tg(t-s)Δu(s)ds+ut=|u|p-1uwith Dirichlet boundary conditions, where∥⋅∥pdenotes the norm in the Lebesgue spaceLp. Under some suitable assumptions ongand the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense thatlim⁡t→T*-(∥u(t)∥22+∫0t∥u(s)∥22ds)=∞for some0<T*<+∞.

scholarly journals Blow-up and energy decay for a class of wave equations with nonlocal Kirchhoff-type diffusion and weak damping

2021 ◽  
Author(s):  
Menglan Liao ◽  
Zhong Tan
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Initial Data ◽  
Dirichlet Boundary Condition ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Kirchhoff Type ◽  
Weak Damping

The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator $\mathcal{L}_K$: \[u_{tt}+[u]_s^{2(\theta-1)}\mathcal{L}_Ku+a|u_t|^{m-1}u_t=b|u|^{p-1}u\] with homogeneous Dirichlet boundary condition and initial data, where $[u]^2_s$ is the Gagliardo seminorm, $a\geq 0,~b>0,~0

A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations

2017 ◽  
Vol 34 (4) ◽  
pp. 1257-1276 ◽  
Author(s):  
Ali Saleh Alshomrani ◽  
Sapna Pandit ◽  
Abdullah K. Alzahrani ◽  
Metib Said Alghamdi ◽  
Ram Jiwari
Keyword(s):  
Numerical Algorithm ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Computational Modelling ◽  
Hyperbolic Type ◽  
Dirichlet Boundary Conditions ◽  
Spline Functions ◽  
Content Type ◽  
B Spline ◽  
Type Wave

Purpose The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc. Design/methodology/approach Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed. Findings A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations. Originality/value To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

scholarly journals Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems withp-Laplacian with Nonlocal Sources

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
Zhoujin Cui ◽  
Zuodong Yang
Keyword(s):  
Global Existence ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Local Theory ◽  
Smooth Bounded Domain ◽  
Nonlocal Sources ◽  
Nonexistence Result ◽  
Evolution Systems

This paper deals withp-Laplacian systemsut−div(|∇u|p−2∇u)=∫Ωvα(x,t)dx,x∈Ω,t>0,vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx,x∈Ω, t>0,with null Dirichlet boundary conditions in a smooth bounded domainΩ⊂ℝN, wherep,q≥2,α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈ℝN:|x|<R} (R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

scholarly journals Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yuping Cao ◽  
Chuanzhi Bai
Keyword(s):  
Dirichlet Boundary ◽  
Main Tool ◽  
Dirichlet Boundary Conditions ◽  
Type Problem ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Integrodifferential Operators ◽  
Homogeneous Dirichlet Boundary Conditions

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.

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