scholarly journals A blow-up result in a system of nonlinear viscoelastic wave equations with arbitrary positive initial energy

2013 ◽  
Vol 24 (3) ◽  
pp. 602-612 ◽  
Author(s):  
Mohammad Kafini ◽  
Salim A. Messaoudi
Keyword(s):  
Wave Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Positive Initial Energy ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Wave

scholarly journals A blow-up result for a system of coupled viscoelastic equations with arbitrary positive initial energy

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qian Li
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Wave Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Auxiliary Function ◽  
Initial Energy ◽  
Positive Initial Energy ◽  
Viscoelastic Wave ◽  
Viscoelastic Equations

AbstractThis article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature.

scholarly journals On Asymptotic Behavior and Blow-Up of Solutions for a Nonlinear Viscoelastic Petrovsky Equation with Positive Initial Energy

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Gang Li ◽  
Yun Sun ◽  
Wenjun Liu
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Positive Initial Energy ◽  
Nonlinear Viscoelastic ◽  
Petrovsky Equation ◽  
Initial Boundary Value

This paper deals with the initial boundary value problem for the nonlinear viscoelastic Petrovsky equationutt+Δ2u−∫0tgt−τΔ2ux,τdτ−Δut−Δutt+utm−1ut=up−1u. Under certain conditions ongand the assumption thatm<p, we establish some asymptotic behavior and blow-up results for solutions with positive initial energy.

scholarly journals Nonexistence of global solutions for a class of viscoelastic wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge A. Esquivel-Avila
Keyword(s):  
Wave Equations ◽  
Evolution Equations ◽  
Blow Up ◽  
Global Solutions ◽  
Initial Energy ◽  
Nonlinear Evolution Equations ◽  
Material Behavior ◽  
Memory Term ◽  
Viscoelastic Wave ◽  
Deformable Bodies

<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

scholarly journals On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Kafini
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Blow Up ◽  
Initial Energy ◽  
Higher Order ◽  
Viscoelastic Wave Equation ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Wave ◽  
The Cauchy Problem ◽  
Whole Space

<p style='text-indent:20px;'>In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.</p>

scholarly journals A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 203 ◽  
Author(s):  
Khaled Zennir ◽  
Sultan S. Alodhaibi
Keyword(s):  
Decay Rate ◽  
Wave Equations ◽  
Local Existence ◽  
Coupled System ◽  
Initial Energy ◽  
Energy Estimates ◽  
Contraction Mapping Theorem ◽  
Strong Damping ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Wave

The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with the contraction mapping theorem, we established a local existence in [ 0 , T ] . The local solution was made global in time by using appropriate a priori energy estimates. The key to obtaining a novel decay rate is the convexity of the function χ , under the special condition of the initial energy E ( 0 ) . The condition of the weights of weak and strong damping has a fundamental role in the proof. The existence of both three different damping mechanisms and strong nonlinear sources make the paper very interesting from a mathematics point of view, especially when it comes to unbounded spaces such as R n .

Blow Up of Solutions for a Viscoelastic System with Damping and Source Terms in IRn

2010 ◽  
Vol 65 (5) ◽  
pp. 392-400 ◽  
Author(s):  
Wenjun Liu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Coupled System ◽  
Initial Energy ◽  
Source Terms ◽  
Viscoelastic System ◽  
Positive Initial Energy ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Equations ◽  
Damping And Source Terms

This paper deals with a Cauchy problem for the coupled system of nonlinear viscoelastic equations with damping and source terms. We prove a new finite time blow-up result for compactly supported initial data with non-positive initial energy as well as positive initial energy by using the modified energy method and the compact support technique.

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