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 Nonexistence of global solutions for damped abstract wave equations with memory

2021 ◽  
Vol 2090 (1) ◽  
pp. 012117
Author(s):  
Jorge A. Esquivel-Avila
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Initial Energy ◽  
Nonlinear Wave Equations ◽  
Positive Initial Energy ◽  
Viscoelastic Wave ◽  
With Memory

Abstract We consider a class of abstract nonlinear wave equations with memory and linear dissipation. We give sufficient conditions in terms of the nitial data to prove the nonexistence of global solutions. We improve recent results that have studied this problem for viscoelastic wave, Kirchhoff and Petrovsky equations with positive initial energy values.

scholarly journals Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Jorge A. Esquivel-Avila
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Coupled System ◽  
Initial Energy ◽  
Evolution System ◽  
High Energies ◽  
Second Order In Time

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.

Blow Up of Solutions for a System of Nonlinear Viscoelastic Equations with Damping Terms in Rn

2009 ◽  
Vol 64 (3-4) ◽  
pp. 180-184
Author(s):  
Wenjun Liu ◽  
Shengqi Yub
Keyword(s):  
Initial Data ◽  
Differential Inequality ◽  
Blow Up ◽  
Coupled System ◽  
Initial Energy ◽  
Relaxation Functions ◽  
Nonlinear Viscoelastic ◽  
Linear Damping ◽  
Viscoelastic Equations ◽  
Damping And Source Terms

Abstract We consider a coupled system of nonlinear viscoelastic equations with linear damping and source terms. Under suitable conditions of the initial data and the relaxation functions, we prove a finitetime blow-up result with vanishing initial energy by using the modified energy method and a crucial lemma on differential inequality

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>

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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