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 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 A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

2014 ◽  
Vol 1 (2) ◽  
pp. 140038 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Antonio Mastroberardino
Keyword(s):  
Riccati Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions

The purpose of this article is to present an analytical method, namely the improved F -expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

scholarly journals Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun
Keyword(s):  
Evolution Equations ◽  
Global Solutions ◽  
Global Attractors ◽  
Nonlinear Evolution Equations ◽  
Time Behavior ◽  
Asymptotic Regularity ◽  
Compact Attractor ◽  
Long Time ◽  
Critical Exponential Growth ◽  
Semilinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

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