Blow-Up of Solutions for a System of Petrovsky Equations with an Indirect Linear Damping

2013 ◽  
Vol 68 (5) ◽  
pp. 343-349
Author(s):  
Wenjun Liu
Keyword(s):  
Boundary Conditions ◽  
Qualitative Analysis ◽  
Blow Up ◽  
Coupled System ◽  
Initial Energy ◽  
Point Of View ◽  
Damping Term ◽  
Rate Of Decay ◽  
Linear Damping ◽  
Indirect Damping

In this paper, we consider a coupled system of Petrovsky equations in a bounded domain with clamped boundary conditions. Due to several physical considerations, a linear damping which is distributed everywhere in the domain under consideration appears only in the first equation whereas no damping term is applied to the second one (this is indirect damping). Many studies show that the solution of this kind of system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. For four different ranges of initial energy, we show here the blow-up of solutions and give the lifespan estimates by improving the method of Wu (Electron. J. Diff. Equ. 105, 1 (2009)) and Li et al. (Nonlin. Anal. 74, 1523 (2011)). From the applications point of view, our results may provide some qualitative analysis and intuition for the researchers in other fields such as engineering and mechanics when they study the concrete models of Petrovsky type.

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 Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Ülkü Dinlemez ◽  
Esra Aktaş
Keyword(s):  
Blow Up ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Nonlinear Hyperbolic Equations ◽  
Damping Term ◽  
Linear Damping ◽  
Initial Boundary Value ◽  
Nonlinear String

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

Blow up of Solution for the Generalized Boussinesq Equation with Damping Term

2006 ◽  
Vol 61 (5-6) ◽  
pp. 235-238
Author(s):  
Necat Polat ◽  
Doğan Kaya
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Damping Term ◽  
Initial Boundary Value ◽  
Generalized Boussinesq Equation

We consider the blow up of solution to the initial boundary value problem for the generalized Boussinesq equation with damping term. Under some assumptions we prove that the solution with negative initial energy blows up in finite time

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 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.

scholarly journals General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mengxian Lv ◽  
Jianghao Hao
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Decay Rates ◽  
General Decay ◽  
Dynamic Boundary Conditions ◽  
Positive Initial Energy ◽  
Relaxation Functions ◽  
Dynamic Boundary

<p style='text-indent:20px;'>In this paper we consider a system of viscoelastic wave equations of Kirchhoff type with dynamic boundary conditions. Supposing the relaxation functions <inline-formula><tex-math id="M1">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (i = 1, 2, \cdots, l) $\end{document}</tex-math></inline-formula> satisfy <inline-formula><tex-math id="M3">\begin{document}$ g_i(t)\leq-\xi_i(t)G(g_i(t)) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is an increasing and convex function near the origin and <inline-formula><tex-math id="M5">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> are nonincreasing, we establish some optimal and general decay rates of the energy using the multiplier method and some properties of convex functions. Moreover, we obtain the finite time blow-up result of solution with nonpositive or arbitrary positive initial energy. The results in this paper are obtained without imposing any growth condition on weak damping term at the origin. Our results improve and generalize several earlier related results in the literature.</p>

scholarly journals Blowup for nonlinearly damped viscoelastic equations with logarithmic source and delay terms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Energy Method ◽  
Blow Up ◽  
Nonlinear Damping ◽  
Initial Energy ◽  
Damping Term ◽  
Finite Point ◽  
Source Effect ◽  
Positive Initial Energy ◽  
Viscoelastic Equations

AbstractIn this work, we investigate blowup phenomena for nonlinearly damped viscoelastic equations with logarithmic source effect and time delay in the velocity. Owing to the nonlinear damping term instead of strong or linear dissipation, we cannot apply the concavity method introduced by Levine. Thus, utilizing the energy method, we show that the solutions with not only non-positive initial energy but also some positive initial energy blow up at a finite point in time.

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