FINITE TIME BLOW-UP FOR THE NONLINEAR FOURTH-ORDER DISPERSIVE-DISSIPATIVE WAVE EQUATION AT HIGH ENERGY LEVEL

2012 ◽  
Vol 23 (05) ◽  
pp. 1250060 ◽  
Author(s):  
RUNZHANG XU ◽  
YANBING YANG
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Energy ◽  
Initial Energy ◽  
High Energy Level ◽  
Positive Initial Energy ◽  
Dissipative Wave Equation

In this paper, we investigate the initial boundary value problem of the nonlinear fourth-order dispersive-dissipative wave equation. By using the concavity method, we establish a blow-up result for certain solutions with arbitrary positive initial energy.

scholarly journals Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yue Pang ◽  
Xingchang Wang ◽  
Furong Wu
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Contraction Mapping Principle ◽  
Logarithmic Strain ◽  
Infinite Time ◽  
Positive Initial Energy ◽  
Fourth Order Wave Equation

<p style='text-indent:20px;'>We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.</p>

A Class of Fourth Order Damped Wave Equations with Arbitrary Positive Initial Energy

2018 ◽  
Vol 62 (1) ◽  
pp. 165-178
Author(s):  
Yang Liu ◽  
Jia Mu ◽  
Yujuan Jiao
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Positive Initial Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. In the framework of the energy method, we further exploit the properties of the Nehari functional. Finally, the global existence and finite time blow-up of solutions are obtained.

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.

Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Lian ◽  
Vicenţiu D. Rădulescu ◽  
Runzhang Xu ◽  
Yanbing Yang ◽  
Nan Zhao
Keyword(s):  
Fourth Order ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Point Theory ◽  
Initial Energy ◽  
Damping Term ◽  
Blowup Of Solutions ◽  
Positive Initial Energy ◽  
Initial Boundary Value

Abstract In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.

FINITE TIME BLOW UP OF FOURTH-ORDER WAVE EQUATIONS WITH NONLINEAR STRAIN AND SOURCE TERMS AT HIGH ENERGY LEVEL

2013 ◽  
Vol 24 (06) ◽  
pp. 1350043 ◽  
Author(s):  
JIHONG SHEN ◽  
YANBING YANG ◽  
SHAOHUA CHEN ◽  
RUNZHANG XU
Keyword(s):  
Energy Level ◽  
Fourth Order ◽  
Finite Time ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
High Energy ◽  
Source Terms ◽  
High Energy Level ◽  
Nonlinear Strain

In this paper, we study the initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms at high energy level. We prove that, for certain initial data in the unstable set, the solution with arbitrarily positive initial energy blows up in finite time.

scholarly journals Blow-up of solutions of a semilinear heat equation with a memory term

2005 ◽  
Vol 2005 (2) ◽  
pp. 87-94 ◽  
Author(s):  
Salim A. Messaoudi
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Memory Term ◽  
Semilinear Heat Equation ◽  
Positive Initial Energy ◽  
Initial Boundary Value

We consider an initial boundary value problem related to the equationut−Δu+∫0tg(t−s)Δu(x,s)ds=|u|p−2uand prove, under suitable conditions ongandp, a blow-up result for certain solutions with positive initial energy.

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 and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Fosheng Wang ◽  
Chengqiang Wang
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Viscoelastic Wave Equation ◽  
Frictional Damping ◽  
Viscoelastic Wave ◽  
Initial Boundary Value

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.

scholarly journals Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yang Cao ◽  
Qiuting Zhao
Keyword(s):  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Potential Well ◽  
Kirchhoff Equations ◽  
Exponential Decay Rate ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy <inline-formula><tex-math id="M1">\begin{document}$ J(u_0)\leq d $\end{document}</tex-math></inline-formula>. When the initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_0)&gt;d $\end{document}</tex-math></inline-formula>, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.</p>

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