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 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 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>

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.

Blow-Up Of Solutions For The Damped Boussinesq Equation

2005 ◽  
Vol 60 (7) ◽  
pp. 473-476 ◽  
Author(s):  
Necat Polat ◽  
Doğan Kaya ◽  
H. Ilhan Tutalar
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Initial Boundary Value

We consider the blow-up of solutions as a function of time to the initial boundary value problem for the damped Boussinesq equation. Under some assumptions we prove that the solutions with vanishing initial energy blow up in finite time

On strong well-posedness of initial-boundary value problems for higher order nonlinear hyperbolic equations with two independent variables

2017 ◽  
Vol 24 (3) ◽  
pp. 409-428 ◽  
Author(s):  
Tariel Kiguradze ◽  
Raja Ben-Rabha
Keyword(s):  
Hyperbolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Well Posedness ◽  
Initial Boundary Value Problems ◽  
Nonlinear Hyperbolic Equations ◽  
Independent Variables ◽  
Initial Boundary Value

AbstractProblems with linear initial-boundary conditions for higher order nonlinear hyperbolic equations are investigated. The concept of strong well-posedness of an initial-boundary value problem is introduced, and conditions guaranteeing solvability and strong well-posedness of the problem under consideration are established.

EXISTENCE AND BLOW-UP OF SOLUTIONS TO A PARABOLIC EQUATION WITH NONSTANDARD GROWTH CONDITIONS

2018 ◽  
Vol 99 (2) ◽  
pp. 242-249
Author(s):  
YANG LIU
Keyword(s):  
Parabolic Equation ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Growth Conditions ◽  
Initial Energy ◽  
Nonstandard Growth ◽  
Nonstandard Growth Conditions ◽  
Initial Boundary Value

We study the initial boundary value problem for a fourth-order parabolic equation with nonstandard growth conditions. We establish the local existence of weak solutions and derive the finite time blow-up of solutions with nonpositive initial energy.

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.

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.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

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