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.

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.

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.

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 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>

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

scholarly journals Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation

1996 ◽  
Vol 39 (1) ◽  
pp. 81-96
Author(s):  
D. E. Tzanetis
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Heat Equation ◽  
Unbounded Solutions ◽  
Semilinear Heat Equation ◽  
Initial Boundary Value

The initial-boundary value problem for the nonlinear heat equation u1 = Δu + λf(u) might possibly have global classical unbounded solutions, for some “critical” initial data . The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;λ) for some values of λ We find, for radial symmetric solutions, that u*(r, t)→w(r) for any 0<r≤l but supu*(·, t) = u*(0, t)→∞, as t→∞. Furthermore, if , where is some such critical initial data, then û = u(x, t; û0) blows up in finite time provided that f grows sufficiently fast.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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.

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