scholarly journals Extinction and Positivity of the Solutions for a -Laplacian Equation with Absorption on Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Qiao Xin ◽  
Chunlai Mu ◽  
Dengming Liu
Keyword(s):  
Boundary Value Problem ◽  
Numerical Experiment ◽  
Finite Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Simple Graph ◽  
Weighted Graphs ◽  
Laplacian Equation ◽  
Initial Boundary Value ◽  
Standard Weight

We deal with the extinction of the solutions of the initial-boundary value problem of the discretep-Laplacian equation with absorption withp> 1,q> 0, which is said to be the discretep-Laplacian equation on weighted graphs. For 0 <q< 1, we show that the nontrivial solution becomes extinction in finite time while it remains strictly positive for , and . Finally, a numerical experiment on a simple graph with standard weight is given.

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 Global Well-Posedness for a Class of Kirchhoff-Type Wave System

2017 ◽  
Vol 2017 ◽  
pp. 1-18
Author(s):  
Xiaoli Jiang ◽  
Xiaofeng Wang
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Wave System ◽  
Well Posedness ◽  
Initial Value ◽  
Kirchhoff Type ◽  
Type Wave ◽  
Initial Boundary Value

In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars’ attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.

scholarly journals A Study of Deformations in a Thermoelastic Dipolar Body with Voids

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 267 ◽  
Author(s):  
Marin Marin ◽  
Ibrahim Abbas ◽  
Sorin Vlase ◽  
Eduard M. Craciun
Keyword(s):  
Boundary Value Problem ◽  
Porous Body ◽  
Finite Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Body ◽  
Decay Of Solutions ◽  
Rate Of Decay ◽  
Initial Boundary Value ◽  
Dipolar Structure

In this paper, we consider the mixed initial boundary value problem in the context of a thermoelastic porous body having a dipolar structure. We intend to analyze the rate of decay of solutions to this problem to ensure that in a finite time, they become null. In our main result, we find that the combined contribution of the dipolar constitution of the body together with voids dissipation and thermal behavior cannot cause vanishing of the deformations in a finite time.

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 Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Zhao Wang ◽  
Changchun Liu
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sixth Order ◽  
Classical Solutions ◽  
Oil Water ◽  
Initial Boundary Value

We study the initial boundary value problem for a sixth-order Cahn-Hilliard-type equation which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. We show that the solutions might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time. We also discuss the existence of global attractor.

scholarly journals Global Existence and Blow-Up for the Pseudo-parabolic p(x)-Laplacian Equation with Logarithmic Nonlinearity

2021 ◽  
Author(s):  
Fugeng Zeng ◽  
Qigang Deng ◽  
Dongxiu Wang
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Potential Well ◽  
Laplacian Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.

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.

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