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.

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>

Existence and Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

2013 ◽  
Vol 785-786 ◽  
pp. 1454-1458
Author(s):  
Yan Ping Ran ◽  
Cong Ming Peng
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Semilinear Parabolic Equation ◽  
Initial Boundary Value ◽  
Semilinear Parabolic

This article considers the following degenerate semilinear parabolic initial-boundary value problem,where be constants. We obtained the conditions of global existence and blow-up.

scholarly journals Critical convective-type equations on a half-line

2006 ◽  
Vol 2006 ◽  
pp. 1-24
Author(s):  
Elena I. Kaikina
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Behavior ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equationsut+ℕ(u,ux)+(an∂xn+am∂xm)u=0,(x,t)∈ℝ+×ℝ+,u(x,0)=u0(x),x∈ℝ+,∂xj−1u(0,t)=0forj=1,…,m/2, where the constantsan,am∈ℝ,n,mare integers, the nonlinear termℕ(u,ux)depends on the unknown functionuand its derivativeuxand satisfies the estimate|ℕ(u,v)|≤C|u|ρ|v|σwithσ≥0,ρ≥1, such that((n+2)/2n)(σ+ρ−1)=1,ρ≥1,σ∈[0,m). Also we suppose that∫ℝ+xn/2ℕdx=0. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.

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 Application of Initial Boundary Value Problem on Logarithmic Wave Equation in Dynamics

2018 ◽  
Author(s):  
Shkelqim Hajrulla ◽  
Leonard Bezati ◽  
Fatmir Hoxha
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Sobolev Inequality ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Initial Boundary Value ◽  
The Galerkin Method

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

Global existence, nonexistence, and decay of solutions for a wave equation of p-Laplacian type with weak and p-Laplacian damping, nonlinear boundary delay and source terms

2021 ◽  
pp. 1-16
Author(s):  
Nouri Boumaza ◽  
Billel Gheraibia
Keyword(s):  
Global Existence ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Source Terms ◽  
Delay Term ◽  
Laplacian Equation ◽  
Decay Of Solutions

In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2. To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term.

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 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.

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