scholarly journals STOCHASTIC WAVE EQUATIONS WITH NONLINEAR DAMPING AND SOURCE TERMS

2013 ◽  
Vol 16 (02) ◽  
pp. 1350013 ◽  
Author(s):  
HONGJUN GAO ◽  
FEI LIANG ◽  
BOLING GUO
Keyword(s):  
Wave Equations ◽  
Blow Up ◽  
Energy Inequality ◽  
Local Existence ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Damping ◽  
Local Existence And Uniqueness ◽  
Stochastic Wave Equations

In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term |ut|q–2utand a source term of the type|u|p–2u. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and show that the solution is global for q ≥ p. Secondly, by an appropriate energy inequality, the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q.

scholarly journals Stochastic Viscoelastic Wave Equations with Nonlinear Damping and Source Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Shuilin Cheng ◽  
Yantao Guo ◽  
Yanbin Tang
Keyword(s):  
Local Existence ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Random Force ◽  
Nonlinear Damping ◽  
Source Terms ◽  
Local Existence And Uniqueness ◽  
Viscoelastic Wave ◽  
Damping And Source Terms

The goal of this paper is to study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data: the relaxation function, the indices of nonlinear damping, and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions, respectively.

scholarly journals On a system of nonlinear wave equations of Kirchhoff type with a strong dissipation

2007 ◽  
Vol 38 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Shun-Tang Wu ◽  
Long-Yi Tsai
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Wave Equations ◽  
Banach Fixed Point Theorem ◽  
Kirchhoff Type ◽  
Initial Boundary Value

The initial boundary value problem for systems of nonlinear wave equations of Kirchhoff type with strong dissipation in a bounded domain is considered. We prove the local existence of solutions by Banach fixed point theorem and blow-up of solutions by energy method. Some estimates for the life span of solutions are given.

scholarly journals Dynamics of solutions to a semilinear plate equation with memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinxing Liu ◽  
Xiongrui Wang ◽  
Jun Zhou ◽  
Xu Liu
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plate Equation ◽  
Energy Estimation ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value ◽  
With Memory

<p style='text-indent:20px;'>In this paper we consider an initial-boundary value problem of a semilinear regularity-loss-type plate equation with memory in a bounded domain of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ n = 1,2,\cdots $\end{document}</tex-math></inline-formula>). By using the Faedo-Galërkin method and some theories of ordinary differential equations, we obtain the local existence and uniqueness of weak solutions. Then, we study the dynamics of the weak solutions, such as global existence and finite time blow-up, by energy estimation and some ordinary differential inequalities. Moreover, the upper bound of blow-up time for the blow-up solutions is also considered.</p>

Global existence and explosive solution for stochastic viscoelastic wave equation with nonlinear damping

2014 ◽  
Vol 26 (07) ◽  
pp. 1450013 ◽  
Author(s):  
Fei Liang ◽  
Hongjun Gao
Keyword(s):  
Wave Equation ◽  
Source Term ◽  
Blow Up ◽  
Local Existence ◽  
Galerkin Approximation ◽  
Nonlinear Damping ◽  
Stochastic Equations ◽  
Viscoelastic Wave Equation ◽  
Local Existence And Uniqueness ◽  
Viscoelastic Wave

In this paper, we discuss an initial boundary value problem for the stochastic viscoelastic wave equation involving the nonlinear damping term |ut|q-2utand a source term of the type |u|p-2u. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and an elementary measure-theoretic argument. Moreover, we also show that the solution is global for q ≥ p. Secondly, by the technique of [Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations 109 (1994) 295–308] with a modification in the energy functional, we prove that the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q. This result extends earlier ones obtained by Liang and Gao [Explosive solutions of stochastic viscoelastic wave equations with damping, Rev. Math. Phys. 23(8) (2011) 883–902] in which only linear damping is considered. Furthermore, upon comparing our stochastic equations with their deterministic counterparts, we find that our results indicates that the presence of noise might affect the occurrence of blow-up.

INITIAL-BOUNDARY VALUE PROBLEMS AND FORMATION OF SINGULARITIES FOR ONE-DIMENSIONAL NON-RELATIVISTIC RADIATION HYDRODYNAMIC EQUATIONS

2012 ◽  
Vol 09 (04) ◽  
pp. 711-738 ◽  
Author(s):  
PENG JIANG ◽  
YAGUANG WANG
Keyword(s):  
Smooth Solution ◽  
Blow Up ◽  
Boundary Value ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boltzmann Equations ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value

In this paper, we investigate the well-posedness for the Euler–Boltzmann equations of radiation hydrodynamics in one spatial variable. We obtain the local existence and uniqueness of smooth solution to the initial-boundary value problem. Then, we show that a smooth solution will blow up in finite time regardless of the size of the initial disturbance.

scholarly journals Local and Global Existence of Solution for Love Type Waves with Past History

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1998
Author(s):  
Mohamed Biomy ◽  
Khaled Zennir ◽  
Ahmed Himadan
Keyword(s):  
Existence And Uniqueness ◽  
Past History ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linearization Method ◽  
Weak Compactness ◽  
Existence Of Solution ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value

In this paper, we consider an initial boundary value problem for nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo–Galerkin method, and the weak compactness method, the local existence and uniqueness of weak solution is proved. Using the potential well method, it is shown that the solution for a class of Love-equation exists globally under some conditions on the initial datum and kernel function.

scholarly journals Global Existence and Blow-Up for the Euler-Bernoulli Plate Equation with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Jianghao Hao ◽  
Jie Lan
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Stability Of Solutions ◽  
Plate Equation ◽  
Existence And Stability ◽  
Initial Boundary Value

We prove the local existence, blow-up, global existence, and stability of solutions to the initial boundary value problem for Euler-Bernoulli plate equation with variable coefficients.

scholarly journals EXISTENCE RESULTS FOR A POLYMER MELT WITH AN EVOLVING NATURAL CONFIGURATION

2012 ◽  
Vol 22 (01) ◽  
pp. 1150004
Author(s):  
J. K. DJOKO ◽  
B. D. REDDY
Keyword(s):  
Viscoelastic Fluid ◽  
Local Existence ◽  
Polymer Melt ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Deformation Tensor ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
Natural Configuration ◽  
Proper Orthogonal

We consider a set of equations governing the behavior of a polymer melt which is modeled as a viscoelastic fluid possessing a natural, or stress-free state. The natural configuration is characterized through a symmetric, proper orthogonal intermediate deformation tensor, analogous to the left Cauchy–Green deformation tensor in continuum mechanics. This tensor is required to satisfy an evolution equation. It is shown that the constraint that the intermediate tensor be proper orthogonal is satisfied provided that its initial value satisfies this constraint. Local existence and uniqueness of solutions to the initial boundary value problem of the resulting viscoelastic fluid system are established. It is also shown that the local solutions can be extended globally provided that the data are small enough.

Non-existence of global solutions for a class of wave equations with nonlinear damping and source terms

2011 ◽  
Vol 141 (4) ◽  
pp. 865-880 ◽  
Author(s):  
Shun-Tang Wu
Keyword(s):  
Wave Equations ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Damping ◽  
Critical Value ◽  
Initial Energy ◽  
Source Terms ◽  
Initial Boundary Value ◽  
Damping And Source Terms

An initial–boundary-value problem for a class of wave equations with nonlinear damping and source terms in a bounded domain is considered. We establish the non-existence result of global solutions with the initial energy controlled above by a critical value via the method introduced in a work by Autuori et al. in 2010. This improves the 2009 result of Liu and Wang.

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