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 Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yaojun Ye
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Decay Estimate ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Wave Equations ◽  
Stable Set ◽  
Initial Boundary Value ◽  
Damping And Source Terms

The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions through the use of a difference inequality.

Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

2013 ◽  
Vol 14 (3) ◽  
pp. 574-598 ◽  
Author(s):  
Hermann Brunner ◽  
Hongwei Li ◽  
Xiaonan Wu
Keyword(s):  
Numerical Solution ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Blow Up ◽  
Splitting Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Absorbing Boundary Conditions ◽  
Nonlinear Wave Equations ◽  
Computational Domain

AbstractThe numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.

scholarly journals Existence, Asymptotic Behaviour, and Blow up of Solutions for a Class of Nonlinear Wave Equations with Dissipative and Dispersive Terms

2009 ◽  
Vol 64 (5-6) ◽  
pp. 315-326
Author(s):  
Necat Polat ◽  
Doğan Kaya
Keyword(s):  
Asymptotic Behaviour ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Term ◽  
Blow Up ◽  
Suitable Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Nonlinear Wave Equations

Abstract We consider the existence, both locally and globally in time, the asymptotic behaviour, and the blow up of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative and dispersive terms. Under rather mild conditions on the nonlinear term and the initial data we prove that the above-mentioned problem admits a unique local solution, which can be continued to a global solution, and the solution decays exponentially to zero as t →+∞. Finally, under a suitable condition on the nonlinear term, we prove that the local solutions with negative and nonnegative initial energy blow up in finite time.

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.

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