scholarly journals Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yu-Zhu Wang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Hyperbolic Equation ◽  
Contraction Mapping ◽  
Dimensional Space ◽  
Contraction Mapping Principle ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped nonlinear hyperbolic equation inn-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

scholarly journals GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CAUCHY PROBLEM OF A DISSIPATIVE BOUSSINESQ-TYPE EQUATION

2017 ◽  
Vol 22 (4) ◽  
pp. 441-463 ◽  
Author(s):  
Amin Esfahani ◽  
Hamideh B. Mohammadi
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Contraction Mapping ◽  
Type Equation ◽  
Global Solutions ◽  
Contraction Mapping Principle ◽  
Equation Modeling ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle.

scholarly journals Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yinxia Wang ◽  
Hengjun Zhao
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Global Existence ◽  
Decay Estimate ◽  
Boussinesq Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
Linear Solution ◽  
Optimal Decay ◽  
The Cauchy Problem

We investigate the Cauchy problem for the generalized damped Boussinesq equation. Under small condition on the initial value, we prove the global existence and optimal decay estimate of solutions for all space dimensionsn≥1. Moreover, whenn≥2, we show that the solution can be approximated by the linear solution as time tends to infinity.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CAHN–HILLIARD EQUATION WITH INERTIAL TERM

2012 ◽  
Vol 23 (09) ◽  
pp. 1250087 ◽  
Author(s):  
YIN-XIA WANG ◽  
ZHIQIANG WEI
Keyword(s):  
Asymptotic Behavior ◽  
Global Existence ◽  
Dimensional Space ◽  
Solution Space ◽  
Contraction Mapping Principle ◽  
Asymptotic Behavior Of Solutions ◽  
Inertial Term ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

In this paper, we investigate the Cauchy problem for Cahn–Hilliard equation with inertial term in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a solution space. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.

scholarly journals Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

2017 ◽  
Vol 2017 ◽  
pp. 1-21
Author(s):  
Fernando Bernal-Vílchis ◽  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Pseudodifferential Operator ◽  
Critical Case ◽  
Large Time ◽  
Asymptotic Behavior Of Solutions ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Time Asymptotic Behavior

We consider the Cauchy problem for the Ostrovsky-Hunter equation ∂x∂tu-b/3∂x3u-∂xKu3=au, t,x∈R2,  u0,x=u0x, x∈R, where ab>0. Define ξ0=27a/b1/4. Suppose that K is a pseudodifferential operator with a symbol K^ξ such that K^±ξ0=0, Im K^ξ=0, and K^ξ≤C. For example, we can take K^ξ=ξ2-ξ02/ξ2+1. We prove the global in time existence and the large time asymptotic behavior of solutions.

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