Complicated asymptotic behavior of solutions for the Cauchy problem of Cahn–Hilliard equation

2019 ◽  
Vol 98 ◽  
pp. 95-100
Author(s):  
Yuqiu Wu ◽  
Liangwei Wang ◽  
Langhao Zhou
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Behavior Of Solutions ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

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 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 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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