scholarly journals Asymptotic Behavior of Solutions Toward a Multiwave Pattern to the Cauchy Problem for the Dissipative Wave Equation with Partially Linearly Degenerate Flux

2021 ◽  
Vol 64 (1) ◽  
pp. 49-73
Author(s):  
Natsumi Yoshida
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Wave Equation ◽  
Asymptotic Behavior Of Solutions ◽  
The Cauchy Problem ◽  
Dissipative Wave Equation ◽  
Linearly Degenerate

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.

Sign in / Sign up

Export Citation Format

Share Document