Asymptotic Behavior of Solutions to Euler Equations with Time-Dependent Damping in Critical Case

2020 ◽  
Vol 52 (2) ◽  
pp. 1463-1488 ◽  
Author(s):  
Shifeng Geng ◽  
Yanping Lin ◽  
Ming Mei
Keyword(s):  
Asymptotic Behavior ◽  
Euler Equations ◽  
Critical Case ◽  
Time Dependent ◽  
Asymptotic Behavior Of Solutions

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.

scholarly journals Boundary Effect on Asymptotic Behavior of Solutions to the p-System with Time-Dependent Damping

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Ran Duan ◽  
Mina Jiang ◽  
Yinghui Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Boundary ◽  
Boundary Effect ◽  
Initial Perturbation ◽  
Time Dependent ◽  
Asymptotic Behavior Of Solutions ◽  
P System ◽  
Weighted Energy Method ◽  
Weighted Energy ◽  
Constant State

In this paper, we consider the asymptotic behavior of solutions to the p-system with time-dependent damping on the half-line R+=0,+∞, vt−ux=0,ut+pvx=−α/1+tλu with the Dirichlet boundary condition ux=0=0, in particular, including the constant and nonconstant coefficient damping. The initial data v0,u0x have the constant state v+,u+ at x=+∞. We prove that the solutions time-asymptotically converge to v+,0 as t tends to infinity. Compared with previous results about the p-system with constant coefficient damping, we obtain a general result when the initial perturbation belongs to H3R+×H2R+. Our proof is based on the time-weighted energy method.

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