scholarly journals Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type

2021 ◽  
Vol 11 (04) ◽  
pp. 346-368
Author(s):  
Qi Zhang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Global Existence ◽  
Strong Solution ◽  
Large Time ◽  
Boussinesq Equations ◽  
Density Dependent ◽  
The Cauchy Problem ◽  
Time Asymptotic Behavior

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 Global existence and asymptotic behavior for a generalized Boussinesq type equation without dissipation

2021 ◽  
Author(s):  
Guowei Liu ◽  
Wei Wang ◽  
Qiuju Xu
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Initial Data ◽  
Global Existence ◽  
Linear Group ◽  
Type Equation ◽  
Small Initial Data ◽  
Dispersive Estimate ◽  
The Cauchy Problem

In this paper, we study the Cauchy problem for a generalized Boussinesq type equation in $\mathbb{R}^n$. We establish a dispersive estimate for the linear group associated with the generalized Boussinesq type equation. As applications, the global existence, decay and scattering of solutions are established for small initial data.

scholarly journals Riemann-Hilbert approach for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data

2018 ◽  
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Large Time ◽  
Transform Method ◽  
Factorization Problem ◽  
Scattering Transform ◽  
The Cauchy Problem ◽  
Hilbert Type ◽  
Time Asymptotic Behavior ◽  
Nonlocal Nonlinear

We study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0\] with a step-like initial data: $q(x,0)=o(1)$ as $x\to-\infty$ and $q(x,0)=A+o(1)$ as $x\to\infty$, where $A>0$ is an arbitrary constant. We develop the inverse scattering transform method for this problem in the form of the Riemann-Hilbert approach and obtain the representation of the solution of the Cauchy problem in terms of the solution of an associated Riemann-Hilbert-type analytic factorization problem, which can be efficiently used for further studying the properties of the solution, including the large time asymptotic behavior.

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