scholarly journals Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data

2014 ◽  
Vol 139 (2) ◽  
pp. 353-361 ◽  
Author(s):  
Daisuke Shimotsuma ◽  
Tomomi Yokota ◽  
Kentarou Yoshii
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Type Equation ◽  
Ginzburg Landau

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.

Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Qingying Hu ◽  
Chenxia Zhang ◽  
Hongwei Zhang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Initial Data ◽  
Global Existence ◽  
Exponential Growth ◽  
Type Equation ◽  
Global Weak Solution ◽  
Existence Of Solution ◽  
Two Dimensional ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equations utt-Δu-Δutt+Δ2u=Δfu. Under the assumptions that fu is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.

Cauchy problem for the time-dependent Ginzburg–Landau model of superconductivity

2000 ◽  
Vol 130 (6) ◽  
pp. 1383-1404 ◽  
Author(s):  
A. Rodriguez-Bernal ◽  
B. Wang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Result ◽  
Uniqueness Result ◽  
Time Dependent ◽  
Ginzburg Landau ◽  
Landau Model ◽  
The Cauchy Problem ◽  
Ginzburg Landau Equations ◽  
Well Posed

The Cauchy problem for the time-dependent Ginzburg–Landau equations of superconductivity in Rd (d = 2, 3) is investigated in this paper. When d = 2, we show that the Cauchy problem for this model is well posed in L2. When d = 3, we establish the existence result of solutions for L3 initial data and the uniqueness result for L4 initial data.

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

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