scholarly journals General decay and well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation with memory

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1745-1773
Author(s):  
Salah Boulaaras ◽  
Abdelbaki Choucha ◽  
Djamel Ouchenane
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Generalized Solution ◽  
General Decay ◽  
Small Data ◽  
Well Posedness ◽  
Third Order ◽  
The Cauchy Problem ◽  
With Memory

In this paper, we consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan-Moore-Gibson-Thompson (JMGT) equation with the presence of both memory. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result, and we show a local existence result in appropriate function spaces. Finally, we prove a global existence result for small data, and we prove the uniqueness of the generalized solution.

Global well-posedness of the Cauchy problem for the 3D Jordan–Moore–Gibson–Thompson equation

2020 ◽  
pp. 2050069
Author(s):  
Reinhard Racke ◽  
Belkacem Said-Houari
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Small Data ◽  
Third Order ◽  
Bootstrap Argument ◽  
The Cauchy Problem ◽  
Linear Decay

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained.

Well-posedness and limit behaviors for a stochastic higher order modified Camassa–Holm equation

2016 ◽  
Vol 16 (06) ◽  
pp. 1650019
Author(s):  
Lin Lin ◽  
Guangying Lv ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Higher Order ◽  
Well Posedness ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
The Cauchy Problem

This paper is devoted to the Cauchy problem for a stochastic higher order modified-Camassa–Holm equation [Formula: see text] The local existence and uniqueness with initial data [Formula: see text], [Formula: see text] and [Formula: see text], is established. The limit behaviors of the solution are examined as [Formula: see text].

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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.

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the 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].

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

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