Long-time behavior of the solution of the Cauchy problem for the third-order Airy equation

2020 ◽  
Vol 116 (2) ◽  
pp. 139-148
Author(s):  
Sergei V. Zakharov
Keyword(s):  
Cauchy Problem ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Third Order ◽  
The Third ◽  
The Cauchy Problem ◽  
Long Time ◽  
Airy Equation

scholarly journals Cauchy problem for hyperbolic equations of third order with variable exponent of nonlinearity

2020 ◽  
Vol 12 (2) ◽  
pp. 419-433
Author(s):  
O.M. Buhrii ◽  
O.T. Kholyavka ◽  
P.Ya. Pukach ◽  
M.I. Vovk
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Lebesgue Spaces ◽  
Third Order ◽  
The Third ◽  
The Cauchy Problem

We investigate weak solutions of the Cauchy problem for the third order hyperbolic equations with variable exponent of the nonlinearity. The problem is considered in some classes of functions namely in Lebesgue spaces with variable exponents. The sufficient conditions of the existence and uniqueness of the weak solutions to given problem are found.

Holder Estimates for General Models

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
General Model ◽  
Resolvent Operator ◽  
Heat Equations ◽  
Second Derivative ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
The Resolvent Operator ◽  
Hölder Estimates

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining first and second derivative estimates in the latter case, before discussing a general result describing the off-diagonal and long-time behavior of the solution kernel for the general model. It also states a proposition summarizing the properties of the resolvent operator as an operator on the Hölder spaces. In contrast to the case of the heat equation, there is no need to assume that the data has compact support in the x-variables to prove estimates when k > 0.

scholarly journals Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in ℝ 3

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Yunshun Wu ◽  
Yong Wang ◽  
Rong Shen
Keyword(s):  
Heat Conduction ◽  
Euler Equations ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Algebraic Decay ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Long Time

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

scholarly journals Cauchy problem for general time fractional diffusion equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1545-1559
Author(s):  
Chung-Sik Sin
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Behavior ◽  
Analysis Technique ◽  
The Cauchy Problem ◽  
Long Time ◽  
Time Fractional Diffusion Equation

Abstract In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general Caputo-type operator, the general diffusion equation with source term is studied.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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