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
2020 ◽  
Vol 12 (2) ◽  
pp. 419-433
Author(s):  
O.M. Buhrii ◽  
O.T. Kholyavka ◽  
P.Ya. Pukach ◽  
M.I. Vovk

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.


Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo

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.


2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Yunshun Wu ◽  
Yong Wang ◽  
Rong Shen

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.


2020 ◽  
Vol 23 (5) ◽  
pp. 1545-1559
Author(s):  
Chung-Sik Sin

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.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao

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 ) .


2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu

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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