Global Attractor for the Convective Cahn-Hilliard Equation in $H^k$

Xiaopeng Zhao; Ning Duan

Displaying similar documents to “Global Attractor for the Convective Cahn-Hilliard Equation in $H^{k}$ ”

Existence and global attractivity of periodic solutions in a higher order difference equation

Chuanxi Qian, Justin Smith (2018)

Archivum Mathematicum

Similarity:

Consider the following higher order difference equation $x (n + 1) = f (n, x (n)) + g (n, x (n - k)), n = 0, 1, \dots$ where $f (n, x)$ and $g (n, x) : {0, 1, \dots} \times [0, \infty) \to [0, \infty)$ are continuous functions in $x$ and periodic functions in $n$ with period $p$ , and $k$ is a nonnegative integer. We show the existence of a periodic solution ${\tilde{x} (n)}$ under certain conditions, and then establish a sufficient condition for ${\tilde{x} (n)}$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also...

A new look at an old comparison theorem

Jaroslav Jaroš (2021)

Archivum Mathematicum

Similarity:

We present an integral comparison theorem which guarantees the global existence of a solution of the generalized Riccati equation on the given interval $[a, b)$ when it is known that certain majorant Riccati equation has a global solution on $[a, b)$ .

On the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

Similarity:

Let $a ₁, . . ., a_{k}$ be arbitrary nonzero real numbers. An $(a ₁, . . ., a_{k})$ -decomposition of a function f:ℝ → ℝ is a sum $f ₁ + \dots + f_{k} = f$ where $f_{i} : ℝ \to ℝ$ is an $a_{i}$ -periodic function. Such a decomposition is not unique because there are several solutions of the equation $h ₁ + \dots + h_{k} = 0$ with $h_{i} : ℝ \to ℝ a_{i}$ -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a ₁, . . ., a_{k})$ -decomposition is essentially unique. We characterize those periods for which essential...

Existence and uniqueness of periodic solutions for odd-order ordinary differential equations

Yongxiang Li, He Yang (2011)

Annales Polonici Mathematici

Similarity:

The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2 n + 1)} = f (t, u, u^{'}, . . ., u^{(2 n)})$ , where $f : ℝ \times ℝ^{2 n + 1} \to ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f (t, x ₀, x ₁, . . ., x_{2 n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].

Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

Lingbin Kong, Daqing Jiang (1998)

Annales Polonici Mathematici

Similarity:

The fourth order periodic boundary value problem $u^{(4)} - m ⁴ u + F (t, u) = 0$ , 0 < t < 2π, with $u^{(i)} (0) = u^{(i)} (2 π)$ , i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $\pm 10^{- 7}$ .

Attractor of a semi-discrete Benjamin-Bona-Mahony equation on ℝ¹

Chaosheng Zhu (2015)

Annales Polonici Mathematici

Similarity:

This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on ℝ¹. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H¹(ℝ¹). Then we prove that this system possesses a global attractor $_{τ}$ in H¹(ℝ¹). In addition, we show that the global attractor $_{τ}$ is regular, i.e., $_{τ}$ ...

Global behavior of the difference equation $x_{n + 1} = \frac{a x_{n - 3}}{b + c x_{n - 1} x_{n - 3}}$

Raafat Abo-Zeid (2015)

Archivum Mathematicum

Similarity:

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $x_{n + 1} = \frac{a x_{n - 3}}{b + c x_{n - 1} x_{n - 3}}, n = 0, 1, \dots$ where $a, b, c$ are positive real numbers and the initial conditions $x_{- 3}$ , $x_{- 2}$ , $x_{- 1}$ , $x_{0}$ are real numbers.

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

Similarity:

We study the Cauchy problem for the 3D MHD system with damping terms ${ε | u |}^{α - 1} u$ and ${δ | b |}^{β - 1} b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

Similarity:

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system $Δ_{p} u = g (u - α v), Δ_{p} v = f (v - β u)$ in a smooth bounded domain of $ℝ^{N}$ , where $Δ_{p}$ is the p-Laplacian operator defined by $Δ_{p} u = {d i v (| \nabla u |}^{p - 2} \nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

Probabilistic properties of a Markov-switching periodic $G A R C H$ process

Billel Aliat, Fayçal Hamdi (2019)

Kybernetika

Similarity:

In this paper, we propose an extension of a periodic $G A R C H$ ( $P G A R C H$ ) model to a Markov-switching periodic $G A R C H$ ( $M S$ - $P G A$ $R C H$ ), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

Similarity:

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial_{t} u + {div}_{y} A (y, u) - Δ_{y} u = 0$ , where $y \in ℝ^{N}$ and the flux $A$ is periodic in $y$ . More specifically, we consider the case when the initial data is an $L^{1}$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^{1}$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

Periodic solutions of $x " + f (x) x^{' m} + g (x) = 0$

W. R. Utz (1971)

Annales Polonici Mathematici

Similarity:

A class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

Similarity:

Three periodic solutions for a class of higher-dimensional functional differential equations with impulses

Yongkun Li, Changzhao Li, Juan Zhang (2010)

Annales Polonici Mathematici

Similarity:

By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t \neq t_{j}$ , j ∈ ℤ, ⎨ ⎩ $y (t ⁺_{j}) = y (t ¯_{j}) + I_{j} (y (t_{j}))$ , where $A (t) = {(a_{i j} (t))}_{n \times n}$ is a nonsingular matrix with continuous real-valued entries.

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

Similarity:

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n, ν}$ be the space of piecewise linear continuous functions on the torus with knots $t_{j} : 0 \leq j \leq N - 1$ . Finally, let $P_{n, ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n, ν}$ . The main result is $l i m_{n \to \infty, ν = 1} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = s u p_{n \in ℕ, 0 \leq ν \leq n} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = 2 + (33 - 18 \sqrt 3) / 13$ . This shows in particular that the Lebesgue constant of the classical...

Vectorial quasilinear diffusion equation with dynamic boundary condition

Nakayashiki, Ryota

Similarity:

In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S) $_{ε}$ with a nonnegative constant $ε$ , and for any $ε \geq 0$ , (S) $_{ε}$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $Ω$ , and the parabolic equation on the boundary $Γ : = \partial Ω$ , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $Δ u + ρ (\frac{e^{u}}{\int_{T} e^{u}} - \frac{1}{| T |}) = 0, \int_{T} u = 0 .$ It is well-known that the associated energy functional admits a minimizer for each $ρ \leq 8 π$ . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting $λ_{1} (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $ρ \leq min {8 π, λ_{1} (T) | T |}$ . Our results hold more generally for solutions that...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

Similarity:

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_{t} = Δ u - χ \nabla \cdot (u \nabla v / v) + r u - μ u^{k}$ , $0 = Δ v - v + u$ under the non-flux boundary conditions in a smooth bounded domain $Ω \subset ℝ^{n}$ , $χ, r, μ > 0$ , $k > 1$ and $n \geq 1$ . It is shown with $k \in (1, 2)$ that the system possesses a global generalized solution for $n \geq 2$ which is bounded when $χ > 0$ is suitably small related to $r > 0$ and the initial datum is properly small, and a global bounded classical solution for $n = 1$ .

Displaying similar documents to “Global Attractor for the Convective Cahn-Hilliard Equation in H k ”

Displaying similar documents to “Global Attractor for the Convective Cahn-Hilliard Equation in $H^{k}$ ”