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

Chaosheng Zhu

Displaying similar documents to “Attractor of a semi-discrete Benjamin-Bona-Mahony equation on ℝ¹”

On $(n, m)$ - $A$ -normal and $(n, m)$ - $A$ -quasinormal semi-Hilbertian space operators

Samir Al Mohammady, Sid Ahmed Ould Beinane, Sid Ahmed Ould Ahmed Mahmoud (2022)

Mathematica Bohemica

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The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let $ℋ$ be a Hilbert space and let $A$ be a positive bounded operator on $ℋ$ . The semi-inner product ${〈 h ∣ k 〉}_{A} : = 〈 A h ∣ k 〉$ , $h, k \in ℋ$ , induces a semi-norm ${∥ \cdot ∥}_{A}$ . This makes $ℋ$ into a semi-Hilbertian space. An operator $T \in ℬ_{A} (ℋ)$ is said to be $(n, m)$ - $A$ -normal if $[T^{n}, {(T^{♯_{A}})}^{m}] : = T^{n} {(T^{♯_{A}})}^{m} - {(T^{♯_{A}})}^{m} T^{n} = 0$ for some positive integers $n$ and $m$ .

Migrativity properties of 2-uninorms over semi-t-operators

Ying Li-Jun, Qin Feng (2022)

Kybernetika

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In this paper, we analyze and characterize all solutions about $α$ -migrativity properties of the five subclasses of 2-uninorms, i. e. $C^{k}$ , $C_{k}^{0}$ , $C_{k}^{1}$ , $C_{1}^{0}$ , $C_{0}^{1}$ , over semi-t-operators. We give the sufficient and necessary conditions that make these $α$ -migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for $G \in C^{k}$ , the $α$ -migrativity of $G$ over a semi-t-operator $F_{μ, ν}$ is closely related to the $α$ -section of $F_{μ, ν}$ or the ordinal sum representation...

A new look at an old comparison theorem

Jaroslav Jaroš (2021)

Archivum Mathematicum

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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 $[a, b)$ .

Some results on semi-stratifiable spaces

Wei-Feng Xuan, Yan-Kui Song (2019)

Mathematica Bohemica

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We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $D C (ω_{1})$ ; (2) If $X$ is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then $X$ is separable; (3) Let $X$ be a $ω$ -monolithic star countable extent semi-stratifiable space. If $t (X) = ω$ and $d (X) \leq ω_{1}$ , then $X$ is hereditarily separable. Finally, we prove that for...

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

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

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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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 $n = 1$ .

When $C_{p} (X)$ is domain representable

William Fleissner, Lynne Yengulalp (2013)

Fundamenta Mathematicae

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Let M be a metrizable group. Let G be a dense subgroup of $M^{X}$ . We prove that if G is domain representable, then $G = M^{X}$ . The following corollaries answer open questions. If X is completely regular and $C_{p} (X)$ is domain representable, then X is discrete. If X is zero-dimensional, T₂, and $C_{p} (X,)$ is subcompact, then X is discrete.

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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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.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials $P_{c} (z) = z^{d} + c$ for c ∈ ℂ. Denote by $J_{c}$ the Julia set of $P_{c}$ and let $ℳ_{d} = c | J_{c} i s c o n n e c t e d$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c ₀ \in \partial ℳ_{d}$ : those for which the critical point 0 is not recurrent by $P_{c ₀}$ and without parabolic cycles. The Hausdorff dimension of $J_{c}$ , denoted by $H D (J_{c})$ , does not depend continuously on c at such $c ₀ \in \partial ℳ_{d}$ ; on the other hand the function $c \mapsto H D (J_{c})$ is analytic in $ℂ - ℳ_{d}$ . Our first result asserts that there is still some...

Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Given a coarse space $(X, ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X \times X$ to the natural coarse structure on $(ℬ ∖ {\emptyset}) \times (ℬ ∖ {\emptyset})$ and say that a macro-uniform mapping $f : (ℬ ∖ {\emptyset}) \to X$ (or $f : {[X]}^{2} \to X$ ) is a selector (or 2-selector) of $(X, ℰ)$ if $f (A) \in A$ for each $A \in ℬ ∖ {\emptyset}$ ( $A \in {[X]}^{2}$ , respectively). We prove that a discrete coarse space $(X, ℰ)$ admits a selector if and only if $(X, ℰ)$ admits a 2-selector if and only if there exists a linear order “ $\leq$ " on $X$ such that the family of intervals ${[a, b] : a, b \in X, a \leq b}$ is a base for the bornology $ℬ$ .

Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, K. P. Rybakowski (2002)

Studia Mathematica

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Consider the family uₜ = Δu + G(u), t > 0, $x \in Ω_{ε}$ , $\partial_{ν_{ε}} u = 0$ , t > 0, $x \in \partial Ω_{ε}$ , $(E_{ε})$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set $Ω_{ε}$ is a thin domain in $ℝ^{l}$ , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of $ℝ^{l}$ . If G is dissipative, then equation $(E_{ε})$ has a global attractor $_{ε}$ . We identify a “limit” equation for the family $(E_{ε})$ , prove convergence of trajectories and establish an upper semicontinuity result for the family $_{ε}$ as ε → 0⁺. ...

An improved regularity criteria for the MHD system based on two components of the solution

Zujin Zhang, Yali Zhang (2021)

Applications of Mathematics

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As observed by Yamazaki, the third component $b_{3}$ of the magnetic field can be estimated by the corresponding component $u_{3}$ of the velocity field in $L^{λ}$ $(2 \leq λ \leq 6)$ norm. This leads him to establish regularity criterion involving $u_{3}, j_{3}$ or $u_{3}, ω_{3}$ . Noticing that $λ$ can be greater than 6 in this paper, we can improve previous results.

On the distribution of $(k, r)$ -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number $n$ is said to be a $(k, r)$ -integer if $n = a^{k} b$ , where $k > r > 1$ and $b$ is not divisible by the $r$ th power of any prime. We study the distribution of such $(k, r)$ -integers in the Piatetski-Shapiro sequence ${⌊ n^{c} ⌋}$ with $c > 1$ . As a corollary, we also obtain similar results for semi- $r$ -free integers.

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process $ℐ^{u} \subset ℤ^{d}$ at level $u$ , $d \geq 3$ . Roughly speaking, we show that observations of $ℐ^{u}$ restricted to two disjoint subsets $A_{1}$ and $A_{2}$ of $ℤ^{d}$ are approximately independent, once we add a sprinkling to the process $ℐ^{u}$ by slightly increasing the parameter $u$ . Our results differ from previous ones in that we allow the mutual distance between the sets $A_{1}$ and $A_{2}$ to be much smaller than their diameters. We then provide an important application...

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

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In this paper we examine self-similar solutions to the system $u_{i t} - d_{i} Δ u_{i} = \prod_{k = 1}^{m} u_{k}^{p_{k}^{i}}$ , i = 1,…,m, $x \in ℝ^{N}$ , t > 0, $u_{i} (0, x) = u_{0 i} (x)$ , i = 1,…,m, $x \in ℝ^{N}$ , where m > 1 and $p_{k}^{i} > 0$ , to describe asymptotics near the blow up point.

Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling, Masahiko Shimojō (2019)

Mathematica Bohemica

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We consider solutions of quasilinear equations $u_{t} = Δ u^{m} + u^{p}$ in $ℝ^{N}$ with the initial data $u_{0}$ satisfying $0 < u_{0} < M$ and ${lim}_{| x | \to \infty} u_{0} (x) = M$ for some constant $M > 0$ . It is known that if $0 < m < p$ with $p > 1$ , the blow-up set is empty. We find solutions $u$ that blow up throughout $ℝ^{N}$ when $m > p > 1$ .