Abstract : We study the Dirichlet problem for the degenerate nonlocal parabolic equation \[ u_{t}-a\left(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2}\right)\Delta u=C_b\left\Vert u\right\Vert _{L^2(\Omega)}^{\beta}\left\vert u \right\vert^{q\left(x,t\right)-2}u\log|u|+f \quad \text{in $Q_T$}, \] where $Q_{T}:=\Omega \times (0,T)$, $T>0$, $\Omega \subset \mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with a sufficiently smooth boundary, $q(x,t)$ is a measurable function in $Q_{T}$ with values in an interval $[q^{-},q^{+}]\subset(1,\infty)$ and the diffusion coefficient $a(\cdot)$ is a continuous function defined on $\mathbb{R}_+$. It is assumed that $a(s)\to 0$ or $a(s)\to \infty$ as $s\to 0^+$, therefore the equation degenerates or becomes singular as $\|\nabla u(t)\|_{2}\to 0$. For both cases, we show that under appropriate conditions on $a$, $\beta$, $q$, $f$ the problem has a global in time strong solution which possesses the following global regularity property: $\Delta u\in L^2(Q_T)$ and $a(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2})\Delta u\in L^2(Q_T)$.
Abstract : Let $f(z)$ be a meromorphic function in several variables satisfying $$\limsup\limits_{r\rightarrow\infty}\frac{\log T(r,f)}{r}=0.$$ We mainly investigate the uniqueness problem on $f$ in $\mathbb{C}^{m}$ sharing polynomial or periodic small function with its difference polynomials from a new perspective. Our main theorems can be seen as the improvement and extension of previous results.
Abstract : Let $\alpha\in(0,\infty)$, $p\in(0,\infty)$ and $q(\cdot): {{\mathbb R}}^{n}\rightarrow[1,\infty)$ satisfy the globally log-H\"{o}lder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator $M$ and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley $g$-function and the Littlewood-Paley $g^{\ast}_{\lambda}$-function.
Abstract : Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we introduce and study the notions of $u$-$S$-pure $u$-$S$-exact sequences and uniformly $S$-absolutely pure modules which extend the classical notions of pure exact sequences and absolutely pure modules. And then we characterize uniformly $S$-von Neumann regular rings and uniformly $S$-Noetherian rings using uniformly $S$-absolutely pure modules.
Abstract : Let $R$ be a commutative ring with identity and $S$ be a multiplicatively closed subset of $R$. In this article we introduce a new class of ring, called $S$-multiplication rings which are $S$-versions of multiplication rings. An $R$-module $M$ is said to be $S$-multiplication if for each submodule $N$ of $M$, $sN\subseteq JM\subseteq N$ for some $s\in S$ and ideal $J$ of $R$ (see for instance [4, Deﬁnition 1]). An ideal $I$ of $R$ is called $S$-multiplication if $I$ is an $S$-multiplication $R$-module. A commutative ring $R$ is called an $S$-multiplication ring if each ideal of $R$ is $S$-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and $S$-$PIR$. Moreover, we generalize some properties of multiplication rings to $S$-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.
Abstract : In this paper we study a pointwise version of Walters topological stability in the class of homeomorphisms on a compact metric space. We also show that if a sequence of homeomorphisms on a compact metric space is uniformly expansive with the uniform shadowing property, then the limit is expansive with the shadowing property and so topologically stable. Furthermore, we give examples to illustrate our results.
Abstract : In this paper, we consider the existence of time periodic solutions for the compressible magneto-micropolar fluids in the whole space $\mathbb{R}^3$. In particular, we first solve the problem in a sequence of bounded domains by the topological degree theory. Then we obtain the existence of time periodic solutions in $\mathbb{R}^3$ by a limiting process.
Abstract : In this paper, we present a method of characterizing minimal polynomials on the ring ${\mathbf Z}_p$ of $p$-adic integers in terms of their coefficients for an arbitrary prime $p$. We first revisit and provide alternative proofs of the known minimality criteria given by Larin [11] for $p=2$ and Durand and Paccaut [6] for $p=3$, and then we show that for any prime $p\geq 5,$ the proposed method enables us to classify all possible minimal polynomials on ${\mathbf Z}_p$ in terms of their coefficients, provided that two prescribed prerequisites for minimality are satisfied.
Abstract : In this paper, we introduce the concepts of (quasi-)weakly almost periodic point and minimal center of attraction for $\mathbb{Z}^d_+$-actions, explore the connections of levels of the topological structure the orbits of (quasi-)weakly almost periodic points and discuss the relations between (quasi-)weakly almost periodic point and minimal center of attraction. Especially, we investigate the chaotic dynamics near or inside the minimal center of attraction of a point in the cases of $S$-generic setting and non $S$-generic setting, respectively. Actually, we show that weakly almost periodic points and quasi-weakly almost periodic points have distinct topological structures of the orbits and we prove that if the minimal center of attraction of a point is non $S$-generic, then there exist certain Li-Yorke chaotic properties inside the involved minimal center of attraction and sensitivity near the involved minimal center of attraction; if the minimal center of attraction of a point is $S$-generic, then there exist stronger Li-Yorke chaotic (Auslander-Yorke chaotic) dynamics and sensitivity ($\aleph_0$-sensitivity) in the involved minimal center of attraction.
Abstract : As an analogy of the Rogers-Ramanujan continued fraction, we define a Ramanujan continued fraction of order eighteen. There are essentially three Ramanujan continued fractions of order eighteen, and we study them using the theory of modular functions. First, we prove that they are modular functions and find the relations with the Ramanujan cubic continued fraction $C(\tau)$. We can then obtain that their values are algebraic numbers. Finally, we evaluate them at some imaginary quadratic quantities.
Kitae Kim, Hyang-Sook Lee, Seongan Lim, Jeongeun Park, Ikkwon Yie
J. Korean Math. Soc. 2022; 59(6): 1047-1065
https://doi.org/10.4134/JKMS.j210496
Hyunjin Lee, Young Jin Suh, Changhwa Woo
J. Korean Math. Soc. 2022; 59(2): 255-278
https://doi.org/10.4134/JKMS.j200614
Jyunji Inoue, Sin-Ei Takahasi
J. Korean Math. Soc. 2022; 59(2): 367-377
https://doi.org/10.4134/JKMS.j210290
Yuhui Liu
J. Korean Math. Soc. 2022; 59(3): 439-448
https://doi.org/10.4134/JKMS.j190679
Xing Yu Song, Ling Wu
J. Korean Math. Soc. 2023; 60(5): 1023-1041
https://doi.org/10.4134/JKMS.j220589
Chunfang Gao
J. Korean Math. Soc. 2022; 59(2): 235-254
https://doi.org/10.4134/JKMS.j200484
J. Korean Math. Soc. 2022; 59(1): 193-204
https://doi.org/10.4134/JKMS.j210357
Jongsu Kim
J. Korean Math. Soc. 2022; 59(3): 649-650
https://doi.org/10.4134/JKMS.j210761
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