Issue 1, Volume 50, March 2025

2025

1. Blow-up and Decay of Global Solutions for a Free Boundary Problem with Competing Nonlocal Nonlinearity and Absorption Hoang Huy Truong , Dung Tien Nguyen , Hoang-Hung Vo PDF

   ▸ Abstract

   In this paper, we are concerned with the characterization of the blow-up and global solutions for free boundary parabolic equation with competing nonlocal nonlinearity and absorption $$\begin{aligned} u_t(t,x) = u_{xx}(t,x) + u^{p}(t,x) \int _0^{s(t)}u^{q}(t,x)dx -\gamma u^\alpha (t,x), t>0,\ 0<x <s(t), \end{aligned}$$where $q, \alpha \ge 1$, $p=0$,  or $p\ge 1$ and $\gamma > 0$ are given constants. This study is motivated from the works [Abdelhedi and Zaag: J. Differential Equations 272, 1–45, (2021); Souplet: SIAM J. Math. Anal. 29, 1301–1334, (1998); Zhou and Lin: J. Funct. Anal. 262, 3409–3429, (2012)] arisen from the investigation of many physical and biological phenomena such as population dynamics, combustion theory, phase separation in binary mixtures, theory of nuclear reactor dynamics... We first prove the local existence, uniqueness and stability of solution thanks to the “extension trick" introduced in [Du et al.: Math. Ann. 386(3-4), 2061–2106, (2023); Wang and Du: Discrete Contin. Dyn. Syst. Ser. B 26(4), 2201–2238, (2021)]. Second, by improving the comparison principle used in [Souplet: SIAM J. Math. Anal. 29, 1301–1334, (1998); Zhou and Lin: J. Funct. Anal. 262, 3409–3429, (2012)], we find a sharp criterion characterizing the blow-up and global solutions of (1) in term of power coefficients and initial data. We further show that there exists a threshold for the initial data that determines whether blow-up, global fast, or global slow solutions occur and find an upper bound for the existence time of blow-up solutions in two different cases $\alpha >1$ and $\alpha =1$. Our proofs are mainly based on the comparison principle by improving several techniques in previous works, combined new idea to handle differential inequalities and unified local existence theory for nonlocal semilinear parabolic equations.

   1-31
2. Integral Closure of Powers of Edge Ideals of Weighted Oriented Graphs Arindam Banerjee , Kanoy Kumar Das , Sirajul Haque PDF

   ▸ Abstract

   In this article, we study monomial ideals associated with a simple graph, namely edge ideals of weighted oriented graphs. Let $D$ be a weighted oriented graph. Assuming that all the vertices of $D$ have weights greater than 1, we completely characterize weighted oriented graphs $D$ for which $I(D)$ is integrally closed, and show that this is equivalent to $I(D)$ being normal. We also give an equivalent condition for $\(\overline{I(D)}=I(D)\)$ when the underlying simple graph of $D$ is a complete graph. Finally, we give a necessary and sufficient condition when the edge ideal of a uniform whiskered graph is integrally closed.

   33-49
3. General Form of Second Main Theorem on Generalized $p$-Parabolic Manifolds for Arbitrary Closed Subschemes Duc Quang Si PDF

   ▸ Abstract

   By introducing the notion of distributive constant for a family of closed subschemes, we establish a general form of the second main theorem for algebraic non-degenerate meromorphic mappings from a generalized $p$-Parabolic manifold into a projective variety with arbitrary families of closed subschemes. As its consequence, we give a second main theorem for such meromorphic mappings intersecting arbitrary hypersurfaces with an explicitly truncation level for the counting functions.

   51-66
4. Upper Bound for the Conjunction Probability of Smooth Stationary Two-dimensional Gaussian Fields Viet-Hung Pham PDF

   ▸ Abstract

   In this paper, we provide an upper bound for the conjunction probability of smooth stationary two-dimensional Gaussian fields. The key ingredient is the record method, which is generalized from the problem of estimating the distribution of the maximum. The given upper bound is also a good asymptotic formula.

   67-77
5. Asymptotically Almost Periodic and Almost Automorphic Solutions to the Non-autonomous Oseen-Navier-Stokes Equations Ngoc Huy Nguyen , Thieu Huy Nguyen , Thi Ngoc Ha Vu PDF

   ▸ Abstract

   In this paper, we investigate the existence, uniqueness and stability of asymptotically almost periodic and almost automorphic solutions to the non-autonomous Oseen-Navier-Stokes Equations (ONSE) in an unbounded domain $\varOmega \subset \mathbb {R}^{3}$ exterior to a rigid body $D$, i.e., $\varOmega =\mathbb {R}^{3}\backslash D$, with the data belonging to $L^p$-spaces and with the asymptotically almost periodic or almost automorphic external forces, respectively. Our method is based on the $L^p-L^q$ smoothness of the evolution family corresponding to linearized equations in combination with interpolation functors and fixed-point arguments.

   79-99
6. Dual-ADS, ADS$^\#$ and ADS* Modules Abyzov Adel Nailevich , Bui Tien Dat , Truong Cong Quynh PDF

   ▸ Abstract

   A right $R$-module $M$ is said to be dual-ADS if for every decomposition $M=A\oplus B$ then $A$ and $B$ are mutually projective. The class of ADS*-modules contains the class of dual-ADS modules. In this article, we study several properties of these modules. It is shown that a module $M$ is dual-ADS if and only if for any direct summand $S$ and $T^\prime \le M$ with $T^\prime +S$ a direct summand of $M$, then $T^\prime$ contains a direct complement of $S$ in $T^\prime +S$. A generalization of dual-ADS modules is considered, namely, ADS$^\#$-modules. It is shown that a module $M$ is ADS$^\#$ if and only if for any direct summand $S$ of $M$, and any weak supplement $T^\prime$ of $S$ in $T^\prime +S$ such that $T^\prime +S$ is a direct summand of $M$, then $T^\prime$ contains a direct complement of $S$ in $T^\prime +S$.

   101-121
7. Differences of Products of Volterra Type Operators and Composition Operators Between Fock Spaces Pham The Hao , Pham Trong Tien PDF

   ▸ Abstract

   We characterize boundedness and compactness of the differences of products of Volterra integral operators $V_g$ (Volterra companion operators $J_g$) and composition operators $C_{\varphi }$ between different Fock spaces $\mathcal {F}^p(\mathbb {C})$ and $\mathcal {F}^q(\mathbb {C})$ for every $p, q \in (0, \infty ]$.

   123-141
8. Measure Pressure for Measure Preserving Maps and an Upper Bound for the Case of $C^2$ Endomorphisms Sanaz Lamei , Pouya Mehdipour , Maryam Razi PDF

   ▸ Abstract

   A measure-theoretic pressure was defined by [L. He, J. Lv and L. Zhou: Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica (English Series) 20, 709–718 (2004)] based on the Katok entropy formula. For a measure preserving map f, we generalized this definition to define a measure-theoretic pressure by using both $(n,\epsilon )$-spanning and $(n,\epsilon )$-separated sets. A variational principle for this pressure is established. Furthermore, we investigate an upper bound for the measure theoretic pressure of a $C^{2}$ endomorphism preserving a hyperbolic measure.

   143-157