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The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping 具有非线性阻尼的三维准线性双曲方程的最佳大时间行为
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-14 DOI: 10.1007/s10473-024-0317-6
Han Wang, Yinghui Zhang
{"title":"The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping","authors":"Han Wang, Yinghui Zhang","doi":"10.1007/s10473-024-0317-6","DOIUrl":"https://doi.org/10.1007/s10473-024-0317-6","url":null,"abstract":"<p>We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular, are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal <i>L</i><sup>2</sup> convergence rate of the <i>k</i> (∈ [0, 3])-order spatial derivatives of the solution is <span>({(1 + t)^{ - {{3 + 2k} over 4}}})</span>. Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"17 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a universal inequality for approximate phase isometries 关于近似相等距的普遍不等式
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-14 DOI: 10.1007/s10473-024-0303-z
Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng
{"title":"On a universal inequality for approximate phase isometries","authors":"Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng","doi":"10.1007/s10473-024-0303-z","DOIUrl":"https://doi.org/10.1007/s10473-024-0303-z","url":null,"abstract":"<p>Let <i>X</i> and <i>Y</i> be two normed spaces. Let <span>({cal U})</span> be a non-principal ultrafilter on ℕ. Let g: <i>X</i> → <i>Y</i> be a standard <i>ε</i>-phase isometry for some <i>ε</i> ≥ 0, i.e., <i>g</i>(0) = 0, and for all <i>u, v ϵ X</i>, </p><span>$$|,,|,||g(u) + g(v)|| pm ||g(u) - g(v)||,| - |,||u + v|| pm ||u - v||,|,,|, le varepsilon .$$</span><p>The mapping <i>g</i> is said to be a phase isometry provided that <i>ε</i> = 0. In this paper, we show the following universal inequality of <i>g</i>: for each <span>({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}})</span>, there exist a phase function <span>({sigma _{{u^ * }}}:X to { - 1,1} )</span> and <i>φ</i> ϵ <i>Y</i>* with <span>(||varphi || = ||{u^ * }|| equiv alpha )</span> satisfying that </p><span>$$|leftlangle {{u^ * },u} rightrangle - {sigma _{{u^ * }}}(u)leftlangle {varphi ,g(u)} rightrangle | le {5 over 2}varepsilon alpha ,,,,{rm{for}},{rm{all}},u in X.$$</span><p>In particular, let <i>X</i> be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all <i>u</i>* ∈ <i>X</i>*; (2) the constant <span>({5 over 2})</span> can be reduced to <span>({3 over 2})</span> provided that <i>Y</i>* is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: <i>X</i> → <i>Y</i> such that <span>(Theta (u) = mathop {lim }limits_{n,{cal U}} {{g(nu)} over n})</span> provided that <i>Y</i>** has the <i>w</i>*-Kadec-Klee property (for example, <i>Y</i> is both reflexive and locally uniformly convex).</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The global existence and analyticity of a mild solution to the 3D regularized MHD equations 三维正则化多流体力学方程温和解的全局存在性和解析性
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-14 DOI: 10.1007/s10473-024-0311-z
Cuntao Xiao, Hua Qiu, Zheng-an Yao
{"title":"The global existence and analyticity of a mild solution to the 3D regularized MHD equations","authors":"Cuntao Xiao, Hua Qiu, Zheng-an Yao","doi":"10.1007/s10473-024-0311-z","DOIUrl":"https://doi.org/10.1007/s10473-024-0311-z","url":null,"abstract":"<p>In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core 具有收缩自聚焦核心的半线性椭圆系统解的极限轮廓
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0212-1
Ke Jin, Ying Shi, Huafei Xie
{"title":"The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core","authors":"Ke Jin, Ying Shi, Huafei Xie","doi":"10.1007/s10473-024-0212-1","DOIUrl":"https://doi.org/10.1007/s10473-024-0212-1","url":null,"abstract":"<p>In this paper, we consider the semilinear elliptic equation systems </p><span>$$left{ {matrix{{ - Delta u + u = alpha {Q_n}(x)|u{|^{alpha - 2}}|v{|^beta }u,,{rm{in}},{mathbb{R}^N},} hfill cr { - Delta v + v = beta Q(x)|u{|^alpha }|v{|^{beta - 2}}v,,,,{rm{in}},{mathbb{R}^N},} hfill cr } } right.$$</span><p>\u0000where <span>(Ngeqslant 3,,,alpha ,,,beta &gt; 1,,alpha + beta &lt; {2^ * },,{2^ * } = {{2N} over {N - 2}})</span> and <i>Q</i><sub><i>n</i></sub> are bounded given functions whose self-focusing cores {<i>x</i> ∈ ℍ<sup><i>N</i></sup><i>Q</i><sub><i>n</i></sub>(<i>x</i>) &gt; 0} shrink to a set with finitely many points as <i>n</i> → ∞. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"31 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The extremes of dependent chi-processes attracted by the Brown-Resnick process 布朗-雷斯尼克过程所吸引的依存戚过程的极值
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0217-9
Junjie Sun, Zhongquan Tan
{"title":"The extremes of dependent chi-processes attracted by the Brown-Resnick process","authors":"Junjie Sun, Zhongquan Tan","doi":"10.1007/s10473-024-0217-9","DOIUrl":"https://doi.org/10.1007/s10473-024-0217-9","url":null,"abstract":"<p>Motivated by some recent works on the topic of the Brown-Resnick process, we study the functional limit theorem for normalized pointwise maxima of dependent chi-processes. It is proven that the properly normalized pointwise maxima of those processes are attracted by the Brown-Resnick process.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The sparse representation related with fractional heat equations 与分数热方程有关的稀疏表示
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0211-2
Wei Qu, Tao Qian, Ieng Tak Leong, Pengtao Li
{"title":"The sparse representation related with fractional heat equations","authors":"Wei Qu, Tao Qian, Ieng Tak Leong, Pengtao Li","doi":"10.1007/s10473-024-0211-2","DOIUrl":"https://doi.org/10.1007/s10473-024-0211-2","url":null,"abstract":"<p>This study introduces a pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). As a first step, the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as a second step, makes use of the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show the effectiveness and efficiency of the proposed series solutions.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"39 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift 具有线性自斥漂移的分数奥恩斯坦-乌伦贝克过程的长期行为
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0216-x
Xiaoyu Xia, Litan Yan, Qing Yang
{"title":"The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift","authors":"Xiaoyu Xia, Litan Yan, Qing Yang","doi":"10.1007/s10473-024-0216-x","DOIUrl":"https://doi.org/10.1007/s10473-024-0216-x","url":null,"abstract":"<p>Let <i>B</i><sup><i>H</i></sup> be a fractional Brownian motion with Hurst index <span>({1 over 2} le H &lt; 1)</span>. In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift) </p><span>$${rm{d}}X_t^H = dB_t^H + sigma X_t^H{rm{d}}t + nu {rm{d}}t - theta left( {int_0^t {(X_{^t}^H - X_s^H){rm{d}}s} } right){rm{d}}t,$$</span><p> where θ &lt; 0, <i>σ, v</i> ∈ ℝ. The process is an analogue of self-attracting diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87–93). Our main aim is to study the large time behaviors of the process. We show that the solution <i>X</i><sup><i>H</i></sup> diverges to infinity as t tends to infinity, and obtain the speed at which the process <i>X</i><sup><i>H</i></sup> diverges to infinity as <i>t</i> tends to infinity.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"93 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations 时变金兹堡-朗道方程的广义标量辅助变量法
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0215-y
Zhiyong Si
{"title":"A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations","authors":"Zhiyong Si","doi":"10.1007/s10473-024-0215-y","DOIUrl":"https://doi.org/10.1007/s10473-024-0215-y","url":null,"abstract":"<p>This paper develops a generalized scalar auxiliary variable (SAV) method for the time-dependent Ginzburg-Landau equations. The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations. In this method, the system is decoupled and linearized to avoid solving the non-linear equation at each step. The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability, and this is confirmed by the numerical result, and also shows that the numerical algorithm is stable.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The absence of singular continuous spectrum for perturbed Jacobi operators 扰动雅可比算子不存在奇异连续谱
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0208-x
Zhengqi Fu, Xiong Li
{"title":"The absence of singular continuous spectrum for perturbed Jacobi operators","authors":"Zhengqi Fu, Xiong Li","doi":"10.1007/s10473-024-0208-x","DOIUrl":"https://doi.org/10.1007/s10473-024-0208-x","url":null,"abstract":"<p>This paper is mainly about the spectral properties of a class of Jacobi operators </p><span>$$({H_{c,b}}u)(n) = {c_n}u(n + 1) + {c_{n - 1}}u(n - 1) + {b_n}u(n),$$</span><p> where ∣<i>c</i><sub><i>n</i></sub> − 1∣ = <i>O</i>(<i>n</i><sup><i>−α</i></sup>) and <i>b</i><sub><i>n</i></sub> = <i>O</i>(<i>n</i><sup>−1</sup>). We will show that, for <i>α</i> ≥ 1, the singular continuous spectrum of the operator is empty.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"79 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A flexible objective-constraint approach and a new algorithm for constructing the Pareto front of multiobjective optimization problems 构建多目标优化问题帕累托前沿的灵活目标约束方法和新算法
IF 1 4区 数学
Acta Mathematica Scientia Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0218-8
N. Hoseinpoor, M. Ghaznavi
{"title":"A flexible objective-constraint approach and a new algorithm for constructing the Pareto front of multiobjective optimization problems","authors":"N. Hoseinpoor, M. Ghaznavi","doi":"10.1007/s10473-024-0218-8","DOIUrl":"https://doi.org/10.1007/s10473-024-0218-8","url":null,"abstract":"<p>In this article, a novel scalarization technique, called the improved objective-constraint approach, is introduced to find efficient solutions of a given multiobjective programming problem. The presented scalarized problem extends the objective-constraint problem. It is demonstrated that how adding variables to the scalarized problem, can lead to find conditions for (weakly, properly) Pareto optimal solutions. Applying the obtained necessary and sufficient conditions, two algorithms for generating the Pareto front approximation of bi-objective and three-objective programming problems are designed. These algorithms are easy to implement and can achieve an even approximation of (weakly, properly) Pareto optimal solutions. These algorithms can be generalized for optimization problems with more than three criterion functions, too. The effectiveness and capability of the algorithms are demonstrated in test problems.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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