We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct. For more information see https://ejde.math.txstate.edu/Volumes/2024/04/abstr.html
{"title":"Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusion","authors":"Yuequn Li, Hui Liu, Fei Guo","doi":"10.58997/ejde.2024.04","DOIUrl":"https://doi.org/10.58997/ejde.2024.04","url":null,"abstract":"We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct. \u0000For more information see https://ejde.math.txstate.edu/Volumes/2024/04/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"15 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139445558","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}
Jesus Ildefonso Diaz, T. Shaposhnikova, Alexander V. Podolskiy
We study the homogenization of a nonlinear problem given by the Poisson equation, in a domain with arbitrarily shaped perforations (or particles) and with a dynamic unilateral boundary condition (of Signorini type), with a large coefficient, on the boundary of these perforations (or particles). This problem arises in the study of chemical reactions of zero order. The consideration of a possible asymmetry in the perforations (or particles) is fundamental for considering some applications in nanotechnology, where symmetry conditions are too restrictive. It is important also to consider perforations (or particles) constituted by small different parts and then with several connected components. We are specially concerned with the so-called critical case in which the relation between the coefficient in the boundary condition, the period of the basic structure, and the size of the holes (or particles) leads to the appearance of an unexpected new term in the effective homogenized equation. Because of the dynamic nature of the boundary condition this ``strange term'' becomes now a non-local in time and non-linear operator. We prove a convergence theorem and find several properties of the ``strange operator'' showing that there is a kind of regularization through the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2024/03/abstr.html
{"title":"Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical size","authors":"Jesus Ildefonso Diaz, T. Shaposhnikova, Alexander V. Podolskiy","doi":"10.58997/ejde.2024.03","DOIUrl":"https://doi.org/10.58997/ejde.2024.03","url":null,"abstract":"We study the homogenization of a nonlinear problem given by the Poisson equation, in a domain with arbitrarily shaped perforations (or particles) and with a dynamic unilateral boundary condition (of Signorini type), with a large coefficient, on the boundary of these perforations (or particles). This problem arises in the study of chemical reactions of zero order. The consideration of a possible asymmetry in the perforations (or particles) is fundamental for considering some applications in nanotechnology, where symmetry conditions are too restrictive. It is important also to consider perforations (or particles) constituted by small different parts and then with several connected components. We are specially concerned with the so-called critical case in which the relation between the coefficient in the boundary condition, the period of the basic structure, and the size of the holes (or particles) leads to the appearance of an unexpected new term in the effective homogenized equation. Because of the dynamic nature of the boundary condition this ``strange term'' becomes now a non-local in time and non-linear operator. We prove a convergence theorem and find several properties of the ``strange operator'' showing that there is a kind of regularization through the homogenization process. \u0000For more information see https://ejde.math.txstate.edu/Volumes/2024/03/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"63 9","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139387168","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}
In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results. For more information see https://ejde.math.txstate.edu/Volumes/2024/01/abstr.html
{"title":"Stability and rate of decay for solutions to stochastic differential equations with Markov switching","authors":"Shuaishuai Lu, Xue Yang","doi":"10.58997/ejde.2024.01","DOIUrl":"https://doi.org/10.58997/ejde.2024.01","url":null,"abstract":"In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results.\u0000For more information see https://ejde.math.txstate.edu/Volumes/2024/01/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"128 42","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139387708","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}
In this article, we establish a Kolmogorov-Arnold-Moser (KAM) theorem for degenerate infinite-dimensional reversible systems under a non-degenerate condition of Russmann type. This theorem broadens the scope of applicability of degenerate KAM theory, previously confined to Hamiltonian systems, by incorporating infinite-dimensional reversible systems. Using this theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of non-Hamiltonian but reversible beam equations with non-linearities in derivatives. For more information see https://ejde.math.txstate.edu/Volumes/2024/02/abstr.html
在这篇文章中，我们在鲁斯曼类型的非退化条件下，建立了退化无限维可逆系统的科尔莫戈罗夫-阿诺德-莫泽尔（KAM）定理。该定理拓宽了退化 KAM 理论的适用范围，将无限维可逆系统纳入其中，而该理论以前仅限于哈密尔顿系统。利用该定理，我们得到了一类非哈密尔顿但可逆的、导数非线性的梁方程的准周期解的存在性和线性稳定性。更多信息，请参见 https://ejde.math.txstate.edu/Volumes/2024/02/abstr.html。
{"title":"KAM theorem for degenerate infinite-dimensional reversible systems","authors":"Zhaowei Lou, Youchao Wu","doi":"10.58997/ejde.2024.02","DOIUrl":"https://doi.org/10.58997/ejde.2024.02","url":null,"abstract":"In this article, we establish a Kolmogorov-Arnold-Moser (KAM) theorem for degenerate infinite-dimensional reversible systems under a non-degenerate condition of Russmann type. This theorem broadens the scope of applicability of degenerate KAM theory, previously confined to Hamiltonian systems, by incorporating infinite-dimensional reversible systems. Using this theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of non-Hamiltonian but reversible beam equations with non-linearities in derivatives.\u0000For more information see https://ejde.math.txstate.edu/Volumes/2024/02/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"15 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139389609","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}
In this article, we study the asymptotic stability of Bresse transmission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index (thetain[0,1]) and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with (thetaneq 0), the system is not exponential stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with (theta=0), the system is exponential stable. For more information see https://ejde.math.txstate.edu/Volumes/2023/87/abstr.html
{"title":"Asymptotic stabilization for Bresse transmission systems with fractional damping","authors":"Jianghao Hao, Dingkun Wang","doi":"10.58997/ejde.2023.87","DOIUrl":"https://doi.org/10.58997/ejde.2023.87","url":null,"abstract":"In this article, we study the asymptotic stability of Bresse transmission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index (thetain[0,1]) and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with (thetaneq 0), the system is not exponential stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with (theta=0), the system is exponential stable. For more information see https://ejde.math.txstate.edu/Volumes/2023/87/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"41 18","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139151298","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}
This article concerns the weak solutions of a 3D Cauchy problem of compressible magneto-micropolar fluids with discontinuous initial data. Under the assumption that the initial data are of small energy and the initial density is positive and essentially bounded, we establish the existence of weak solutions that are global-in-time. Moreover, we obtain the large-time behavior of such solutions. For moreinformation see https://ejde.math.txstate.edu/Volumes/2023/86/abstr.html
{"title":"Global low-energy weak solutions for compressible magneto-micropolar fluids with discontinuous initial data in R^3","authors":"Wanping Wu, Yinghui Zhang","doi":"10.58997/ejde.2023.86","DOIUrl":"https://doi.org/10.58997/ejde.2023.86","url":null,"abstract":"This article concerns the weak solutions of a 3D Cauchy problem of compressible magneto-micropolar fluids with discontinuous initial data. Under the assumption that the initial data are of small energy and the initial density is positive and essentially bounded, we establish the existence of weak solutions that are global-in-time. Moreover, we obtain the large-time behavior of such solutions. \u0000For moreinformation see https://ejde.math.txstate.edu/Volumes/2023/86/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"20 5","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138956127","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}
This work creates a version of the periodic unfolding method suitable for domains with very small inclusions in (mathbb{R}^N) for (Ngeq 3). In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter ( gamma < 0). In particular, we consider the cases when (gamma in (-1,0)), ( gamma < -1) and (gamma = -1). We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2023/85/abstr.html
{"title":"Periodic unfolding method for domains with very small inclusions","authors":"J. Avila, Bituin C. Cabarrubias","doi":"10.58997/ejde.2023.85","DOIUrl":"https://doi.org/10.58997/ejde.2023.85","url":null,"abstract":"This work creates a version of the periodic unfolding method suitable for domains with very small inclusions in (mathbb{R}^N) for (Ngeq 3). In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter ( gamma < 0). In particular, we consider the cases when (gamma in (-1,0)), ( gamma < -1) and (gamma = -1). We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2023/85/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139168709","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}
In this article, we investigate the relationship between growth and value distribution of meromorphic solutions for the higher-order complex linear difference equations $$ A_n(z)f(z+n)+dots+A_1(z)f(z+1)+A_0(z)f(z)=0 quad text{and } =F(z), $$ and for the linear difference polynomial $$ g(z)=alpha_n(z)f(z+n)+dots+alpha_1(z)f(z+1)+alpha_0(z)f(z) $$ generated by (f(z)) where (A_j(z)), (alpha_j(z)) ((j=0,1,ldots,n)), (F(z)) ((notequiv0)) are meromorphic functions. We improve some previous results due to Belaidi, Chen and Zheng and others. For more information see https://ejde.math.txstate.edu/Volumes/2023/84/abstr.html
{"title":"Growth and value distribution of linear difference polynomials generated by meromorphic solutions of higher-order linear difference equations","authors":"Yi Xin Luo, Xiu Min Zheng","doi":"10.58997/ejde.2023.84","DOIUrl":"https://doi.org/10.58997/ejde.2023.84","url":null,"abstract":"In this article, we investigate the relationship between growth and value distribution of meromorphic solutions for the higher-order complex linear difference equations $$ A_n(z)f(z+n)+dots+A_1(z)f(z+1)+A_0(z)f(z)=0 quad text{and } =F(z), $$ and for the linear difference polynomial $$ g(z)=alpha_n(z)f(z+n)+dots+alpha_1(z)f(z+1)+alpha_0(z)f(z) $$ generated by (f(z)) where (A_j(z)), (alpha_j(z)) ((j=0,1,ldots,n)), (F(z)) ((notequiv0)) are meromorphic functions. We improve some previous results due to Belaidi, Chen and Zheng and others. \u0000For more information see https://ejde.math.txstate.edu/Volumes/2023/84/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"38 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138966904","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}
This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincare disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus. For more information see https://ejde.math.txstate.edu/Volumes/2023/83/abstr.html
{"title":"Global analysis on a continuous planar piecewise linear differential system with three zones","authors":"Man Jia, Youfeng Su, Hebai Chen","doi":"10.58997/ejde.2023.83","DOIUrl":"https://doi.org/10.58997/ejde.2023.83","url":null,"abstract":"This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincare disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus. \u0000For more information see https://ejde.math.txstate.edu/Volumes/2023/83/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"991 ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138982600","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}
We construct non-homogeneous polynomial lump wave solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation, based on a bilinear approach, enriching the formal diversity of lump waves. By studying the interaction between the lump solutions of the YTSF equation and the solitary wave solutions, we find a new aggregation effect and elastic collision effect. We obtain exact solutions, such as the solution of separated variables and periodic nonlinear wave solutions, by applying the Lie symmetry group method and the bilinear method. For more information see https://ejde.math.txstate.edu/Volumes/2023/82/abstr.html
{"title":"Variety of solutions and dynamical behavior for YTSF equations","authors":"Wei Chen","doi":"10.58997/ejde.2023.82","DOIUrl":"https://doi.org/10.58997/ejde.2023.82","url":null,"abstract":"We construct non-homogeneous polynomial lump wave solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation, based on a bilinear approach, enriching the formal diversity of lump waves. By studying the interaction between the lump solutions of the YTSF equation and the solitary wave solutions, we find a new aggregation effect and elastic collision effect. We obtain exact solutions, such as the solution of separated variables and periodic nonlinear wave solutions, by applying the Lie symmetry group method and the bilinear method. \u0000For more information see https://ejde.math.txstate.edu/Volumes/2023/82/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"53 20","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138982422","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}
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