This paper is concerned with the existence, stability and almost periodicity of global bounded solutions of linear and semilinear parabolic equations. We give a rather complete characterization to the linear equation ut Du 1⁄4 luþ f ðx; tÞ, in term of the parameter l, of whether or not the global bounded solutions exist. Also, we obtain a complete characterization on the stability of global bounded solutions.
研究了一类线性和半线性抛物型方程整体有界解的存在性、稳定性和概周期性。我们给出了线性方程ut Du 1 / 4 luþ f ðx的较为完整的表征;TÞ，用参数l表示，全局有界解是否存在。同时，我们也得到了全局有界解稳定性的完整表征。
{"title":"On Global Boundedness, Stability and Almost Periodicity of Solutions for Heat Equations","authors":"Yi Xie, Peidong Lei","doi":"10.1619/FESI.62.191","DOIUrl":"https://doi.org/10.1619/FESI.62.191","url":null,"abstract":"This paper is concerned with the existence, stability and almost periodicity of global bounded solutions of linear and semilinear parabolic equations. We give a rather complete characterization to the linear equation ut Du 1⁄4 luþ f ðx; tÞ, in term of the parameter l, of whether or not the global bounded solutions exist. Also, we obtain a complete characterization on the stability of global bounded solutions.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431481","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}
{"title":"Lp Boundedness of Higher Order Schrödinger Type Operators","authors":"Satoko Sugano","doi":"10.1619/fesi.62.409","DOIUrl":"https://doi.org/10.1619/fesi.62.409","url":null,"abstract":"","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431571","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 paper, we study the spectral structure of periodic Schro¨dinger operators on a generalization of carbon nanotubes from the point of view of the quantum graphs. Since there exist chemical double bonds between carbon atoms on a hexagonal lattice with a cylindrical structure corresponding to carbon nanotubes, we study the spectral structure of periodic Schro¨dinger operators on zigzag nanotubes with multiple bonds of atoms in this paper. Utilizing the Floquet–Bloch theory, the spectrum of the operator consists of the absolutely continuous spectral bands and the flat band. We study the relationship between the number of the chemical bonds and the existence of spectral gaps.
{"title":"On the Spectra of Schrödinger Operators on Zigzag Nanotubes with Multiple Bonds","authors":"Hiroaki Niikuni","doi":"10.1619/FESI.62.255","DOIUrl":"https://doi.org/10.1619/FESI.62.255","url":null,"abstract":". In this paper, we study the spectral structure of periodic Schro¨dinger operators on a generalization of carbon nanotubes from the point of view of the quantum graphs. Since there exist chemical double bonds between carbon atoms on a hexagonal lattice with a cylindrical structure corresponding to carbon nanotubes, we study the spectral structure of periodic Schro¨dinger operators on zigzag nanotubes with multiple bonds of atoms in this paper. Utilizing the Floquet–Bloch theory, the spectrum of the operator consists of the absolutely continuous spectral bands and the flat band. We study the relationship between the number of the chemical bonds and the existence of spectral gaps.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.62.255","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431756","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}
{"title":"Bifurcating Solutions of a Nonlinear Elliptic Neumann Problem on Large Spherical Caps","authors":"C. Bandle, Yoshitsugu Kabeya, H. Ninomiya","doi":"10.1619/fesi.62.285","DOIUrl":"https://doi.org/10.1619/fesi.62.285","url":null,"abstract":"","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431518","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}
Stokes phenomena with respect to a large parameter are investigated for Shrödinger-type ordinary di¤erential equations having a Stokes curve of loop-type. For this purpose, we employ a Bessel-type equation as a canonical form and compute the Voros coe‰cient of the equation. Combining the formula describing the Stokes automorphism for the Voros coe‰cient and the formal coordinate transformation connecting the Shrödinger-type equation and the Bessel-type equation, we have some formulas describing the action of alien derivatives and Stokes automorphism for WKB solutions of the Shrödinger-type equation.
{"title":"Exact WKB Analysis of Schrödinger Equations with a Stokes Curve of Loop Type","authors":"T. Aoki, Kohei Iwaki, Toshinori Takahashi","doi":"10.1619/FESI.62.1","DOIUrl":"https://doi.org/10.1619/FESI.62.1","url":null,"abstract":"Stokes phenomena with respect to a large parameter are investigated for Shrödinger-type ordinary di¤erential equations having a Stokes curve of loop-type. For this purpose, we employ a Bessel-type equation as a canonical form and compute the Voros coe‰cient of the equation. Combining the formula describing the Stokes automorphism for the Voros coe‰cient and the formal coordinate transformation connecting the Shrödinger-type equation and the Bessel-type equation, we have some formulas describing the action of alien derivatives and Stokes automorphism for WKB solutions of the Shrödinger-type equation.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.62.1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431870","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}
Okaie et al. [8] utilized the Keller-Segel model for mobile bionanosensor networks for target tracking. They introduced a mathematical formulation and described numerical results. In this paper, we would like to study analytically their model. We first construct a unique local solution for model equations. Second, we establish a priori estimates for local solutions to obtain a global solution. Finally, after constructing a non-autonomous dynamical system, we will show existence of exponential attractors.
{"title":"Exponential Attractor for One-Dimensional Self-Organizing Target-Detection Model","authors":"S. Iwasaki","doi":"10.1619/FESI.62.75","DOIUrl":"https://doi.org/10.1619/FESI.62.75","url":null,"abstract":"Okaie et al. [8] utilized the Keller-Segel model for mobile bionanosensor networks for target tracking. They introduced a mathematical formulation and described numerical results. In this paper, we would like to study analytically their model. We first construct a unique local solution for model equations. Second, we establish a priori estimates for local solutions to obtain a global solution. Finally, after constructing a non-autonomous dynamical system, we will show existence of exponential attractors.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67432064","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}
Lotka-Volterra systems associated with skew-symmetric interaction between species are studied. We pick up some form of this model provided with conserved quantities, which makes all the solutions to be periodic-in-time except for the equilibrium. This class is explicitly given by a set of algebraic conditions on coe‰cients. If the system takes N components, we have 2N 3 and 2N 1 degrees of freedom without and with linear terms, respectively.
{"title":"Lotka-Volterra Systems with Periodic Orbits","authors":"Manami Kobayashi, Takashi Suzuki, Yoshio Yamada","doi":"10.1619/FESI.62.129","DOIUrl":"https://doi.org/10.1619/FESI.62.129","url":null,"abstract":"Lotka-Volterra systems associated with skew-symmetric interaction between species are studied. We pick up some form of this model provided with conserved quantities, which makes all the solutions to be periodic-in-time except for the equilibrium. This class is explicitly given by a set of algebraic conditions on coe‰cients. If the system takes N components, we have 2N 3 and 2N 1 degrees of freedom without and with linear terms, respectively.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.62.129","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431414","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 show that solutions to a classical Fokker-Plank equation can be approximated by solutions to nonlocal evolution problems when a rescaling parameter that controls the size of the nonlocality goes to zero.
{"title":"Nonlocal Approximations to Fokker-Planck Equations","authors":"A. Molino, J. Rossi","doi":"10.1619/FESI.62.35","DOIUrl":"https://doi.org/10.1619/FESI.62.35","url":null,"abstract":"We show that solutions to a classical Fokker-Plank equation can be approximated by solutions to nonlocal evolution problems when a rescaling parameter that controls the size of the nonlocality goes to zero.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431529","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 study conditions for the abstract periodic linear functional differential equation $dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.
{"title":"Almost Periodic Solutions of Periodic Linear Partial Functional Differential Equations","authors":"V. Luong, N. Minh","doi":"10.1619/FESI.62.209","DOIUrl":"https://doi.org/10.1619/FESI.62.209","url":null,"abstract":"We study conditions for the abstract periodic linear functional differential equation $dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.62.209","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43960775","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 note we establish existence and uniqueness of weak solutions of linear elliptic equation $text{div}[mathbf{A}(x) nabla u] = text{div}{mathbf{F}(x)}$, where the matrix $mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse Holder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $mathbf{A}$ is the identity matrix. In addition, the example also shows the necessity of the dependence of $alpha$ in the Holder $C^alpha$-regularity theory on the textup{BMO}-semi norm of the skew-symmetric part of $mathbf{A}$. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of $mathbf{A}$ is assumed to be zero.
本文建立了线性椭圆方程$text{div}[mathbf{A}(x) nabla u] = text{div}{mathbf{F}(x)}$弱解的存在唯一性，其中矩阵$mathbf{A}$刚好可测且其偏对称部分可以无界。得到了弱解梯度的全局逆Holder正则性估计。最重要的是，我们通过一个例子证明，即使$mathbf{A}$的对称部分是单位矩阵，$mathbf{A}$的有界性和椭圆性对于更高的可积性估计也是不够的。此外，实例还说明了Holder $C^alpha$ -正则性理论中$alpha$依赖于$mathbf{A}$的偏对称部分的textup{bmo} -半范数的必要性。本文是N. G. Meyers(1963)的经典结果的推广，其中假设$mathbf{A}$的偏对称部分为零。
{"title":"On Higher Integrability Estimates for Elliptic Equations with Singular Coefficients","authors":"J. Foldes, T. Phan","doi":"10.1619/fesi.66.1","DOIUrl":"https://doi.org/10.1619/fesi.66.1","url":null,"abstract":"In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $text{div}[mathbf{A}(x) nabla u] = text{div}{mathbf{F}(x)}$, where the matrix $mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse Holder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $mathbf{A}$ is the identity matrix. In addition, the example also shows the necessity of the dependence of $alpha$ in the Holder $C^alpha$-regularity theory on the textup{BMO}-semi norm of the skew-symmetric part of $mathbf{A}$. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of $mathbf{A}$ is assumed to be zero.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43396837","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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