{"title":"Well-Posedness for the Two-Dimensional Zakharov-Kuznetsov Equation","authors":"Minjie Shan","doi":"10.1619/FESI.63.67","DOIUrl":"https://doi.org/10.1619/FESI.63.67","url":null,"abstract":"","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67432172","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 consider the initial value {problem $partial_{t} u- Delta u=f(u),$ $u(0)=u_0in exp,L^p(mathbb{R}^N),$} where $p>1$ and $f : mathbb{R}tomathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|sim mbox{e}^{|u|^q}$ as $|u|to infty$,} $|f(u)|sim |u|^{m}$ as $uto 0,$ $0 1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$
{"title":"Global Existence and Decay Estimates for the Heat Equation with Exponential Nonlinearity","authors":"M. Majdoub, S. Tayachi","doi":"10.1619/fesi.64.237","DOIUrl":"https://doi.org/10.1619/fesi.64.237","url":null,"abstract":"In this paper we consider the initial value {problem $partial_{t} u- Delta u=f(u),$ $u(0)=u_0in exp,L^p(mathbb{R}^N),$} where $p>1$ and $f : mathbb{R}tomathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|sim mbox{e}^{|u|^q}$ as $|u|to infty$,} $|f(u)|sim |u|^{m}$ as $uto 0,$ $0 1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41387671","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 paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(mathbb{R}^2)$ for $s geq 1/4$. If $d geq 3$, by employing $U^p$ and $V^p$ spaces, we establish the small data global well-posedness in the scaling critical Sobolev space $H^{s_c}(mathbb{R}^d)$ where $s_c = d/2-1$.
{"title":"Well-posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation","authors":"S. Kinoshita","doi":"10.1619/fesi.65.139","DOIUrl":"https://doi.org/10.1619/fesi.65.139","url":null,"abstract":"This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(mathbb{R}^2)$ for $s geq 1/4$. If $d geq 3$, by employing $U^p$ and $V^p$ spaces, we establish the small data global well-posedness in the scaling critical Sobolev space $H^{s_c}(mathbb{R}^d)$ where $s_c = d/2-1$.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42224725","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}
N. Hatano, Ryuya Matsunawa, Tomoki Sato, K. Takemura
We introduce two variants of $q$-hypergeometric equation. We obtain several explicit solutions of variants of $q$-hypergeometric equation. We show that a variant of $q$-hypergeometric equation can be obtained by a restriction of $q$-Appell equation of two variables.
{"title":"Variants of q-Hypergeometric Equation","authors":"N. Hatano, Ryuya Matsunawa, Tomoki Sato, K. Takemura","doi":"10.1619/fesi.65.159","DOIUrl":"https://doi.org/10.1619/fesi.65.159","url":null,"abstract":"We introduce two variants of $q$-hypergeometric equation. We obtain several explicit solutions of variants of $q$-hypergeometric equation. We show that a variant of $q$-hypergeometric equation can be obtained by a restriction of $q$-Appell equation of two variables.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46314776","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 consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.
{"title":"Gevrey Regularity for a System Coupling the Navier-Stokes System with a Beam: the Non-Flat Case","authors":"Mehdi Badra, Takéo Takahashi","doi":"10.1619/fesi.65.63","DOIUrl":"https://doi.org/10.1619/fesi.65.63","url":null,"abstract":"We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43833748","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 consider the energy-critical stochastic cubic nonlinear Schrodinger equation on $mathbb R^4$ with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrodinger equation on $mathbb R^4$, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.
{"title":"Global Well-Posedness of the 4-D Energy-Critical Stochastic Nonlinear Schrödinger Equations with Non-Vanishing Boundary Condition","authors":"Kelvin Cheung, Guopeng Li","doi":"10.1619/fesi.65.287","DOIUrl":"https://doi.org/10.1619/fesi.65.287","url":null,"abstract":"We consider the energy-critical stochastic cubic nonlinear Schrodinger equation on $mathbb R^4$ with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrodinger equation on $mathbb R^4$, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43250963","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}
Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning the linearized problem, the decay estimate with diffusion wave property for an initial data is derived. As an application, the time decay estimate of solutions to the nonlinear problem is given. In contrast to the compressible Navier-Stokes system, for linear system regularities of the initial data are lower and independent of the order of derivative of solutions owing to smoothing effect from the Korteweg tensor. Furthermore, for the nonlinear system diffusion wave property is obtained with an initial data having lower regularity than that of study of the compressible Navier-Stokes system.
{"title":"Time Decay Estimate with Diffusion Wave Property and Smoothing Effect for Solutions to the Compressible Navier-Stokes-Korteweg System","authors":"Takayuki Kobayashi, Kazuyuki Tsuda","doi":"10.1619/fesi.64.163","DOIUrl":"https://doi.org/10.1619/fesi.64.163","url":null,"abstract":"Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning the linearized problem, the decay estimate with diffusion wave property for an initial data is derived. As an application, the time decay estimate of solutions to the nonlinear problem is given. In contrast to the compressible Navier-Stokes system, for linear system regularities of the initial data are lower and independent of the order of derivative of solutions owing to smoothing effect from the Korteweg tensor. Furthermore, for the nonlinear system diffusion wave property is obtained with an initial data having lower regularity than that of study of the compressible Navier-Stokes system.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42950076","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 solve connection problem between fundamental solutions at singular points $0$ and $1$ for the generalized hypergeometric function, using analytic continuation of the integral representation. All connection coefficients are products of the sine and the cosecant.
{"title":"Connection Problem for the Generalized Hypergeometric Function","authors":"Y. Matsuhira, H. Nagoya","doi":"10.1619/fesi.64.323","DOIUrl":"https://doi.org/10.1619/fesi.64.323","url":null,"abstract":"We solve connection problem between fundamental solutions at singular points $0$ and $1$ for the generalized hypergeometric function, using analytic continuation of the integral representation. All connection coefficients are products of the sine and the cosecant.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43248286","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 show the existence of a classical solution to a class of fractional logistic equations in an open bounded subset with smooth boundary. We use the method of sub- and super-solutions with variational arguments to establish the existence of a unique positive solution. We also establish the stability and nondegeneracy of the positive solution.
{"title":"Stability of Positive Solution to Fractional Logistic Equations","authors":"G. Dwivedi, J. Tyagi, R. B. Verma","doi":"10.1619/FESI.62.61","DOIUrl":"https://doi.org/10.1619/FESI.62.61","url":null,"abstract":". In this paper, we show the existence of a classical solution to a class of fractional logistic equations in an open bounded subset with smooth boundary. We use the method of sub- and super-solutions with variational arguments to establish the existence of a unique positive solution. We also establish the stability and nondegeneracy of the positive solution.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.62.61","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43775975","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 the theory of linear autonomous neutral functional di¤erential equations with infinite delay, the spectrum distribution of the infinitesimal generator of its solution operators is studied under a certain phase space. Thereafter, we prove the representation theorem of the solution operators, which is later employed to obtain exponential dichotomy properties in terms of semigroup theory. Formal adjoint theory for linear autonomous NFDEs with infinite delay is established including such topics as formal adjoint equations, the relationship between the formal adjoint and true adjoint, and decomposing the phase space with formal adjoint equation. Finally, the algorithm for calculating the Hopf bifurcation properties for nonlinear NFDEs with infinite delay is presented based on the theory of linear equations.
{"title":"Hopf Bifurcations for Neutral Functional Differential Equations with Infinite Delays","authors":"Chuncheng Wang, Junjie Wei","doi":"10.1619/FESI.62.95","DOIUrl":"https://doi.org/10.1619/FESI.62.95","url":null,"abstract":"In the theory of linear autonomous neutral functional di¤erential equations with infinite delay, the spectrum distribution of the infinitesimal generator of its solution operators is studied under a certain phase space. Thereafter, we prove the representation theorem of the solution operators, which is later employed to obtain exponential dichotomy properties in terms of semigroup theory. Formal adjoint theory for linear autonomous NFDEs with infinite delay is established including such topics as formal adjoint equations, the relationship between the formal adjoint and true adjoint, and decomposing the phase space with formal adjoint equation. Finally, the algorithm for calculating the Hopf bifurcation properties for nonlinear NFDEs with infinite delay is presented based on the theory of linear equations.","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.62.95","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67432124","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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