Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|to +infty$ (for example, solution's growth restriction as $|x|to +infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|to +infty$ to solvability of the initial-boundary value problems for parabolic equations considered above. However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity. Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations. In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.
{"title":"INITIAL-BOUNDARY VALUE PROBLEM FOR HIGHER-ORDERS NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENTS OF THE NONLINEARITY IN UNBOUNDED DOMAINS WITHOUT CONDITIONS AT INFINITY","authors":"M. Bokalo","doi":"10.31861/bmj2022.02.05","DOIUrl":"https://doi.org/10.31861/bmj2022.02.05","url":null,"abstract":"Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the\u0000uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|to +infty$ (for example, solution's growth restriction as $|x|to +infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|to +infty$ to solvability of the initial-boundary value problems for parabolic equations considered above.\u0000\u0000However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity.\u0000\u0000Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations.\u0000\u0000In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115850112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Uniqueness theorems are considered for various types of almost periodic objects: functions, measures, distributions, multisets, holomorphic and meromorphic functions.
研究了各种类型的概周期对象的唯一性定理:函数、测度、分布、多集、全纯函数和亚纯函数。
{"title":"UNIQUENESS THEOREMS FOR ALMOST PERIODIC OBJECTS","authors":"S. Favorov, O. Udodova","doi":"10.31861/bmj2021.01.03","DOIUrl":"https://doi.org/10.31861/bmj2021.01.03","url":null,"abstract":"Uniqueness theorems are considered for various types of almost periodic objects: functions, measures, distributions, multisets, holomorphic and meromorphic functions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124606601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a topological ring R, we study semitopological R-modules, construct their completions, Bohr and borno modications. For every topological space X, we construct the free (semi)topological R-module over X and prove that for a k-space X its free semitopological R-module is a topological R-module. Also we construct a Tychono space X whose free semitopological R-module is not a topological R-module.
{"title":"SEMITOPOLOGICAL MODULES","authors":"T. Banakh, A. Ravsky","doi":"10.31861/bmj2021.01.01","DOIUrl":"https://doi.org/10.31861/bmj2021.01.01","url":null,"abstract":"Given a topological ring R, we study semitopological R-modules, construct their completions, Bohr and borno modications. For every topological space X, we construct the free (semi)topological R-module over X and prove that for a k-space X its free semitopological R-module is a topological R-module. Also we construct a Tychono space X whose free semitopological R-module is not a topological R-module.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121831956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Differential equations for moments and the generating function of number of transformations for branching process with continuous time and migration","authors":"H. Yakymyshyn, I. Bazylevych","doi":"10.31861/bmj2019.01.003","DOIUrl":"https://doi.org/10.31861/bmj2019.01.003","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121786901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By $mathcal{A}^2$ denote the class of analytic functions of the formBy $mathcal{A}^2$ denote the class of analytic functions of the form$f(z)=sum_{n+m=0}^{+infty}a_{nm}z_1^nz_2^m,$with {the} domain of convergence $mathbb{T}={z=(z_1,z_2)inmathbb C^2colon|z_1|<1, |z_2|<+infty}=mathbb{D}timesmathbb{C}$ and$frac{partial}{partial z_2}f(z_1,z_2)notequiv0$ in $mathbb{T}.$ In this paper we prove some analogue of Wiman's inequalityfor analytic functions $finmathcal{A}^2$. Let a function $hcolon mathbb R^2_+to mathbb R_+$ be such that$h$ is nondecreasing with respect to each variables and $h(r)geq 10$ for all $rin T:=(0,1)times (0,+infty)$and $iint_{Delta_varepsilon}frac{h(r)dr_1dr_2}{(1-r_1)r_2}=+infty$ for some $varepsilonin(0,1)$, where $Delta_{varepsilon}={(t_1, t_2)in Tcolon t_1>varepsilon, t_2> varepsilon}$.We say that $Esubset T$ is a set of asymptotically finite $h$-measure on ${T}$if $nu_{h}(E){:=}iintlimits_{EcapDelta_{varepsilon}}frac{h(r)dr_1dr_2}{(1-r_1)r_2}<+infty$ for some $varepsilon>0$. For $r=(r_1,r_2)in T$ and a function $finmathcal{A}^2$ denotebegin{gather*}M_f(r)=max {|f(z)|colon |z_1|leq r_1,|z_2|leq r_2},mu_f(r)=max{|a_{nm}|r_1^{n} r_2^{m}colon(n,m)in{mathbb{Z}}_+^2}.end{gather*}We prove the following theorem:{sl Let $finmathcal{A}^2$. For every $delta>0$ there exists a set $E=E(delta,f)$ of asymptotically finite $h$-measure on ${T}$ such that for all $rin (TcapDelta_{varepsilon})backslash E$ we have begin{equation*} M_f(r)leqfrac{h^{3/2}(r)mu_f(r)}{(1-r_1)^{1+delta}}ln^{1+delta} Bigl(frac{h(r)mu_f(r)}{1-r_1}Bigl)cdotln^{1/2+delta}frac{er_2}{varepsilon}. end{equation*}}
{"title":"WIMAN’S TYPE INEQUALITY FOR SOME DOUBLE POWER SERIES","authors":"A. Kuryliak, L. O. Shapovalovska, O. Skaskiv","doi":"10.31861/bmj2021.01.05","DOIUrl":"https://doi.org/10.31861/bmj2021.01.05","url":null,"abstract":"By $mathcal{A}^2$ denote the class of analytic functions of the formBy $mathcal{A}^2$ denote the class of analytic functions of the form$f(z)=sum_{n+m=0}^{+infty}a_{nm}z_1^nz_2^m,$with {the} domain of convergence $mathbb{T}={z=(z_1,z_2)inmathbb C^2colon|z_1|<1, |z_2|<+infty}=mathbb{D}timesmathbb{C}$ and$frac{partial}{partial z_2}f(z_1,z_2)notequiv0$ in $mathbb{T}.$ In this paper we prove some analogue of Wiman's inequalityfor analytic functions $finmathcal{A}^2$. Let a function $hcolon mathbb R^2_+to mathbb R_+$ be such that$h$ is nondecreasing with respect to each variables and $h(r)geq 10$ for all $rin T:=(0,1)times (0,+infty)$and $iint_{Delta_varepsilon}frac{h(r)dr_1dr_2}{(1-r_1)r_2}=+infty$ for some $varepsilonin(0,1)$, where $Delta_{varepsilon}={(t_1, t_2)in Tcolon t_1>varepsilon, t_2> varepsilon}$.We say that $Esubset T$ is a set of asymptotically finite $h$-measure on ${T}$if $nu_{h}(E){:=}iintlimits_{EcapDelta_{varepsilon}}frac{h(r)dr_1dr_2}{(1-r_1)r_2}<+infty$ for some $varepsilon>0$. For $r=(r_1,r_2)in T$ and a function $finmathcal{A}^2$ denotebegin{gather*}M_f(r)=max {|f(z)|colon |z_1|leq r_1,|z_2|leq r_2},mu_f(r)=max{|a_{nm}|r_1^{n} r_2^{m}colon(n,m)in{mathbb{Z}}_+^2}.end{gather*}We prove the following theorem:{sl Let $finmathcal{A}^2$. For every $delta>0$ there exists a set $E=E(delta,f)$ of asymptotically finite $h$-measure on ${T}$ such that for all $rin (TcapDelta_{varepsilon})backslash E$ we have begin{equation*} M_f(r)leqfrac{h^{3/2}(r)mu_f(r)}{(1-r_1)^{1+delta}}ln^{1+delta} Bigl(frac{h(r)mu_f(r)}{1-r_1}Bigl)cdotln^{1/2+delta}frac{er_2}{varepsilon}. end{equation*}}","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115699081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For entire functions $F$ and $G$ defined by Dirichlet series with exponents increasing to $+infty$ formulas are found for the finding the generalized order $displaystyle varrho_{alpha,beta}[F]_G = varlimsuplimits_{sigmato=infty} frac{alpha(M^{-1}_G(M_F(sigma)))}{beta(sigma)}$ and the generalized lower order $displaystyle lambda_{alpha,beta}[F]_G=varliminflimits_{sigmato+infty} frac{alpha(M^{-1}_G(M_F(sigma)))}{beta(sigma)}$ of $F$ with respect to $G$, where $M_F(sigma)=sup{|F(sigma+it)|:,tin{Bbb R}}$ and $alpha$ and $beta$ are positive increasing to $+infty$ functions.
{"title":"RELATIVE GROWTH OF ENTIRE DIRICHLET SERIES WITH DIFFERENT GENERALIZED ORDERS","authors":"M. Sheremeta, O. Mulyava","doi":"10.31861/bmj2021.02.02","DOIUrl":"https://doi.org/10.31861/bmj2021.02.02","url":null,"abstract":"For entire functions $F$ and $G$ defined by Dirichlet series with exponents increasing to $+infty$ formulas are found for the finding the generalized order $displaystyle varrho_{alpha,beta}[F]_G = varlimsuplimits_{sigmato=infty} frac{alpha(M^{-1}_G(M_F(sigma)))}{beta(sigma)}$ and the generalized lower order $displaystyle lambda_{alpha,beta}[F]_G=varliminflimits_{sigmato+infty} frac{alpha(M^{-1}_G(M_F(sigma)))}{beta(sigma)}$ of $F$ with respect to $G$, where $M_F(sigma)=sup{|F(sigma+it)|:,tin{Bbb R}}$ and $alpha$ and $beta$ are positive increasing to $+infty$ functions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125941270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a (cid:28)nite system of rays in terms of Fourier coe(cid:30)cients of their logarithmic derivative.
{"title":"Regular growth of Fourier coefficients of the logarithmic derivative of entire functions of improved regular growth","authors":"R. Khats","doi":"10.31861/bmj2019.01.114","DOIUrl":"https://doi.org/10.31861/bmj2019.01.114","url":null,"abstract":"We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a (cid:28)nite system of rays in terms of Fourier coe(cid:30)cients of their logarithmic derivative.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115331791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Bodnaruk, V. Gorodetskyi, R. Kolisnyk, N. Shevchuk
In the theory of fractional integro-differentiation the operator $A := displaystyle Big(I-frac{partial^2}{partial x^2}Big)$ is often used. This operator called the Bessel operator of fractional differentiation of the order of $ 1/2 $. This paper investigates the properties of the operator $B := displaystyle Big(I-frac{partial^2}{partial x^2}+frac{partial^4}{partial x^4}Big)$, which can be understood as a certain analogue of the operator $A$. It is established that $B$ is a self-adjoint operator in Hilbert space $L_2(mathbb{R})$, the narrowing of which to a certain space of $S$ type (such spaces are introduced in cite{lit_bodn_2}) matches the pseudodifferential operator $F_{sigma to x}^{-1}[a(sigma) F_{xto sigma}]$ constructed by the function-symbol $a(sigma) = (1+sigma^2+sigma^4)^{1/4}$, $sigma in mathbb{R}$ (here $F$, $F^{-1}$ are the Fourier transforms). This approach allows us to apply effectively the Fourier transform method in the study of the correct solvability of a nonlocal by time problem for the evolution equation with the specified operator. The correct solvability for the specified equation is established in the case when the initial function, by means of which the nonlocal condition is given, is an element of the space of the generalized function of the Gevrey ultradistribution type. The properties of the fundamental solution of the problem was studied, the representation of the solution in the form of a convolution of the fundamental solution of the initial function is given.
在分数阶积分微分理论中，经常使用算子$A := displaystyle Big(I-frac{partial^2}{partial x^2}Big)$。这个算子叫做分数阶微分的贝塞尔算子$ 1/2 $的阶。本文研究了算子$B := displaystyle Big(I-frac{partial^2}{partial x^2}+frac{partial^4}{partial x^4}Big)$的性质，它可以理解为算子$A$的某种类似物。建立了$B$是Hilbert空间$L_2(mathbb{R})$中的自共轭算子，将其缩小到某个$S$类型的空间(这种空间在cite{lit_bodn_2}中介绍)与由函数符号$a(sigma) = (1+sigma^2+sigma^4)^{1/4}$、$sigma in mathbb{R}$(这里$F$、$F^{-1}$是傅里叶变换)构造的伪微分算子$F_{sigma to x}^{-1}[a(sigma) F_{xto sigma}]$相匹配。这种方法使我们能够有效地应用傅里叶变换方法来研究具有特定算子的演化方程的非局部随时间问题的正确可解性。当给定非局部条件的初始函数是Gevrey超分布型广义函数空间中的一个元素时，建立了给定方程的正确可解性。研究了该问题的基本解的性质，给出了其解的卷积形式。
{"title":"NONLOCAL BY TIME PROBLEM FOR SOME DIFFERENTIAL-OPERATOR EQUATION IN SPACES OF S AND S TYPES","authors":"S. Bodnaruk, V. Gorodetskyi, R. Kolisnyk, N. Shevchuk","doi":"10.31861/bmj2021.02.04","DOIUrl":"https://doi.org/10.31861/bmj2021.02.04","url":null,"abstract":"In the theory of fractional integro-differentiation the operator $A := displaystyle Big(I-frac{partial^2}{partial x^2}Big)$ is often used. This operator called the Bessel operator of fractional differentiation of the order of $ 1/2 $. This paper investigates the properties of the operator $B := displaystyle Big(I-frac{partial^2}{partial x^2}+frac{partial^4}{partial x^4}Big)$, which can be understood as a certain analogue of the operator $A$. It is established that $B$ is a self-adjoint operator in Hilbert space $L_2(mathbb{R})$, the narrowing of which to a certain space of $S$ type (such spaces are introduced in cite{lit_bodn_2}) matches the pseudodifferential operator $F_{sigma to x}^{-1}[a(sigma) F_{xto sigma}]$ constructed by the function-symbol $a(sigma) = (1+sigma^2+sigma^4)^{1/4}$, $sigma in mathbb{R}$ (here $F$, $F^{-1}$ are the Fourier transforms).\u0000\u0000This approach allows us to apply effectively the Fourier transform method in the study of the correct solvability of a nonlocal by time problem for the evolution equation with the specified operator. The correct solvability for the specified equation is established in the case when the initial function, by means of which the nonlocal condition is given, is an element of the space of the generalized function of the Gevrey ultradistribution type. The properties of the fundamental solution of the problem was studied, the representation of the solution in the form of a convolution of the fundamental solution of the initial function is given.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115552572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The concept of parabolicity by Shilov generalizes the concept of parabolicity by Petrovsky of equations with partial derivatives and leads to a significant expansion of the known Petrovsky class with those parabolic equations, the order of which may not coincide with the parabolicity index. Generally speaking, such an extension deprives of the parabolic stability сoncerning the change of the coefficients of parabolic Shilov equations, which is inherent to the Petrovsky class equations. As a result, significant difficulties arise in the study of the Cauchy problem for parabolic Shilov equations with variable coefficients. In the 60s of the last century, Y.I. Zhytomyrsky defined a special class of parabolic Shilov equations, which extends the Shilov class and at the same time is parabolically resistant to changes in the junior coefficients. For this class, by the method of successive approximations, he established the correct solvability of the Cauchy problem in the class of bounded initial functions of finite smoothness. However, to obtain more general results, it is important to know the Green’s function of the Cauchy problem. In this publication, for parabolic Shilov equations with bounded smooth variable coefficients and negative genus, estimates of repeated kernels of the Green’s function of the Cauchy problem are established, which allow us to investigate the properties of the density of volume potential of this function. These results are important for the development of the Cauchy problem theory for parabolic Shilov equations by classical means of the Green’s function.
{"title":"REPEATED KERNELS OF THE GREEN’S FUNCTION OF PARABOLIC SHILOV EQUATIONS WITH VARIABLE COEFFICIENTS AND NEGATIVE GENUS","authors":"V. Litovchenko, D. Kharyna","doi":"10.31861/bmj2022.01.07","DOIUrl":"https://doi.org/10.31861/bmj2022.01.07","url":null,"abstract":"The concept of parabolicity by Shilov generalizes the concept of parabolicity by Petrovsky of equations with partial derivatives and leads to a significant expansion of the known Petrovsky class with those parabolic equations, the order of which may not coincide with the parabolicity index. Generally speaking, such an extension deprives of the parabolic stability сoncerning the change of the coefficients of parabolic Shilov equations, which is inherent to the Petrovsky class equations. As a result, significant difficulties arise in the study of the Cauchy problem for parabolic Shilov equations with variable coefficients. In the 60s of the last century, Y.I. Zhytomyrsky defined a special class of parabolic Shilov equations, which extends the Shilov class and at the same time is parabolically resistant to changes in the junior coefficients. For this class, by the method of successive approximations, he established the correct solvability of the Cauchy problem in the class of bounded initial functions of finite smoothness. However, to obtain more general results, it is important to know the Green’s function of the Cauchy problem.\u0000In this publication, for parabolic Shilov equations with bounded smooth variable coefficients and negative genus, estimates of repeated kernels of the Green’s function of the Cauchy problem are established, which allow us to investigate the properties of the density of volume potential of this function. These results are important for the development of the Cauchy problem theory for parabolic Shilov equations by classical means of the Green’s function.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116127038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Such problems are mainly ill possed in the Hadamard sence, and their solvability is connected with the problem of small denominators. A specific feature of the present work is the study of a nonlocal boundary-value problem for partial differential equations with the operator of the generalized differentiation $B=zd/dz$, which operate on functions of scalar complex variable $z$. A criterion for the unique solvability of these problems and a sufficient conditions for the existence of its solutions are established in the spaces of functions, which are Dirichlet-Taylor series. The unity theorem and existence theorems of the solution of problem in these spaces are proved. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the article shown that in the case of one variable the corresponding denominators are not small and are estimated from below by some constants. Correctness after Hadamard of the problem is shown. It distinguishes it from an illconditioned after Hadamard problem with many spatial variables.
{"title":"NONLOCAL BOUNDARY VALUE PROBLEM IN SPACES OF EXPONENTIAL TYPE OF DIRICHLET-TAYLOR SERIES FOR THE EQUATION WITH COMPLEX DIFFERENTIATION OPERATOR","authors":"V. Il'kiv, N. Strap, I. Volyanska","doi":"10.31861/bmj2022.02.04","DOIUrl":"https://doi.org/10.31861/bmj2022.02.04","url":null,"abstract":"Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Such problems are mainly ill possed in the Hadamard sence, and their solvability is connected with the problem of small denominators. A specific feature of the present work is the study of a nonlocal boundary-value problem for partial differential equations with the operator of the generalized differentiation $B=zd/dz$, which operate on functions of scalar complex variable $z$. A criterion for the unique solvability of these problems and a sufficient conditions for the existence of its solutions are established in the spaces of functions, which are Dirichlet-Taylor series. The unity theorem and existence theorems of the solution of problem in these spaces are proved. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the article shown that in the case of one variable the corresponding denominators are not small and are estimated from below by some constants. Correctness after Hadamard of the problem is shown. It distinguishes it from an illconditioned after Hadamard problem with many spatial variables.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129074402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：