. We consider the Cauchy problem for a certain class of anisotropic parabolic second-order equations with double non-power nonlinearities. The equation contains an “inhomogeneity” in the form of a non-divergent term depending on the sought function and spatial variables. Non-linearities are characterized by N -functions, for which ∆ 2 -condition is not imposed. The uniqueness of renormalized solutions in Sobolev-Orlich spases is proved by the S.N.Kruzhkov method of doubling the variables.
{"title":"Uniqueness of the renormalized solutions to the Cauchy problem for an anisotropic parabolic equation","authors":"Farit Khamzaevich Mukminov","doi":"10.13108/2016-8-2-44","DOIUrl":"https://doi.org/10.13108/2016-8-2-44","url":null,"abstract":". We consider the Cauchy problem for a certain class of anisotropic parabolic second-order equations with double non-power nonlinearities. The equation contains an “inhomogeneity” in the form of a non-divergent term depending on the sought function and spatial variables. Non-linearities are characterized by N -functions, for which ∆ 2 -condition is not imposed. The uniqueness of renormalized solutions in Sobolev-Orlich spases is proved by the S.N.Kruzhkov method of doubling the variables.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"20 1","pages":"44-57"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90500334","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}
In the work we consider the scheme of constructing exact solutions to the Sine-Gordon equation based on a restricting the characteristic Lie ring. We study in details the case when the dimension of the space formed by commutators of length 6 is equal to 1.
{"title":"Construction of exact solution to sine-Gordon equation on the base of its characteristic Lie ring","authors":"A. V. Zhiber, Sabina Nazirovna Kamaeva","doi":"10.13108/2016-8-3-49","DOIUrl":"https://doi.org/10.13108/2016-8-3-49","url":null,"abstract":"In the work we consider the scheme of constructing exact solutions to the Sine-Gordon equation based on a restricting the characteristic Lie ring. We study in details the case when the dimension of the space formed by commutators of length 6 is equal to 1.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"5 1","pages":"49-57"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79111638","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":"On solvability by quadratures conditions for second order hyperbolic systems","authors":"E. A. Sozontova","doi":"10.13108/2016-8-3-130","DOIUrl":"https://doi.org/10.13108/2016-8-3-130","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"270 1","pages":"130-135"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82796678","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}
Stationary harmonic functions on homogeneous spaces are considered. A relation to double periodic harmonic functions of three variables is showed.
研究齐次空间上的平稳调和函数。给出了三元双周期调和函数的一个关系式。
{"title":"Stationary harmonic functions on homogeneous spaces","authors":"Khoroshchak Vasylyna Stepanivna, Kondratyuk Andriy Andriyovych","doi":"10.13108/2015-7-4-149","DOIUrl":"https://doi.org/10.13108/2015-7-4-149","url":null,"abstract":"Stationary harmonic functions on homogeneous spaces are considered. A relation to double periodic harmonic functions of three variables is showed.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"39 1","pages":"155-159"},"PeriodicalIF":0.5,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77060667","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}
In the paper we propose an entire function such that the logarithm of its modulus asymptotically approximates the given subharmonic function (Re z ), where is the Legendre transformation of a convex function ℎ(t ) on (−1; 1). Such functions have applications in the issues on representation by exponential series of functions in integral weighted spaces on the interval (−1; 1) with the weight exp ℎ(t ). At that, better the ap- proximation, a finer topology can be used for the representation by exponential series. For functions ℎ obeying (1 − |t |) n = �� (exp(ℎ(t ))), n ∈ N, the corresponding entire func- tions were constructed before. In the present paper we consider the functions satisfying exp(ℎ(t )) = o ((1 − |t |) n ), n ∈ N. In the suggested construction we take into considera- tion the necessary conditions for the distribution of exponents for the exponentials in the unconditional bases obtained in previous works. This is why the main result of the paper (Theorem 1) should be treated not as a tool for constructing unconditional bases but as an argument supporting the absence of such bases.
{"title":"ENTIRE FUNCTIONS WITH FINE ASYMPTOTIC ESTIMATES FOR CONVEX FUNCTIONS","authors":"K. P. Isaev, R. S. Yulmukhametov, A. A. Yunusov","doi":"10.13108/2014-6-2-35","DOIUrl":"https://doi.org/10.13108/2014-6-2-35","url":null,"abstract":"In the paper we propose an entire function such that the logarithm of its modulus asymptotically approximates the given subharmonic function (Re z ), where is the Legendre transformation of a convex function ℎ(t ) on (−1; 1). Such functions have applications in the issues on representation by exponential series of functions in integral weighted spaces on the interval (−1; 1) with the weight exp ℎ(t ). At that, better the ap- proximation, a finer topology can be used for the representation by exponential series. For functions ℎ obeying (1 − |t |) n = �� (exp(ℎ(t ))), n ∈ N, the corresponding entire func- tions were constructed before. In the present paper we consider the functions satisfying exp(ℎ(t )) = o ((1 − |t |) n ), n ∈ N. In the suggested construction we take into considera- tion the necessary conditions for the distribution of exponents for the exponentials in the unconditional bases obtained in previous works. This is why the main result of the paper (Theorem 1) should be treated not as a tool for constructing unconditional bases but as an argument supporting the absence of such bases.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"2012 1","pages":"35-43"},"PeriodicalIF":0.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73950747","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}
In this paper we study the connection between the growth of coefficients of an exponentials series with its convergence domain in finite-dimensional real and complex spaces. Among the first results of the subject is the well-known Cauchy-Hadamard formula. We obtain exact conditions on the exponentials and a convex region in which there is a generalization of the Cauchy-Hadamard theorem. To the sequence of coefficients of exponential series we associate a space of sequences forming a commutative ring with unit. A study of the properties of this ring allows us to obtain the results on solvability of non- homogeneous systems of convolution equations.
{"title":"CAUCHY-HADAMARD THEOREM FOR EXPONENTIAL SERIES","authors":"S. G. Merzlyakov","doi":"10.13108/2014-6-1-71","DOIUrl":"https://doi.org/10.13108/2014-6-1-71","url":null,"abstract":"In this paper we study the connection between the growth of coefficients of an exponentials series with its convergence domain in finite-dimensional real and complex spaces. Among the first results of the subject is the well-known Cauchy-Hadamard formula. We obtain exact conditions on the exponentials and a convex region in which there is a generalization of the Cauchy-Hadamard theorem. To the sequence of coefficients of exponential series we associate a space of sequences forming a commutative ring with unit. A study of the properties of this ring allows us to obtain the results on solvability of non- homogeneous systems of convolution equations.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"229 1","pages":"71-79"},"PeriodicalIF":0.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77698545","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":"On Fourier transformation of a class of entire functions","authors":"Ildar Khamitovich Musin, M. Musin","doi":"10.13108/2014-6-4-108","DOIUrl":"https://doi.org/10.13108/2014-6-4-108","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"12 1","pages":"108-121"},"PeriodicalIF":0.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76912047","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}
In the work we describe the structure of integrals of systems of discrete equations. We consider an example of discrete Toda chain corresponding to Lie algebra of series A2.
{"title":"On structure of integrals for systems of discrete equations","authors":"M. Yangubaeva","doi":"10.13108/2014-6-1-111","DOIUrl":"https://doi.org/10.13108/2014-6-1-111","url":null,"abstract":"In the work we describe the structure of integrals of systems of discrete equations. We consider an example of discrete Toda chain corresponding to Lie algebra of series A2.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"5 1","pages":"111-116"},"PeriodicalIF":0.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84731871","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}
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