arXiv subjects
Publications and source records attributed to Kamal N. Soltanov.
This article examines the smoothness of the solution to the Navier-Stokes equation from a novel perspective. Here, the existence of the smoother solution relative to x and to the time t was shown only for a finite time. Moreover, for each considered case of the problem, a blowup time for its solutions can be demonstrated.
This article studies the uniqueness of the weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case. Here, the investigation is provided using two different approaches. The first (the main) result is obtained for given functions possessing a certain smoothness using the new approach. The second result is without the complementary conditions but is, in some sense, the "local" result investigated by another approach. In addition, here the solvability and uniqueness of the weak solutions to auxiliary problems lead out from the main problem are investigated.
Here we consider a perturbation of continuous mappings on Banach spaces and investigate their image under various conditions. Consequently, we study the solvability of some classes of equations and inclusions. For these, we start by the investigating of local properties of the considered mapping and local comparing this mapping with certain smooth mappings. Moreover, we study different mixed problems.
In this article, we study the properties of a class of functional spaces which arise from the investigation of nonlinear differential equations. We establish some integral inequalities then by applying these inequalities, we prove some lemmas and theorems, which indicate the relation of these spaces (pn-spaces) with the Lebesgue and Sobolev spaces in the case when pn-spaces with constant and variable exponents.
In this article, the existence of the spectrum (the eigenvalues) for the nonlinear continuous operators acting in the Banach spaces is investigated. For the study, this question is used a different approach that allows the studying of all eigenvalues of the nonlinear operator relative to another nonlinear operator. Here shows that in nonlinear operators, case is necessary to seek the spectrum of the given nonlinear operator relative to another nonlinear operator satisfying certain conditions. The different examples, for which eigenvalues can be found are provided. Moreover, the nonlinear problems including parameters are studied.
In this work, using a new geometrical approach we study to the existence of the fixed-point of mappings that independence of the smoothness, and also of their single-values or multi-values. This work proved the theorems that generalize in some sense the Brouwer and Schauder fixed-point theorems, and also such type results in multi-valued cases. One can reckon this approach is based on the generalization of the one theorem Cauchy and on the convexity properties of sets. As the used approach is based on the geometry of the image of the examined mappings that are independent of the topological properties of the space we could to prove the general results for almost every vector space. The general results we applied to the study of the nonlinear equations and inclusions in VTS, and also by applying these results are investigated different concrete nonlinear problems. Here provided also sufficient conditions under which the conditions of the theorems will fulfill.
In this paper, we propose a new mathematical model nonlinear reaction-diffusion PDE's describing the dynamics of propagation of cancer. Here the mixed problem for the proposed PDE's is investigated and by applying obtained results conclusions on the dynamics of propagation of cancer are drawn. These problems have nonlocal nonlinearity with variable exponents and possess special properties: these can be to remain either dissipative all time or become non-dissipative after a finite time. Here the solvability and behavior of solutions both when problems are yet dissipative and when become nondissipative are proved. It is shown that if the studied process gets become nondissipative can have various states, e.g. an infinite number of different unstable solutions with varying speeds, in addition, their propagation can become chaotic. The behavior of these solutions is analyzed in detail and it is explained how space-time chaos can arise. Investigation of this mathematics model allows explaining the dynamics of propagation of cancer, which are provided here as conclusions for each case.
This article proposed a new approach to the determination of the spectrum for nonlinear continuous operators in the Banach spaces and using it investigated the spectrum of some classes of operators. Here shows that in nonlinear operators case is necessary to seek the spectrum of a given nonlinear operator relatively to another nonlinear operator. Moreover, the order of nonlinearity of examined operator and operator relatively to which seek the spectrum must be identical. Here provided different examples relative to how one can find the eigenvalue and also studied solvability problems.
In this paper, we study the mixed problem for new class of nonlinear reaction-diffusion PDEs with the nonlocal nonlinearity with variable exponents. Here we obtain results on solvability and behavior of solutions both when these are yet dissipative and when these get become non-dissipative problems. These problems are possess special properties: these can be to remain dissipative all time or can get become non dissipative after finite time. It is shown that if the studied problems get become non-dissipative can have an infinite number of different unstable solutions with varying speeds and also an infinite number of different states of spatio-temporal (diffusion) chaos that are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and it is explained how space-time chaos can arise. Since these problems desctibe the dynamics of cancer, we explain the behavior of the dynamics of cancer using obtained here results.
In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity for the considered problem is proved for given functions from spaces that posseses some smoothness. Moreover, these spaces are dense in respective spaces of functions, under which were proved existence of the weak solutions. In addition here the solvability and uniqueness of the weak solutions of auxiliary problems associated with the main problem is investigated, and also one conditional result on uniqueness is proved.
In this article we study the uniqueness of the weak solution of the incompressible Navier-Stokes Equation in the 3-dimensional case with use of different approach. Here the uniqueness of the obtained by Leray of the weak solution is proved in the case, when datums from spaces that are densely contained into spaces of datums for which was proved the existence of the weak solution. Moreover we investigate the solvability and uniqueness of the weak solutions of problems associated with investigation of the main problem
Here we study the abstract nonlinear differential equation of second order that in special case is the equation of the type of equation of traffic flow. We prove the solvability theorem for the posed problem under the appropriate conditions and also investigate the behaviour of the solution.
Here we study the nonlinear hyperbolic equations of the type of equations from theory of flows on networks, for which we prove the solvability theorem under the appropriate conditions and also investigate the behaviour of the solution.
Here we investigate 3-dimensional Navier-Stokes Equations in the incompressible case with use of different approach and we prove the uniqueness of the weak solutions for the data from the space, which is dense in usual space of data. Moreover we study the solvability and uniqueness of the weak solutions of problems associated with investigation of the main problem.
In this paper we study a mixed problem for the nonlinear Schrödinger equation globally that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a global solvability theorem of the considered problem with use of the general solvability theorem of the article [30]. Furthermore here is investigated also the behaviour of the solution of the studied problem.
Here a mixed problem for a nonlinear hyperbolic equation with Neumann boundary value condition is investigated, and a priori estimations for the possible solutions of the considered problem are obtained. These results demonstrate that any solution of this problem possess certain smoothness properties.
In this article we discuss the solvability of some class of fully nonlinear equations, and equations with p-Laplacian in more general conditions by using a new approach given in [1] for studying the nonlinear continuous operator. Moreover we reduce certain general results for the continuous operators acting on Banach spaces, and investigate their image. Here we also consider the existence of a fixed-point of the continuous operators under various conditions.
The mixed problem for the implicit degenerating nonlinear parabolic equation is considered, and the solvability and behavior of solutions of this problem are studied. Furthermore, some classes of function spaces and their relations with Sobolev spaces are investigated, embedding and compactness theorems for these spaces are proved