In this paper, we propose a new mathematical model nonlinear reaction-diffusion PDE’s describing ... more 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 expl...
This article proposed a new approach to the determination of the spectrum for nonlinear continuou... more 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 iegenvalue and also studied solvability problems.
In this article we study the uniqueness of the weak solution of the incompressible Navier-Stokes ... more 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
In this paper we study a mixed problem for the nonlinear Schr\"odinger equation globally tha... more In this paper we study a mixed problem for the nonlinear Schr\"odinger 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.
In this article we discuss the solvability of some class of fully nonlinear equations, and equati... more 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.
In this article, we obtain existence and uniqueness results to some problems involving complex no... more In this article, we obtain existence and uniqueness results to some problems involving complex nonlinear fractional differential equations (FDEs) in the closed unit disc of C. By help of these results, we prove that some IVPs for some fractional differential equations with Caputo or Riemann-Liouville derivative admit at least one local (or unique) solution continuous on a closed interval [0,R] and real analytic on (0,R), where 0<R≤ 1.
There is developed a differential-algebraic approach to studying the representations of commuting... more There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg-de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.
In this paper, we propose a new mathematical model nonlinear reaction-diffusion PDE’s describing ... more 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 expl...
This article proposed a new approach to the determination of the spectrum for nonlinear continuou... more 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 iegenvalue and also studied solvability problems.
In this article we study the uniqueness of the weak solution of the incompressible Navier-Stokes ... more 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
In this paper we study a mixed problem for the nonlinear Schr\"odinger equation globally tha... more In this paper we study a mixed problem for the nonlinear Schr\"odinger 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.
In this article we discuss the solvability of some class of fully nonlinear equations, and equati... more 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.
In this article, we obtain existence and uniqueness results to some problems involving complex no... more In this article, we obtain existence and uniqueness results to some problems involving complex nonlinear fractional differential equations (FDEs) in the closed unit disc of C. By help of these results, we prove that some IVPs for some fractional differential equations with Caputo or Riemann-Liouville derivative admit at least one local (or unique) solution continuous on a closed interval [0,R] and real analytic on (0,R), where 0<R≤ 1.
There is developed a differential-algebraic approach to studying the representations of commuting... more There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg-de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.
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