Problem without initial conditions for linear and almost linear degenerate operator differential equations

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.     

A penalty continuation method for the ∓∞ solution of overdetermined linear systems

A new algorithm for the ∓_∞ solution of overdetermined linear systems is given. The algorithm is based on the application of quadratic penalty functions to a primal linear programming formulation of the ∓_∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ∓_∞ solutions. It is shown that the entire set of ∓_∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ∓_∞ problems. The algorithm is implemented and extensively tested using random and function approximation problems. Comparisons with the Barrodale-Phillips simplex based algorithm and the more recent predictor-corrector primal-dual interior point algorithm are given. The results indicate that the new algorithm shows a promising performance on random (non-function approximation) problems.     

One-Dimensional Riemann Problem for Equations of Constant Pressure Fluid Dynamics with Measure Solutions by the Viscosity Method

The Riemann problem for the equations of constant pressure fluid dynamics was considered. Solutions of this problem were constructed by employing the viscosity vanishing approach. For some initial data, solutions can be viewed as bounded functions in L(R × R+) plus bounded linear functionals on C0(R× R+) with nonclassical waves as their supports. Vacuum regions appear so that uniqueness of Riemann solutions fails for some initial data.     

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

Bubbles and droplets in nonlinear physics models: Analysis and numerical simulation of singular nonlinear boundary value problem

For a second-order nonlinear ordinary differential equation (ODE), a singular Boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble-type (droplet-type) solutions are sought. The equation, defined on a semi-infinite interval 0 < r < ∞, possesses a regular singular point as r→ 0 and an irregular one as r→ ∞. We give the restrictions to the parameters for a correct mathematical statement of the limit boundary conditions in singular points and their accurate transfer into the neighborhoods of these points using certain results for singular Cauchy problems and stable initial manifolds. The necessary and sufficient conditions for the existence of bubble-type (droplet-type) solutions are discussed (in the form of additional restrictions to the parameters) and some estimates are obtained. A priori detailed analysis of a singular nonlinear BVP leads to efficient shooting methods for solving it approximately. Some results of the numerical experiments are displayed and their physical interpretation is discussed. This article was submitted by the author in English.     

Concentration Problems for Bandpass Filters in Communication Theory over Disjoint Frequency Intervals and Numerical Solutions

The concentration problem of maximizing signal strength of bandlimited and timelimited nature is important in communication theory. In this paper we consider two types of concentration problems for the signals which are bandlimited in disjoint frequency-intervals, which constitute a band-pass filter. For the first type the problem is to determine which members of L ²(−∞,∞) lose the smallest fraction of their energy when first timelimited and then bandlimited. For the second type the problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted to a given time interval. For both types of problems, basic theoretical properties and numerical algorithms for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L ²(−∞,∞) are also proved. Numerical computations are carried out which corroborate the theory. Relationship between eigenvalues of these two types of problems is also established. Several properties of eigenvalues of both types of problems are proved.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

Regularized Conservation Laws and Nonlinear Geometric Optics

 This paper is devoted to the study of Cauchy problems for regularized conservation laws in Colombeau algebras of generalized functions. The existence and uniqueness of generalized solutions to these Cauchy problems are obtained. Further, we develop a generalized variant of nonlinear geometric optics for the regularized problems. Consistency with the classical results is shown to hold for scalar conservation laws with bounded variation initial data in one space variable. Received 6 November 1996; in revised form 5 August 1997     

Existence and stability of a summable generalized solution to a mathematical model of a catalytic fluidized bed process

For a system of first-order partial differential equations describing a catalytic process in a fluidized bed, we consider a mixed problem in the half-strip 0 ≤ x ≤ h, t ≥ 0. We prove the existence and uniqueness of a bounded summable generalized solution and study its stability. We prove the stabilization as t → ∞ of the values of some physically meaningful functionals of the solution.     

On solvability of an initial-boundary-value problem for equations of magnetohydrodynamics

In this paper, we consider a nonstationary problem of magnetohydrodynamics (MHD) for viscous incompressible fluid under the condition that the medium is poorly conducting. The problem is analyzed in a bounded one-connected domain Ω ⊂ ℝⁿ, n = 2,3, for t > 0 under the condition of ideal conductivity on the boundary. We prove a theorem on the unique solvability of the problem “in the small,” on a small time interval, and on a given time interval ]0, T[ (including T = +∞) when the given data of the problem are sufficiently small (precise formulations are given in Sect. 2). To investigate the nonlinear problem, several auxiliary linear problems are preliminarily considered. The results of this paper were announced by the author in the Trakai Conference on Mathematical Modeling and Analysis in spring of 2005. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 234–279, April–June, 2007.     

The problem of thermo-chemical formation of a composite material. Properties of solutions and homogenization

We study a problem with rapidly oscillating coefficients which arises in describing the process of thermo-chemical formation of a composite material. We homogenize this problem and study the existence and uniqueness of solutions to the original and homogenized problems, as well as properties of the solutions. We estimate an error in homogenization with order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) in the energy norm and with order O(ε) in the L ^∞-norm. Bibliography: 10 titles.     

Nonuniqueness for a bounded, differentiable complete orthonormal system inL 2[0, 1]

The argument of Müsegian and Ovsepjan is adapted to produce a complete orthonormal system on [0, 1] of uniformly bounded functions, differentiable on [0, 1], andC ^∞ on [0, 1], for which the analogue of Cantor's uniqueness theorem is false. We also construct a complete orthonormal system ofC ^∞ functions which vanish to infinite order at both endpoints.     

On a class of second order variational problems with constraints

We study the structure of optimal solutions for a class of constrained, second order variational problems on bounded intervals. We show that, for intervals of length greater than some positive constant, the optimal solutions are bounded inC ¹ by a bound independent of the length of the interval. Furthermore, for sufficiently large intervals, the ‘mass’ and ‘energy’ of optimal solutions are almost uniformly distributed.     

A boundedness theorem for a certain fourth order differential equation

Summary The paper investigates conditions under which solutions of equations of the form(1.1) are bounded as t→+∞. Entrata in Redazione il 31 agosto 1970.     

A note on the boundedness and the stability of solutions of certain third-order differential equations

Summary This paper establishes conditions under which solutions of equations of the form (1.1) are bounded or tend to zero as t → ∞. Entrata in Redazione il 14 aprile 1971.     

A capacitary method for the asymptotic analysis of dirichlet problems for monotone operators

Given a non-linear elliptic equation of monotone type in a bounded open set Ω ⊂ Rⁿ, we prove that the asymptotic behaviour, asj → ∞, of the solutions of the Dirichlet problems corresponding to a sequence (Ω_j) of open sets contained in Ω is uniquely determined by the asymptotic behaviour, asj → ∞, of suitable non-linear capacities of the setsKΩ _j, whereK runs in the family of all compact subsets of Ω.     

On the uniqueness of solutions to some boundary-value problems for differential equations in a domain with algebraic boundary

Classes of differential equations with constant coefficients admitting unique solutions of Dirichlet and Cauchy boundary-value problems are considered in a bounded domain with algebraic boundary. For the Dirichlet problem in a ball, the necessary and sufficient conditions for the uniqueness of the solution are obtained in the form of a countable sequence of inequalities polynomial in the coefficients of the equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 898–906, July, 1993.