On estimates of solutions to systems of differential equations of neutral type with periodic coefficients

We study the systems of differential equations of neutral type with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution and obtain estimates for solutions which characterize the decay rate at infinity.     

Estimates for normal solutions of problems with zone controls in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> for the wave equation

For the wave equation with variable coefficients and homogeneous boundary conditions of the first kind, we consider problems with regular zone controls and dual zone observation problems. For weak generalized solutions of the observation problem on sufficiently large time intervals, we obtain constructive estimates that imply the well-posed solvability of the observation operator. These estimates contain information that permits one to construct stable approximate solutions of both problems with the use of a variational method suggested earlier by the author for linear equations with nonuniformly perturbed operators.     

Stability of solutions to differential equations with several variable delays. I

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.     

On the solvability of the boundary-value problem for second-order equations in Hilbert space with an operator coefficient in the boundary condition

We consider the boundary-value problem on a finite interval for a class of second-order operator-differential equations with a linear operator in one of its boundary conditions. We obtain sufficient conditions for the regular solvability of the boundary-value problem under consideration; these conditions are expressed only in terms of its operator coefficients.     

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

Boundary decay estimates for solutions of fourth-order elliptic equations

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.     

Boundary-Value Problems for a Class of Operator Differential Equations of High Order with Variable Coefficients

In this paper, we obtain sufficient conditions for the existence of a unique regular solution of the boundary-value problems for operator differential equations of order 2k with variable coefficients. These conditions are expressed solely in terms of operator coefficients of the equations under study.     

Estimate for Euclidean parameters of a mixture of two symmetric distributions

A sample from a mixture of two symmetric distributions is observed. The considered distributions differ only by a shift. Estimates are constructed by the method of estimating equations for parameters of mean locations and concentrations (mixing probabilities) of both components. We obtain conditions for the asymptotic normality of these estimates. The greatest lower bounds for the coefficients of dispersion of the estimates are determined.     

Quantitative Stochastic Homogenization of Viscous Hamilton-Jacobi Equations

We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian.     

On estimates for the Besov norms of solutions to 3D subelliptic equations

We consider a class of linear subelliptic equations for functions of three variables. We obtain estimates, similar to the Schauder estimates, for the Besov norms of the derivatives of weak solutions to the equations under consideration with coefficients in suitable Besov spaces.     

Estimates of solutions to a class of systems of nonlinear delay differential equations

Under study are the systems of nonlinear delay differential equations with periodic coefficients of the linear terms. Some sufficient conditions for the asymptotic stability of the zero solution are established. We obtain the estimates that characterize the decay rate of solutions at infinity and describe the attraction sets of the zero solution.     

On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.     

Martingale problems for some degenerate Kolmogorov equations

We obtain Calderón–Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.     

Estimates of solutions to the linear differential equations of neutral type with several delays of the argument

Under study is a systemof linear differential equations of neutral type with several delays of the argument. We obtain the conditions on the matrix coefficients of the system under which all solutions decrease with an exponential rate at infinity. Using some functionals of Lyapunov-Krasovskii type, the uniform estimates of solutions are established.     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion,discontinuous coefficients,and a partly degenerate diffusion operator

We obtain estimates for functionals of solutions of stochastic differential equations with standard and fractional Brownian motion. We prove a theorem on the existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion, discontinuous coefficients, and a partly degenerate diffusion operator.     

Estimates for the embedding operator of a sobolev space of periodic functions and for the solutions of differential equations with periodic coefficients

We obtain estimates for the embedding operator of a Sobolev space in the space of continuous periodic functions and use them to estimate the solutions of differential equations with periodic coefficients. We prove a theorem on a necessary and sufficient condition for the invertibility of a differential operator with unbounded operator coefficients.     

Asymptotics of the eigenvalues of a boundary value problem for a vector singular differential equation

By analyzing a system of integro-differential equations of the Volterra-Stieltjes type, for large parameter values, we obtain asymptotic formulas for a linearly independent system of solutions of ordinary differential equations with generalized functions in coefficients. These solutions permit one to construct asymptotic formulas for the eigenvalues of a boundary value problem in the case of regular boundary conditions.     

On Stability of Solutions to Quasilinear Periodic Systems of Differential Equations

We consider a quasilinear system of differential equations with periodic coefficients in the linear terms. We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity. The results are stated in terms of the integrals of the norm of a periodic solution to the Lyapunov differential equation.