Abstract Based on earlier-established necessary nonlocal conditions that must be satisfied by all regular solutions of a fractional diffusion–wave equation in a rectangular domain, we propose a method for determining the well-posedness of nonlocal boundary value problems for this equation and a method for finding their solutions.

Abstract For an integro-differential equation of the convolution type defined on the half-line \([0,\infty ) \) with a power nonlinearity and an inhomogeneity in the linear part, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space \(C[0,\infty ) \). It is shown that the solution can be found by a

Abstract Bitsadze–Samarskii boundary conditions are obtained for a regular elliptic-parabolic volume potential given in a cylindrical domain based on the spectral decomposition of Green’s function of the Holmgren problem.

Abstract The notions of semilalgebra and its associated subalgebra are introduced. Linear spaces of row-column, skew-angle, and acute-angle matrices that are subalgebras or semialgebras of the full matrix algebra are also defined. Conditions for the splittability and solvability by quadratures of a matrix linear ordinary differential equation of the first order with two-sided multiplication \(dX/dt=\sum

Abstract In the Cartesian product \(\mathbb {R}^n\times (0,+\infty ) \), we consider an integro-differential heat equation with an integral term of the convolution type on the right-hand side. The direct problem is the Cauchy problem about determining the temperature of a medium given a known initial heat distribution (for the zero value of the time variable \(t \)). The inverse problem consists in

Abstract We consider an operator semigroup arising in the study of an integro-differential equation modeling small transverse vibrations of a viscoelastic pipeline with allowance for Kelvin–Voigt damping. The explicit form of the generator of this semigroup is established, its spectrum is studied, and the norm of its resolvent is estimated. Based on these results, the analyticity of this semigroup

Abstract For the beam vibration equation, we study the inverse problems of finding the right-hand side (the source of vibrations) and the initial conditions. The solutions of the problems are constructed in closed form as sums of series, and the corresponding existence and uniqueness theorems are proved. The problem of small denominators arises when justifying the convergence of the series. In this

Abstract In the Cartesian product \((\mathbb {R}^n\setminus B)\times \mathbb {R} \), where \(B \) is a ball in \(\mathbb {R}^n \), \(n\geq 3\), we consider a system of two semilinear parabolic equations with bounded measurable time-periodic coefficients and with power nonlinearities. Exact conditions on the nonlinearity exponents under which this system does not have global positive time-periodic solutions

Abstract Boundary value problems for a second-order ordinary differential equation with a nonsmooth potential are considered. An asymptotic representation for an operator semigroup is established based on a previously derived theorem on the eigenvalue asymptotics for the Hill operator. This semigroup is used to describe weak solutions of a mixed problem for a parabolic equation.

Abstract For a two-dimensional hyperbolic equation with a differential-difference operator acting with respect to the spatial variable, we construct a one-parameter family of global solutions. We prove that these solutions are classical for all values of the real parameter provided that the real part of the symbol of the difference operator occurring in the equation is positive. We present a class

Abstract We consider expressions \(\mathcal {L}_k[y]\) that are ordered products of \(k\), \(k\in \mathbb {N} \), linear symmetric quasidifferential expressions of the second order with matrix coefficients. Asymptotic formulas as \(x\to +\infty \) are derived for one of the fundamental solution systems of the equation \(\mathcal {L}_k[y]=\lambda y \), \(\lambda \in \mathbb {C} \), \(\lambda \ne 0\)

Abstract We propose a new approach to studying the global asymptotic stability of an equilibrium position of a system of ordinary differential equations with respect to part of the variables. New functions, different from Lyapunov ones, are used to study state trajectories of systems of ordinary differential equations and prove the global stability theorem. An application of the theorem to proving

Abstract Under appropriate conditions, we prove that for two arbitrary solutions \(s_1 \) and \(s_2 \) of the Dirichlet problem lying between lower and upper functions there exists a continuous mapping \(z \) of the interval \([0,1] \) into the solution set of the Dirichlet problem such that \(z(0)=s_1 \) and \(z(1)=s_2 \).

Abstract We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions

Abstract We study the Dirichlet problem for a nonlocal integro-functional-differential equation of the composite type. Theorems on the existence and uniqueness of a twice continuously differentiable solution are proved.

Abstract We consider an initial–boundary value problem for the beam vibration equation, which is a fourth-order nonlinear equation with two independent variables. It is shown that under certain conditions on the initial data this problem can be reduced to the Cauchy problem for a countable system of quasilinear ordinary differential equations. Using the method of energy inequalities, we prove that

Abstract We study the Bessel and basis properties of root vector functions of a Dirac type operator with an integrable potential. Criteria for the Bessel and basis properties are established, and a theorem on the equivalent basis property in \(L_{2}^{2m}(0,2\pi )\) is proved.

Abstract We construct a fundamental solution for an ordinary differential equation with a distributed-order differentiation operator. The properties of the solution are studied. A Lagrange formula is obtained for distributed-order differential operators. A well-posed form of the Cauchy problem is found for the equation in question, and the solution of the problem is written using the Lagrange formula

Abstract A boundary control problem is studied for a model loaded equation of the hyperbolic type. Necessary and sufficient conditions on the initial and terminal functions ensuring the existence of boundary controls are established. Under these conditions, an explicit analytical form of the desired controls is found.

Abstract For one-dimensional non-selfadjoint Schrödinger and Dirac operators with periodic complex-valued potentials belonging to the class \(L_2 \), asymptotic representations for spectral gaps are obtained in terms of Fourier coefficients of the potentials and estimates for the gap lengths are given.

Abstract We use a constructive method to obtain sufficient conditions for the unique solvability of the Vallée-Poussin problem for a nonlinear second-order matrix Lyapunov equation. These conditions are efficiently verifiable based on the initial data of the problem. We give an application-friendly algorithm for constructing the solution. An estimate of the solution localization domain is found. To

Abstract We carry out an analytical and numerical analysis of a model of the “predator–prey” system with a lower prey population threshold. The model is described by a system of partial differential equations of the “reaction–diffusion” type. Conditions are found for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions from the system thermodynamic branch. It is shown

Abstract We consider a differential equation of the Klein–Gordon type with an unknown input. The solution of this equation is assumed to be measured with some errors. We indicate a reconstruction algorithm whose input is formed by the feedback principle and serves as approximations to the unknown input of the original equation.

Abstract We study boundary conditions for the diffusion operator defined on a star-shaped geometric graph consisting of three edges with a common vertex. We show that if the edge lengths are pairwise distinct, then there do not exist degenerate boundary conditions for the diffusion operator. If the edge lengths coincide and the potentials are symmetric, then the characteristic determinant of a boundary

Abstract We study the stability and asymptotic stability of the zero solution of an autonomous evolution stochastic differential equation with discontinuous coefficients in a Hilbert space. The second Lyapunov method is applied to this end. An analog of the Lyapunov stability theorem, as well as an analog of the Barbashin–Krasovskii theorem on the asymptotic stability of the zero solution, is resulting

Abstract We consider the problem about a dynamic laminar boundary layer of finite thickness \(\delta (x) \). Formulas for calculating \(\delta (x) \) and the tangential stress \(\tau _0(x) \) are obtained. Based on these formulas, we carry out the error analysis of the approximation to the corresponding velocity distribution.

Abstract For a spectrally observable delay system with one-dimensional output, we construct a differential-difference finite time observer, i.e., an estimation system whose output, starting from a certain time moment, is identically equal to the solution of the original system. The results are illustrated by an example.

Abstract We consider the mixed problem with acoustic transmission conditions for semilinear hyperbolic equations. The existence and uniqueness are proved; a smoothness result is also obtained. The main result is the proof of the existence of a global attractor for the problem in question.

Abstract For the operator defined by the differential operation $$ Lu=-y^{\prime \prime }+x^{-2}(\nu ^2-{1}/{4})y,$$ \(0<\nu <1 \), on the interval \((0,1) \), we study the statements of spectral boundary value problems with general boundary conditions at the singular point \(x=0 \) as well as the basis properties of the systems of cylinder functions arising in these problems. We find sufficient conditions

Abstract For a singularly perturbed system consisting of the two parabolic equations $$ \varepsilon ^4(\Delta u-u_t)=f,\quad \varepsilon ^2(\Delta v-v_t)=g$$ and given in the Cartesian product \(D\times (0,+\infty ) \) of a two-dimensional simply connected domain \(D \) with smooth boundary and the time half-line, where \(\varepsilon >0\) is a small parameter and \(f \) and \(g \) are sufficiently

Abstract For autonomous systems of differential equations with smooth right-hand sides, in a simply connected domain of the real phase plane we consider the problem of finding the exact number of limit cycles surrounding one or several stationary points with total Poincaré index \(+1 \). A two-step method is proposed for solving this problem. At the first step, using the Dulac–Cherkas test, we find

Abstract We consider the Cauchy problem for a linear integro-differential equation whose differential part contains only the first derivative multiplied by a small positive parameter and whose integral part is the sum of two Volterra integral operators, one with slowly and one with rapidly varying kernel. Earlier, this Cauchy problem has only been considered for the case in which the integral part

Abstract We study a linear nonlocal problem for an evolution equation in a Banach space. The standard semigroup approach is used but integral averaging over time is used instead of the traditional initial condition. It is assumed that the evolution semigroup associated with the abstract differential equation is superstable (quasinilpotent), i.e., has an infinite negative exponential type. A theorem

Abstract The uniform convergence of spectral expansions on the real line \(\mathbb {R} \) is established for a self-adjoint operator \(\mathcal {A} \) generated on \(\mathbb {R} \) by the differential operation \(Au\equiv (-1)^n u^{(2n)}+\sum _{k=0}^{n-1} (q_k(x)u^{(k)})^{(k)}\) with uniformly locally integrable coefficients. The uniform convergence of the derivatives of these expansions is studied

Abstract A special method is proposed for reducing an \(n \)-dimensional system of second-order ordinary differential equations to a \(2n\)-dimensional system of first-order ones. Using matrix weight functions (which can be chosen arbitrarily when applying the method), we establish sufficient conditions for the boundedness and power-law absolute integrability on the half-line of the components of all

Abstract The use of special decompositions of nonlinear control systems for simplifying control problems by reducing their dimension is considered.

Abstract We consider wave propagation processes generated by forces of various types in a two-component Biot medium. Based on the Fourier transform of generalized functions, we construct a Green tensor describing the process of propagation of waves in spaces of dimension \(N=1,2,3 \) acted upon by pulsed sources of force that are lumped at the coordinate origin and given by the singular Dirac delta