Vol 207 (2022)

Abstract. In this paper, we examine the existence of solutions of the Poisson equations on a noncompact Riemannian manifold M without boundary. To describe the asymptotic behavior of a solution, we is introduce the notion of φ-equivalence on the set of continuous functions on a Riemannian manifold and establish a relationship between the solvability of boundary-value problems for the Poisson equations on the manifold M and outside some compact subset B ⊂ M with the same growth “at infinity.” Moreover, the notion of φ-equivalence of continuous functions on M allows one to estimate the rate of asymptotic convergence of solutions of boundary-value and outer boundary-value problems to boundary data.

A special class of systems of first-order quasilinear partial differential equations is considered. These divergent-type systems are invariant under time and space translations; they are transformed covariantly under the action of the rotation group. We give a description of the class of nonlinear first-order differential operators corresponding to the systems of the considered class and prove a theorem on the equivalence of the concepts of hyperbolicity and hyperbolicity in the sense of Friedrichs.

Application of the method of separation of variables to problems for the linearly degenerate equation $u_{xx}^{\text{'}\text{'}}+y{u}_{yy}^{\text{'}\text{'}}+c\left(y\right)u_y^{\text{'}}-a\left(x\right)u=f\left(x\mathrm{,}y\right)$ in a rectangle leads to problems for the singularly perturbed ordinary differential equation with degeneration $y{Y}^{\text{'}\text{'}}+c\left(y\right)Y^{\text{'}}-({\pi }^2{k}^2+a(y\left)\right)Y=f_k\left(y\right),$$k\in \mathrm{\mathbb{N}}$. In this paper, we examine the asymptotic behavior of solutions of this equation with given initial data at 0 and zero right-hand side as k → +∞ and obtain the leading term of the asymptotics in the explicit form.

In this paper, we obtain coefficient conditions that guarantee the equiconvergence and Cesaro equisummability almost everywhere of a multiple orthogonal series summed over two different systems of nested sets covering an integer lattice of the arithmetic space.

In this paper, we consider a version of the “reaction-diffusion” system, which can be interpreted as a mathematical model of the Keynes business cycle, taking into account spatial factors. The system is considered together with homogeneous Neumann boundary conditions. For such a nonlinear boundary-value problem, bifurcations in a neighborhood of a spatially homogeneous equilibrium state are studied in the near-critical case of zero and a pair of purely imaginary eigenvalues of the stability spectrum. An analysis of bifurcations allows one to obtain sufficient conditions for the existence and stability of spatially homogeneous and spatially inhomogeneous cycles and a spatially inhomogeneous equilibrium state. The analysis of the problem stated is based on the methods of the theory of infinite-dimensional dynamical systems, namely, the method of integral (invariant) manifolds and the method of normal forms. These methods and asymptotic methods of analysis lead to asymptotic formulas for periodic solutions and inhomogeneous equilibria. For such solutions, we also examine their stability.

We note that in some cases diffusion terms in an ordinary differential equations (for example, the logistic equation) can improve (weaken) sufficient conditions for the stability of a stationary solution. Examples are given.

We discuss regularization of two classical optimality conditions—the Lagrange principle (PL) and the Pontryagin maximum principle (PMP) — in a convex optimal control problem for a parabolic equation with an operator equality constraint and distributed initial and boundary controls. The regularized Lagrange principle and the Pontryagin maximum principle are based on two regularization parameters. These regularized principles are formulated as existence theorems for the original problem of minimizing approximate solutions.